text:
Attic (disambiguation)

An attic is an area under the roof of a house.

Attic may also refer to:

Attica-related

 * The adjectival form of the word Attica, a historical region that encompasses the city of Athens
 * Attic Greek, a dialect of the Greek language spoken in ancient Athens
 * Attic orators, ten ancient Athenian speechwriters
 * Attic peninsula, another name for Attica
 * Attic (architecture), a story or low wall above the cornice of a façade in classical architecture

Other uses

 * Attic (backup software)
 * Apache Attic, a repository for Apache Software Foundation software projects that have been retired
 * Attic Entertainment Software, a defunct German computer game developer and publisher
 * Attic Records, a Canadian record label
 * Attic (or atticus), is the epitympanic recess of the middle ear

title: Attic (disambiguation)


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Very long instruction word

Very long instruction word (VLIW) refers to instruction set architectures that are designed to exploit instruction-level parallelism (ILP). A VLIW processor allows programs to explicitly specify instructions to execute in parallel, whereas conventional central processing units (CPUs) mostly allow programs to specify instructions to execute in sequence only. VLIW is intended to allow higher performance without the complexity inherent in some other designs.

The traditional means to improve performance in processors include dividing instructions into substeps so the instructions can be executed partly at the same time (termed pipelining), dispatching individual instructions to be executed independently, in different parts of the processor (superscalar architectures), and even executing instructions in an order different from the program (out-of-order execution). These methods all complicate hardware (larger circuits, higher cost and energy use) because the processor must make all of the decisions internally for these methods to work.

In contrast, the VLIW method depends on the programs providing all the decisions regarding which instructions to execute simultaneously and how to resolve conflicts. As a practical matter, this means that the compiler (software used to create the final programs) becomes more complex, but the hardware is simpler than in many other means of parallelism.

History
The concept of VLIW architecture, and the term VLIW, were invented by Josh Fisher in his research group at Yale University in the early 1980s. His original development of trace scheduling as a compiling method for VLIW was developed when he was a graduate student at New York University. Before VLIW, the notion of prescheduling execution units and instruction-level parallelism in software was well established in the practice of developing horizontal microcode. Before Fisher the theoretical aspects of what would be later called VLIW were developed by the Soviet computer scientist Mikhail Kartsev based on his Sixties work on military-oriented M-9 and M-10 computers. His ideas were later developed and published as a part of a textbook two years before Fisher's seminal paper, but because of the Iron Curtain and because Kartsev's work was mostly military-related it remained largely unknown in the West.

Fisher's innovations involved developing a compiler that could target horizontal microcode from programs written in an ordinary programming language. He realized that to get good performance and target a wide-issue machine, it would be necessary to find parallelism beyond that generally within a basic block. He also developed region scheduling methods to identify parallelism beyond basic blocks. Trace scheduling is such a method, and involves scheduling the most likely path of basic blocks first, inserting compensating code to deal with speculative motions, scheduling the second most likely trace, and so on, until the schedule is complete.

Fisher's second innovation was the notion that the target CPU architecture should be designed to be a reasonable target for a compiler; that the compiler and the architecture for a VLIW processor must be codesigned. This was inspired partly by the difficulty Fisher observed at Yale of compiling for architectures like Floating Point Systems' FPS164, which had a complex instruction set computing (CISC) architecture that separated instruction initiation from the instructions that saved the result, needing very complex scheduling algorithms. Fisher developed a set of principles characterizing a proper VLIW design, such as self-draining pipelines, wide multi-port register files, and memory architectures. These principles made it easier for compilers to emit fast code.

The first VLIW compiler was described in a Ph.D. thesis by John Ellis, supervised by Fisher. The compiler was named Bulldog, after Yale's mascot.

Fisher left Yale in 1984 to found a startup company, Multiflow, along with cofounders John O'Donnell and John Ruttenberg. Multiflow produced the TRACE series of VLIW minisupercomputers, shipping their first machines in 1987. Multiflow's VLIW could issue 28 operations in parallel per instruction. The TRACE system was implemented in a mix of medium-scale integration (MSI), large-scale integration (LSI), and very large-scale integration (VLSI), packaged in cabinets, a technology obsoleted as it grew more cost-effective to integrate all of the components of a processor (excluding memory) on one chip.

Multiflow was too early to catch the following wave, when chip architectures began to allow multiple-issue CPUs. The major semiconductor companies recognized the value of Multiflow technology in this context, so the compiler and architecture were subsequently licensed to most of these firms.

Motivation
A processor that executes every instruction one after the other (i.e., a non-pipelined scalar architecture) may use processor resources inefficiently, yielding potential poor performance. The performance can be improved by executing different substeps of sequential instructions simultaneously (termed pipelining), or even executing multiple instructions entirely simultaneously as in superscalar architectures. Further improvement can be achieved by executing instructions in an order different from that in which they occur in a program, termed out-of-order execution.

These three methods all raise hardware complexity. Before executing any operations in parallel, the processor must verify that the instructions have no interdependencies. For example, if a first instruction's result is used as a second instruction's input, then they cannot execute at the same time and the second instruction cannot execute before the first. Modern out-of-order processors have increased the hardware resources which schedule instructions and determine interdependencies.

In contrast, VLIW executes operations in parallel, based on a fixed schedule, determined when programs are compiled. Since determining the order of execution of operations (including which operations can execute simultaneously) is handled by the compiler, the processor does not need the scheduling hardware that the three methods described above require. Thus, VLIW CPUs offer more computing with less hardware complexity (but greater compiler complexity) than do most superscalar CPUs. This is also complementary to the idea that as many computations as possible should be done before the program is executed, at compile time.

Design
In superscalar designs, the number of execution units is invisible to the instruction set. Each instruction encodes one operation only. For most superscalar designs, the instruction width is 32 bits or fewer.

In contrast, one VLIW instruction encodes multiple operations, at least one operation for each execution unit of a device. For example, if a VLIW device has five execution units, then a VLIW instruction for the device has five operation fields, each field specifying what operation should be done on that corresponding execution unit. To accommodate these operation fields, VLIW instructions are usually at least 64 bits wide, and far wider on some architectures.

For example, the following is an instruction for the Super Harvard Architecture Single-Chip Computer (SHARC). In one cycle, it does a floating-point multiply, a floating-point add, and two autoincrement loads. All of this fits in one 48-bit instruction:

Since the earliest days of computer architecture, some CPUs have added several arithmetic logic units (ALUs) to run in parallel. Superscalar CPUs use hardware to decide which operations can run in parallel at runtime, while VLIW CPUs use software (the compiler) to decide which operations can run in parallel in advance. Because the complexity of instruction scheduling is moved into the compiler, complexity of hardware can be reduced substantially.

A similar problem occurs when the result of a parallelizable instruction is used as input for a branch. Most modern CPUs guess which branch will be taken even before the calculation is complete, so that they can load the instructions for the branch, or (in some architectures) even start to compute them speculatively. If the CPU guesses wrong, all of these instructions and their context need to be flushed and the correct ones loaded, which takes time.

This has led to increasingly complex instruction-dispatch logic that attempts to guess correctly, and the simplicity of the original reduced instruction set computing (RISC) designs has been eroded. VLIW lacks this logic, and thus lacks its energy use, possible design defects, and other negative aspects.

In a VLIW, the compiler uses heuristics or profile information to guess the direction of a branch. This allows it to move and preschedule operations speculatively before the branch is taken, favoring the most likely path it expects through the branch. If the branch takes an unexpected way, the compiler has already generated compensating code to discard speculative results to preserve program semantics.

Vector processor cores (designed for large one-dimensional arrays of data called vectors) can be combined with the VLIW architecture such as in the Fujitsu FR-V microprocessor, further increasing throughput and speed.

Implementations
Cydrome was a company producing VLIW numeric processors using emitter-coupled logic (ECL) integrated circuits in the same timeframe (late 1980s). This company, like Multiflow, failed after a few years.

One of the licensees of the Multiflow technology is Hewlett-Packard, which Josh Fisher joined after Multiflow's demise. Bob Rau, founder of Cydrome, also joined HP after Cydrome failed. These two would lead computer architecture research at Hewlett-Packard during the 1990s.

Along with the above systems, during the same time (1989–1990), Intel implemented VLIW in the Intel i860, their first 64-bit microprocessor, and the first processor to implement VLIW on one chip. This processor could operate in both simple RISC mode and VLIW mode: In the early 1990s, Intel introduced the i860 RISC microprocessor. This simple chip had two modes of operation: a scalar mode and a VLIW mode. In the VLIW mode, the processor always fetched two instructions and assumed that one was an integer instruction and the other floating-point.

The i860's VLIW mode was used extensively in embedded digital signal processor (DSP) applications since the application execution and datasets were simple, well ordered and predictable, allowing designers to fully exploit the parallel execution advantages enabled by VLIW. In VLIW mode, the i860 could maintain floating-point performance in the range of 20-40 double-precision MFLOPS; a very high value for its time and for a processor running at 25-50Mhz.

In the 1990s, Hewlett-Packard researched this problem as a side effect of ongoing work on their PA-RISC processor family. They found that the CPU could be greatly simplified by removing the complex dispatch logic from the CPU and placing it in the compiler. Compilers of the day were far more complex than those of the 1980s, so the added complexity in the compiler was considered to be a small cost.

VLIW CPUs are usually made of multiple RISC-like execution units that operate independently. Contemporary VLIWs usually have four to eight main execution units. Compilers generate initial instruction sequences for the VLIW CPU in roughly the same manner as for traditional CPUs, generating a sequence of RISC-like instructions. The compiler analyzes this code for dependence relationships and resource requirements. It then schedules the instructions according to those constraints. In this process, independent instructions can be scheduled in parallel. Because VLIWs typically represent instructions scheduled in parallel with a longer instruction word that incorporates the individual instructions, this results in a much longer opcode (termed very long) to specify what executes on a given cycle.

Examples of contemporary VLIW CPUs include the TriMedia media processors by NXP (formerly Philips Semiconductors), the Super Harvard Architecture Single-Chip Computer (SHARC) DSP by Analog Devices, the ST200 family by STMicroelectronics based on the Lx architecture (designed in Josh Fisher's HP lab by Paolo Faraboschi), the FR-V from Fujitsu, the BSP15/16 from Pixelworks, the CEVA-X DSP from CEVA, the Jazz DSP from Improv Systems, the HiveFlex series from Silicon Hive, and the MPPA Manycore family by Kalray. The Texas Instruments TMS320 DSP line has evolved, in its C6000 family, to look more like a VLIW, in contrast to the earlier C5000 family. These contemporary VLIW CPUs are mainly successful as embedded media processors for consumer electronic devices.

VLIW features have also been added to configurable processor cores for system-on-a-chip (SoC) designs. For example, Tensilica's Xtensa LX2 processor incorporates a technology named Flexible Length Instruction eXtensions (FLIX) that allows multi-operation instructions. The Xtensa C/C++ compiler can freely intermix 32- or 64-bit FLIX instructions with the Xtensa processor's one-operation RISC instructions, which are 16 or 24 bits wide. By packing multiple operations into a wide 32- or 64-bit instruction word and allowing these multi-operation instructions to intermix with shorter RISC instructions, FLIX allows SoC designers to realize VLIW's performance advantages while eliminating the code bloat of early VLIW architectures. The Infineon Carmel DSP is another VLIW processor core intended for SoC. It uses a similar code density improvement method called configurable long instruction word (CLIW).

Outside embedded processing markets, Intel's Itanium IA-64 explicitly parallel instruction computing (EPIC) and Elbrus 2000 appear as the only examples of a widely used VLIW CPU architectures. However, EPIC architecture is sometimes distinguished from a pure VLIW architecture, since EPIC advocates full instruction predication, rotating register files, and a very long instruction word that can encode non-parallel instruction groups. VLIWs also gained significant consumer penetration in the graphics processing unit (GPU) market, though both Nvidia and AMD have since moved to RISC architectures to improve performance on non-graphics workloads.

ATI Technologies' (ATI) and Advanced Micro Devices' (AMD) TeraScale microarchitecture for graphics processing units (GPUs) is a VLIW microarchitecture.

In December 2015, the first shipment of PCs based on VLIW CPU Elbrus-4s was made in Russia.

The Neo by REX Computing is a processor consisting of a 2D mesh of VLIW cores aimed at power efficiency.

The Elbrus 2000 (Эльбрус 2000) and its successors are Russian 512-bit wide VLIW microprocessors developed by Moscow Center of SPARC Technologies (MCST) and fabricated by TSMC.

Backward compatibility
When silicon technology allowed for wider implementations (with more execution units) to be built, the compiled programs for the earlier generation would not run on the wider implementations, as the encoding of binary instructions depended on the number of execution units of the machine.

Transmeta addressed this issue by including a binary-to-binary software compiler layer (termed code morphing) in their Crusoe implementation of the x86 architecture. This mechanism was advertised to basically recompile, optimize, and translate x86 opcodes at runtime into the CPU's internal machine code. Thus, the Transmeta chip is internally a VLIW processor, effectively decoupled from the x86 CISC instruction set that it executes.

Intel's Itanium architecture (among others) solved the backward-compatibility problem with a more general mechanism. Within each of the multiple-opcode instructions, a bit field is allocated to denote dependency on the prior VLIW instruction within the program instruction stream. These bits are set at compile time, thus relieving the hardware from calculating this dependency information. Having this dependency information encoded in the instruction stream allows wider implementations to issue multiple non-dependent VLIW instructions in parallel per cycle, while narrower implementations would issue a smaller number of VLIW instructions per cycle.

Another perceived deficiency of VLIW designs is the code bloat that occurs when one or more execution unit(s) have no useful work to do and thus must execute No Operation NOP instructions. This occurs when there are dependencies in the code and the instruction pipelines must be allowed to drain before later operations can proceed.

Since the number of transistors on a chip has grown, the perceived disadvantages of the VLIW have diminished in importance. VLIW architectures are growing in popularity, especially in the embedded system market, where it is possible to customize a processor for an application in a system-on-a-chip.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Very long instruction word


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Talk:Chuck Berry

Is his lawyer notable?
"During this time, Berry began using Wayne T. Schoeneberg as his legal counsel." - why is this important? Seems more like a vanity edit that slipped through, no citation given anyway. — Preceding unsigned comment added by 88.109.197.181 (talk) 23:40, 1 April 2023 (UTC)

Crater
In 2020, the International Astronomical Union named a small crater on Mercury after Chuck Berry. https://planetarynames.wr.usgs.gov/Feature/15937

The Legacy section is semi-protected so I didn't add this myself but someone should. Jstuby (talk) 16:33, 5 July 2023 (UTC)

 * ‑‑Neveselbert (talk · contribs · email) 19:52, 5 July 2023 (UTC)
 * Added by Neveselbert. Msoul13 (talk) 22:09, 5 July 2023 (UTC)

Unfounded allegations not supported by links in article.
There is an allegation on this page stating that Chuck Berry videoed himself urinating on women. Three articles are cited as sources; one is from a satirical magazine similar to MAD Magazine and The Onion, the other two make no mention of these allegations whatsoever. There is no evidence to support this claim. 2601:184:497F:8D80:117A:5CA0:832A:4E8 (talk) 08:44, 28 August 2023 (UTC)
 * I agree, it should be removed. --195.99.227.0 (talk) 01:14, 30 November 2023 (UTC)

Consenual acts don't belong in the "Physical and Sexual Abuse Allegations" section
In the section titled "Physical and sexual abuse allegations", there's a discussion of videos of Berry urinating on a woman and her defecating on him. These acts are specifically described as presumably consensual in the cited source. They don't belong in a section about abuse allegations. There's no section where they do belong, so I'm deleting that sentence. Izzy Grosof (talk) 04:44, 1 June 2024 (UTC)

title: Talk:Chuck Berry


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Bauhin

The Bauhin family is a family of physicians and scientists.

Jean Bauhin (1511–1582): a French physician, who moved with his family to Basel after conversion to Protestantism. His two sons of three were:

 * Gaspard Bauhin (or Caspar Bauhin) (1560–1624): Swiss-French botanist. The ileocecal valve is also called Bauhin's valve, named after him.
 * Johann Bauhin (or Jean Bauhin) (1541–1613): Swiss-French botanist.

Related to Bauhin family

 * Carl Linnaeus named the genus of plants Bauhinia after the Bauhin brothers.

title: Bauhin


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Russulales

The Russulales are an order of the Agaricomycetes, (which include the agaric genera Russula and Lactarius and their polyporoid and corticioid relatives). According to the Dictionary of the Fungi (10th edition, 2008), the order consists of 12 families, 80 genera, and 1767 species. According to Species Fungorum (January 2016), the order contains 13 families, 117 genera (16 not assigned to a family), and 3,060 species.

Russuloid agarics represent an independent evolutionary line of agarics, not directly related to the Agaricales.

This group also includes a number of russuloid hypogeous fungi, polypores such as Bondarzewia, some tooth fungi (e.g. Auriscalpium vulgare), and club fungi e.g. Artomyces. Basidiospores in this group are typically ornamented with amyloid warts or reticulation but a few exceptions are known, e.g. Heterobasidion annosum. The genus Clavicorona was often treated in the Russulales, but its type species, C. taxophila, is in the Agaricales. The remaining species are retained in the Russulales in the genus Artomyces.

Genera Incertae sedis
There are several genera classified in the Russulales that are i) poorly known, ii) have not been subjected to DNA analysis, or iii) if analysed phylogenetically do not group with as yet named or identified families, and have not been assigned to a specific family (i.e., Incertae sedis with respect to familial placement). These include:
 * Aleurocystidiellum
 * Dentipellopsis
 * Gloeoasterostroma
 * Gloeohypochnicium
 * Gloeopeniophorella
 * Haloaleurodiscus
 * Neoalbatrellus
 * Polypus
 * Scopulodontia
 * Scytinostromella
 * Xeroceps

title: Russulales


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Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a $k$ -times differentiable function around a given point by a polynomial of degree $k$, called the $k$ -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order $k$  of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.

Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory.

Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formulas to accurately compute values of many transcendental functions such as the exponential function and trigonometric functions. It is the starting point of the study of analytic functions, and is fundamental in various areas of mathematics, as well as in numerical analysis and mathematical physics. Taylor's theorem also generalizes to multivariate and vector valued functions. It provided the mathematical basis for some landmark early computing machines: Charles Babbage's Difference Engine calculated sines, cosines, logarithms, and other transcendental functions by numerically integrating the first 7 terms of their Taylor series.

Motivation
If a real-valued function $f(x)$ is differentiable at the point $x=a$, then it has a linear approximation near this point. This means that there exists a function h1(x) such that

$$ f(x) = f(a) + f'(a)(x - a) + h_1(x)(x - a), \quad \lim_{x \to a} h_1(x) = 0.$$

Here

$$P_1(x) = f(a) + f'(a)(x - a)$$

is the linear approximation of $f(x)$ for x near the point a, whose graph $y=P_1(x)$  is the tangent line to the graph $y=f(x)$  at x = a. The error in the approximation is: $$R_1(x) = f(x) - P_1(x) = h_1(x)(x - a).$$

As x tends to a, this error goes to zero much faster than $$f'(a)(x{-}a)$$, making $$f(x)\approx P_1(x)$$ a useful approximation.

For a better approximation to $f(x)$, we can fit a quadratic polynomial instead of a linear function:

$$P_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2.$$

Instead of just matching one derivative of $f(x)$ at  $x=a$, this polynomial has the same first and second derivatives, as is evident upon differentiation.

Taylor's theorem ensures that the quadratic approximation is, in a sufficiently small neighborhood of $x=a$, more accurate than the linear approximation. Specifically,

$$f(x) = P_2(x) + h_2(x)(x - a)^2, \quad \lim_{x \to a} h_2(x) = 0.$$

Here the error in the approximation is

$$R_2(x) = f(x) - P_2(x) = h_2(x)(x - a)^2,$$

which, given the limiting behavior of $$h_2$$, goes to zero faster than $$(x - a)^2$$ as x tends to a.

Similarly, we might get still better approximations to f if we use polynomials of higher degree, since then we can match even more derivatives with f at the selected base point.

In general, the error in approximating a function by a polynomial of degree k will go to zero much faster than $$(x-a)^k$$ as x tends to a. However, there are functions, even infinitely differentiable ones, for which increasing the degree of the approximating polynomial does not increase the accuracy of approximation: we say such a function fails to be analytic at x = a: it is not (locally) determined by its derivatives at this point.

Taylor's theorem is of asymptotic nature: it only tells us that the error $R_k$ in an approximation by a $k$ -th order Taylor polynomial Pk tends to zero faster than any nonzero $k$ -th degree polynomial as $x \to a$. It does not tell us how large the error is in any concrete neighborhood of the center of expansion, but for this purpose there are explicit formulas for the remainder term (given below) which are valid under some additional regularity assumptions on f. These enhanced versions of Taylor's theorem typically lead to uniform estimates for the approximation error in a small neighborhood of the center of expansion, but the estimates do not necessarily hold for neighborhoods which are too large, even if the function f is analytic. In that situation one may have to select several Taylor polynomials with different centers of expansion to have reliable Taylor-approximations of the original function (see animation on the right.)

There are several ways we might use the remainder term:

 * 1) Estimate the error for a polynomial Pk(x) of degree k estimating $f(x)$  on a given interval (a – r, a + r). (Given the interval and degree, we find the error.)
 * 2) Find the smallest degree k for which the polynomial Pk(x) approximates $f(x)$  to within a given error tolerance on a given interval (a − r, a + r) . (Given the interval and error tolerance, we find the degree.)
 * 3) Find the largest interval (a − r, a + r) on which Pk(x) approximates $f(x)$  to within a given error tolerance. (Given the degree and error tolerance, we find the interval.)

Statement of the theorem
The precise statement of the most basic version of Taylor's theorem is as follows:

The polynomial appearing in Taylor's theorem is the $\boldsymbol{k}$ -th order Taylor polynomial

$$P_k(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(k)}(a)}{k!}(x-a)^k $$

of the function f at the point a. The Taylor polynomial is the unique "asymptotic best fit" polynomial in the sense that if there exists a function hk : R → R and a $k$ -th order polynomial p such that

$$ f(x) = p(x) + h_k(x)(x-a)^k, \quad \lim_{x\to a} h_k(x) = 0 ,$$

then p = Pk. Taylor's theorem describes the asymptotic behavior of the remainder term

$$ R_k(x) = f(x) - P_k(x),$$

which is the approximation error when approximating f with its Taylor polynomial. Using the little-o notation, the statement in Taylor's theorem reads as

$$R_k(x) = o(|x-a|^{k}), \quad x\to a.$$

Explicit formulas for the remainder
Under stronger regularity assumptions on f there are several precise formulas for the remainder term Rk of the Taylor polynomial, the most common ones being the following.

These refinements of Taylor's theorem are usually proved using the mean value theorem, whence the name. Additionally, notice that this is precisely the mean value theorem when $k=0$. Also other similar expressions can be found. For example, if G(t) is continuous on the closed interval and differentiable with a non-vanishing derivative on the open interval between $a$ and $x$, then

$$ R_k(x) = \frac{f^{(k+1)}(\xi)}{k!}(x-\xi)^k \frac{G(x)-G(a)}{G'(\xi)} $$

for some number $\xi$ between $a$  and $x$. This version covers the Lagrange and Cauchy forms of the remainder as special cases, and is proved below using Cauchy's mean value theorem. The Lagrange form is obtained by taking $$G(t)=(x-t)^{k+1}$$ and the Cauchy form is obtained by taking $$G(t)=t-a$$.

The statement for the integral form of the remainder is more advanced than the previous ones, and requires understanding of Lebesgue integration theory for the full generality. However, it holds also in the sense of Riemann integral provided the (k + 1)th derivative of f is continuous on the closed interval [a,x].

$$

Due to absolute continuity of f on the closed interval between $a$ and $x$, its derivative f exists as an L-function, and the result can be proven by a formal calculation using fundamental theorem of calculus and integration by parts.

Estimates for the remainder
It is often useful in practice to be able to estimate the remainder term appearing in the Taylor approximation, rather than having an exact formula for it. Suppose that f is (k + 1)-times continuously differentiable in an interval I containing a. Suppose that there are real constants q and Q such that

$$q\le f^{(k+1)}(x)\le Q$$

throughout I. Then the remainder term satisfies the inequality

$$q\frac{(x-a)^{k+1}}{(k+1)!}\le R_k(x)\le Q\frac{(x-a)^{k+1}}{(k+1)!},$$

if x > a, and a similar estimate if x < a. This is a simple consequence of the Lagrange form of the remainder. In particular, if

$$|f^{(k+1)}(x)|\le M$$

on an interval I = (a − r,a + r) with some $$r > 0$$, then

$$|R_k(x)|\le M\frac{|x-a|^{k+1}}{(k+1)!}\le M\frac{r^{k+1}}{(k+1)!}$$

for all x∈(a − r,a + r). The second inequality is called a uniform estimate, because it holds uniformly for all x on the interval (a − r,a + r).

Example
Suppose that we wish to find the approximate value of the function $f(x)=e^x$ on the interval $[-1,1]$  while ensuring that the error in the approximation is no more than 10−5. In this example we pretend that we only know the following properties of the exponential function:

From these properties it follows that $f^{(k)}(x)=e^x$ for all $k$, and in particular, $f^{(k)}(0)=1$. Hence the $k$ -th order Taylor polynomial of $f$ at $0$  and its remainder term in the Lagrange form are given by

$$ P_k(x) = 1+x+\frac{x^2}{2!}+\cdots+\frac{x^k}{k!}, \qquad R_k(x)=\frac{e^\xi}{(k+1)!}x^{k+1},$$

where $\xi$ is some number between 0 and x. Since ex is increasing by ($$), we can simply use $e^x \leq 1$ for $x \in [-1,0]$  to estimate the remainder on the subinterval $$[-1,0]$$. To obtain an upper bound for the remainder on $$[0,1]$$, we use the property $e^\xi <e^x$ for $0<\xi<x$  to estimate

$$ e^x = 1 + x + \frac{e^\xi}{2}x^2 < 1 + x + \frac{e^x}{2}x^2, \qquad 0 < x\leq 1 $$

using the second order Taylor expansion. Then we solve for ex to deduce that

$$ e^x \leq \frac{1+x}{1-\frac{x^2}{2}} = 2\frac{1+x}{2-x^2} \leq 4, \qquad 0 \leq x\leq 1 $$

simply by maximizing the numerator and minimizing the denominator. Combining these estimates for ex we see that

$$ |R_k(x)| \leq \frac{4|x|^{k+1}}{(k+1)!} \leq \frac{4}{(k+1)!}, \qquad -1\leq x \leq 1, $$

so the required precision is certainly reached, when

$$ \frac{4}{(k+1)!} < 10^{-5} \quad \Longleftrightarrow \quad 4\cdot 10^5 < (k+1)! \quad \Longleftrightarrow \quad k \geq 9. $$

(See factorial or compute by hand the values $9! =362880$ and $10! =3628800$ .) As a conclusion, Taylor's theorem leads to the approximation

$$ e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^9}{9!} + R_9(x), \qquad |R_9(x)| < 10^{-5}, \qquad -1\leq x \leq 1. $$

For instance, this approximation provides a decimal expression $$e \approx 2.71828$$, correct up to five decimal places.

Taylor expansions of real analytic functions
Let I ⊂ R be an open interval. By definition, a function f : I → R is real analytic if it is locally defined by a convergent power series. This means that for every a ∈ I there exists some r > 0 and a sequence of coefficients ck ∈ R such that (a − r, a + r) ⊂ I and

$$ f(x) = \sum_{k=0}^\infty c_k(x-a)^k = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots, \qquad |x-a|<r. $$

In general, the radius of convergence of a power series can be computed from the Cauchy–Hadamard formula

$$ \frac{1}{R} = \limsup_{k\to\infty}|c_k|^\frac{1}{k}. $$

This result is based on comparison with a geometric series, and the same method shows that if the power series based on a converges for some b ∈ R, it must converge uniformly on the closed interval $[a-r_b,a+r_b]$, where $r_b=\left\vert b-a \right\vert$. Here only the convergence of the power series is considered, and it might well be that (a − R,a + R) extends beyond the domain I of f.

The Taylor polynomials of the real analytic function f at a are simply the finite truncations

$$ P_k(x) = \sum_{j=0}^k c_j(x-a)^j, \qquad c_j = \frac{f^{(j)}(a)}{j!}$$

of its locally defining power series, and the corresponding remainder terms are locally given by the analytic functions

$$ R_k(x) = \sum_{j=k+1}^\infty c_j(x-a)^j = (x-a)^k h_k(x), \qquad |x-a|<r. $$

Here the functions

$$\begin{align} & h_k:(a-r,a+r)\to \R \\[1ex] & h_k(x) = (x-a)\sum_{j=0}^\infty c_{k+1+j} \left(x - a\right)^j \end{align}$$

are also analytic, since their defining power series have the same radius of convergence as the original series. Assuming that [a − r, a + r] ⊂ I and r < R, all these series converge uniformly on (a − r, a + r). Naturally, in the case of analytic functions one can estimate the remainder term $R_k(x)$ by the tail of the sequence of the derivatives f′(a) at the center of the expansion, but using complex analysis also another possibility arises, which is described below.

Taylor's theorem and convergence of Taylor series
The Taylor series of f will converge in some interval in which all its derivatives are bounded and do not grow too fast as k goes to infinity. (However, even if the Taylor series converges, it might not converge to f, as explained below; f is then said to be non-analytic.)

One might think of the Taylor series

$$ f(x) \approx \sum_{k=0}^\infty c_k(x-a)^k = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots $$

of an infinitely many times differentiable function f : R → R as its "infinite order Taylor polynomial" at a. Now the estimates for the remainder imply that if, for any r, the derivatives of f are known to be bounded over (a − r, a + r), then for any order k and for any r > 0 there exists a constant Mk,r > 0 such that

for every x ∈ (a − r,a + r). Sometimes the constants Mk,r can be chosen in such way that Mk,r is bounded above, for fixed r and all k. Then the Taylor series of f converges uniformly to some analytic function

$$\begin{align} & T_f:(a-r,a+r)\to\R \\ & T_f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!} \left(x-a\right)^k \end{align}$$

(One also gets convergence even if Mk,r is not bounded above as long as it grows slowly enough.)

The limit function Tf is by definition always analytic, but it is not necessarily equal to the original function f, even if f is infinitely differentiable. In this case, we say f is a non-analytic smooth function, for example a flat function:

$$\begin{align} & f:\R \to \R \\ & f(x) = \begin{cases} e^{-\frac{1}{x^2}} & x>0 \\ 0 & x \leq 0. \end{cases} \end{align}$$

Using the chain rule repeatedly by mathematical induction, one shows that for any order k,

$$ f^{(k)}(x) = \begin{cases} \frac{p_k(x)}{x^{3k}}\cdot e^{-\frac{1}{x^2}} & x>0 \\ 0 & x \leq 0 \end{cases}$$

for some polynomial pk of degree 2(k − 1). The function $$e^{-\frac{1}{x^2}}$$ tends to zero faster than any polynomial as $x \to 0$, so f is infinitely many times differentiable and f(0) = 0 for every positive integer k. The above results all hold in this case:

However, as k increases for fixed r, the value of Mk,r grows more quickly than rk, and the error does not go to zero.
 * The Taylor series of f converges uniformly to the zero function Tf(x) = 0, which is analytic with all coefficients equal to zero.
 * The function f is unequal to this Taylor series, and hence non-analytic.
 * For any order k ∈ N and radius r > 0 there exists Mk,r > 0 satisfying the remainder bound ($$) above.

Taylor's theorem in complex analysis
Taylor's theorem generalizes to functions f : C → C which are complex differentiable in an open subset U ⊂ C of the complex plane. However, its usefulness is dwarfed by other general theorems in complex analysis. Namely, stronger versions of related results can be deduced for complex differentiable functions f : U → C using Cauchy's integral formula as follows.

Let r > 0 such that the closed disk B(z, r) ∪ S(z, r) is contained in U. Then Cauchy's integral formula with a positive parametrization γ(t) = z + reit of the circle S(z, r) with $$t \in [0,2 \pi]$$ gives

$$f(z) = \frac{1}{2\pi i}\int_\gamma \frac{f(w)}{w-z}\,dw, \quad f'(z) = \frac{1}{2\pi i}\int_\gamma \frac{f(w)}{(w-z)^2} \, dw, \quad \ldots, \quad f^{(k)}(z) = \frac{k!}{2\pi i}\int_\gamma \frac{f(w)}{(w-z)^{k+1}} \, dw.$$

Here all the integrands are continuous on the circle S(z, r), which justifies differentiation under the integral sign. In particular, if f is once complex differentiable on the open set U, then it is actually infinitely many times complex differentiable on U. One also obtains the Cauchy's estimates

$$ |f^{(k)}(z)| \leq \frac{k!}{2\pi}\int_\gamma \frac{M_r}{|w-z|^{k+1}} \, dw = \frac{k!M_r}{r^k}, \quad M_r = \max_{|w-c|=r}|f(w)| $$

for any z ∈ U and r > 0 such that B(z, r) ∪ S(c, r) ⊂ U. These estimates imply that the complex Taylor series

$$ T_f(z) = \sum_{k=0}^\infty \frac{f^{(k)}(c)}{k!}(z-c)^k $$

of f converges uniformly on any open disk $B(c,r) \subset U$ with $S(c,r) \subset U$  into some function Tf. Furthermore, using the contour integral formulas for the derivatives f(c),

$$\begin{align} T_f(z) &= \sum_{k=0}^\infty \frac{(z-c)^k}{2\pi i}\int_\gamma \frac{f(w)}{(w-c)^{k+1}} \, dw \\ &= \frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-c} \sum_{k=0}^\infty  \left(\frac{z-c}{w-c}\right)^k \, dw \\ &= \frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-c}\left( \frac{1}{1-\frac{z-c}{w-c}} \right) \, dw \\ &= \frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-z} \, dw = f(z), \end{align}$$

so any complex differentiable function f in an open set U ⊂ C is in fact complex analytic. All that is said for real analytic functions here holds also for complex analytic functions with the open interval I replaced by an open subset U ∈ C and a-centered intervals (a − r, a + r) replaced by c-centered disks B(c, r). In particular, the Taylor expansion holds in the form

$$ f(z) = P_k(z) + R_k(z), \quad P_k(z) = \sum_{j=0}^k \frac{f^{(j)}(c)}{j!}(z-c)^j, $$

where the remainder term Rk is complex analytic. Methods of complex analysis provide some powerful results regarding Taylor expansions. For example, using Cauchy's integral formula for any positively oriented Jordan curve $\gamma$ which parametrizes the boundary $\partial W \subset U$  of a region $W \subset U$, one obtains expressions for the derivatives f(c) as above, and modifying slightly the computation for Tf(z) = f(z), one arrives at the exact formula

$$ R_k(z) = \sum_{j=k+1}^\infty \frac{(z-c)^j}{2\pi i} \int_\gamma \frac{f(w)}{(w-c)^{j+1}} \, dw = \frac{(z-c)^{k+1}}{2\pi i} \int_\gamma \frac{f(w) \, dw}{(w-c)^{k+1}(w-z)}, \qquad z\in W. $$

The important feature here is that the quality of the approximation by a Taylor polynomial on the region $W \subset U$ is dominated by the values of the function f itself on the boundary $\partial W \subset U$. Similarly, applying Cauchy's estimates to the series expression for the remainder, one obtains the uniform estimates

$$ |R_k(z)| \leq \sum_{j=k+1}^\infty \frac{M_r |z-c|^j}{r^j} = \frac{M_r}{r^{k+1}} \frac{|z-c|^{k+1}}{1-\frac{|z-c|}{r}} \leq \frac{M_r \beta^{k+1}}{1-\beta}, \qquad \frac{|z-c|}{r} \leq \beta < 1. $$

Example
The function

$$\begin{align} & f : \R \to \R \\ & f(x) = \frac{1}{1+x^2} \end{align}$$

is real analytic, that is, locally determined by its Taylor series. This function was plotted above to illustrate the fact that some elementary functions cannot be approximated by Taylor polynomials in neighborhoods of the center of expansion which are too large. This kind of behavior is easily understood in the framework of complex analysis. Namely, the function f extends into a meromorphic function

$$\begin{align} & f:\Complex \cup \{\infty\} \to \Complex \cup \{\infty\} \\ & f(z) = \frac{1}{1+z^2} \end{align}$$

on the compactified complex plane. It has simple poles at $z=i$ and $z=-i$, and it is analytic elsewhere. Now its Taylor series centered at z0 converges on any disc B(z0, r) with r < |z − z0|, where the same Taylor series converges at z ∈ C. Therefore, Taylor series of f centered at 0 converges on B(0, 1) and it does not converge for any z ∈ C with |z| > 1 due to the poles at i and −i. For the same reason the Taylor series of f centered at 1 converges on $B(1, \sqrt{2})$ and does not converge for any z ∈ C with $\left\vert z-1 \right\vert>\sqrt{2}$.

Higher-order differentiability
A function $f: R^{n} → R$ is differentiable at $a ∈ R^{n}$ if and only if there exists a linear functional $L : R^{n} → R$ and a function $h : R^{n} → R$ such that

$$ f(\boldsymbol{x}) = f(\boldsymbol{a}) + L(\boldsymbol{x}-\boldsymbol{a}) + h(\boldsymbol{x})\lVert\boldsymbol{x}-\boldsymbol{a}\rVert, \qquad \lim_{\boldsymbol{x}\to\boldsymbol{a}} h(\boldsymbol{x})=0. $$

If this is the case, then $L = df(\boldsymbol{a})$ is the (uniquely defined) differential of $f$ at the point $a$. Furthermore, then the partial derivatives of $f$ exist at $a$ and the differential of $f$ at $a$ is given by

$$ df( \boldsymbol{a} )( \boldsymbol{v} ) = \frac{\partial f}{\partial x_1}(\boldsymbol{a}) v_1 + \cdots + \frac{\partial f}{\partial x_n}(\boldsymbol{a}) v_n. $$

Introduce the multi-index notation

$$ |\alpha| = \alpha_1+\cdots+\alpha_n, \quad \alpha!=\alpha_1!\cdots\alpha_n!, \quad \boldsymbol{x}^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n} $$

for $α ∈ N^{n}$ and $x ∈ R^{n}$. If all the $k$ -th order partial derivatives of $f : R^{n} → R$ are continuous at $a ∈ R^{n}$, then by Clairaut's theorem, one can change the order of mixed derivatives at $a$, so the notation

$$ D^\alpha f = \frac{\partial^{|\alpha|}f}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}}, \qquad |\alpha|\leq k $$

for the higher order partial derivatives is justified in this situation. The same is true if all the ($k − 1$)-th order partial derivatives of $f$ exist in some neighborhood of $a$ and are differentiable at $a$. Then we say that $f$ is $a$ times differentiable at the point $a$.

Taylor's theorem for multivariate functions
Using notations of the preceding section, one has the following theorem. $$

If the function $a$ is $f$ times continuously differentiable in a closed ball $$B = \{ \mathbf{y} \in \R^n : \left\|\mathbf{a}-\mathbf{y}\right\| \leq r\}$$ for some $$r > 0$$, then one can derive an exact formula for the remainder in terms of ($k$)-th order partial derivatives of f in this neighborhood. Namely,

$$ \begin{align} & f( \boldsymbol{x} ) = \sum_{|\alpha|\leq k} \frac{D^\alpha f(\boldsymbol{a})}{\alpha!} (\boldsymbol{x}-\boldsymbol{a})^\alpha + \sum_{|\beta|=k+1} R_\beta(\boldsymbol{x})(\boldsymbol{x}-\boldsymbol{a})^\beta, \\ & R_\beta( \boldsymbol{x} ) = \frac{|\beta|}{\beta!} \int_0^1 (1-t)^{|\beta|-1}D^\beta f \big(\boldsymbol{a}+t( \boldsymbol{x}-\boldsymbol{a} )\big) \, dt. \end{align} $$

In this case, due to the continuity of ($a$)-th order partial derivatives in the compact set $f : R^{n} → R$, one immediately obtains the uniform estimates

$$ \left|R_\beta(\boldsymbol{x})\right| \leq \frac{1}{\beta!} \max_{|\alpha|=|\beta|} \max_{\boldsymbol{y}\in B} |D^\alpha f(\boldsymbol{y})|, \qquad \boldsymbol{x}\in B. $$

Example in two dimensions
For example, the third-order Taylor polynomial of a smooth function $$f:\mathbb R^2\to\mathbb R$$ is, denoting $$\boldsymbol{x}-\boldsymbol{a}=\boldsymbol{v}$$,

$$ \begin{align} P_3(\boldsymbol{x}) = f ( \boldsymbol{a} ) + {} &\frac{\partial f}{\partial x_1}( \boldsymbol{a} ) v_1 + \frac{\partial f}{\partial x_2}( \boldsymbol{a} ) v_2 + \frac{\partial^2 f}{\partial x_1^2}( \boldsymbol{a} ) \frac {v_1^2}{2!} + \frac{\partial^2 f}{\partial x_1 \partial x_2}( \boldsymbol{a} ) v_1 v_2 + \frac{\partial^2 f}{\partial x_2^2}( \boldsymbol{a} ) \frac{v_2^2}{2!}  \\ & + \frac{\partial^3 f}{\partial x_1^3}( \boldsymbol{a} ) \frac{v_1^3}{3!} + \frac{\partial^3 f}{\partial x_1^2 \partial x_2}( \boldsymbol{a} ) \frac{v_1^2 v_2}{2!} + \frac{\partial^3 f}{\partial x_1 \partial x_2^2}( \boldsymbol{a} ) \frac{v_1 v_2^2}{2!} + \frac{\partial^3 f}{\partial x_2^3}( \boldsymbol{a} ) \frac{v_2^3}{3!} \end{align}$$

Proof for Taylor's theorem in one real variable
Let

$$ h_k(x) = \begin{cases} \frac{f(x) - P(x)}{(x-a)^k} & x\not=a\\ 0&x=a \end{cases} $$

where, as in the statement of Taylor's theorem,

$$ P(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(k)}(a)}{k!}(x-a)^k.$$

It is sufficient to show that

$$ \lim_{x\to a} h_k(x) =0. $$

The proof here is based on repeated application of L'Hôpital's rule. Note that, for each $j=0,1,...,k-1$, $$f^{(j)}(a)=P^{(j)}(a)$$. Hence each of the first $k-1$ derivatives of the numerator in $$h_k(x)$$ vanishes at $$x=a$$, and the same is true of the denominator. Also, since the condition that the function $f$ be $k$  times differentiable at a point requires differentiability up to order $k-1$  in a neighborhood of said point (this is true, because differentiability requires a function to be defined in a whole neighborhood of a point), the numerator and its $k-2$  derivatives are differentiable in a neighborhood of $a$. Clearly, the denominator also satisfies said condition, and additionally, doesn't vanish unless $x=a$, therefore all conditions necessary for L'Hôpital's rule are fulfilled, and its use is justified. So

$$\begin{align} \lim_{x\to a} \frac{f(x) - P(x)}{(x-a)^k} &= \lim_{x\to a} \frac{\frac{d}{dx}(f(x) - P(x))}{\frac{d}{dx}(x-a)^k} \\[1ex] &= \cdots \\[1ex] &= \lim_{x\to a} \frac{\frac{d^{k-1}}{dx^{k-1}}(f(x) - P(x))}{\frac{d^{k-1}}{dx^{k-1}}(x-a)^k}\\[1ex] &= \frac{1}{k!}\lim_{x\to a} \frac{f^{(k-1)}(x) - P^{(k-1)}(x)}{x-a}\\[1ex] &=\frac{1}{k!}(f^{(k)}(a) - P^{(k)}(a)) = 0 \end{align}$$

where the second-to-last equality follows by the definition of the derivative at $ x=a$.

Alternate proof for Taylor's theorem in one real variable
Let $$f(x)$$ be any real-valued continuous function to be approximated by the Taylor polynomial.

Step 1: Let $F$ and $G$  be functions. Set $F$ and $G$  to be

$$\begin{align} F(x) = f(x) - \sum^{n-1}_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^{k} \end{align}$$

$$\begin{align} G(x) = (x-a)^{n} \end{align}$$

Step 2: Properties of $F$ and $G$ :

$$\begin{align} F(a) & = f(a) - f(a) - f'(a)(a - a) - ... - \frac{f^{(n-1)}(a)(a-a)^{n-1}}{(n-1)!} = 0 \\ G(a) & = (a-a)^n = 0 \end{align}$$

Similarly,

$$\begin{align} F'(a) = f'(a) - f'(a) - \frac{2f''(a)(a-a)}{1!} - ... - \frac{f^{(n-2)}(a)(n-1)(a-a)^{n-2}}{(n-1)!} = 0 \end{align}$$

$$\begin{align} G'(a) &= n(a-a)^{n-1} = 0\\ &\qquad \vdots\\ G^{(n-1)}(a) &= F^{(n-1)}(a) = 0 \end{align}$$

Step 3: Use Cauchy Mean Value Theorem

Let $$f_{1}$$ and $$g_{1}$$ be continuous functions on $$[a, b]$$. Since $$a < x < b$$ so we can work with the interval $$[a, x]$$. Let $$f_{1}$$ and $$g_{1}$$ be differentiable on $$(a, x)$$. Assume $$g_{1}'(x) \neq 0$$ for all $$x \in (a, b)$$. Then there exists $$c_{1} \in (a, x)$$ such that

$$\begin{align} \frac{f_{1}(x) - f_{1}(a)}{g_{1}(x) - g_{1}(a)} = \frac{f_{1}'(c_{1})}{g_{1}'(c_{1})} \end{align}$$

Note: $$G'(x) \neq 0$$ in $$(a, b)$$ and $$F(a), G(a) = 0$$ so

$$\begin{align} \frac{F(x)}{G(x)} = \frac{F(x) - F(a)}{G(x) - G(a)} = \frac{F'(c_{1})}{G'(c_{1})} \end{align}$$

for some $$c_{1} \in (a, x)$$.

This can also be performed for $$(a, c_{1})$$:

$$\begin{align} \frac{F'(c_{1})}{G'(c_{1})} = \frac{F'(c_{1}) - F'(a)}{G'(c_{1}) - G'(a)} = \frac{F(c_{2})}{G(c_{2})} \end{align}$$

for some $$c_{2} \in (a, c_{1})$$. This can be continued to $$c_{n}$$.

This gives a partition in $$(a, b)$$:

$$a < c_{n} < c_{n-1} < \dots < c_{1} < x$$

with

$$\frac{F(x)}{G(x)} = \frac{F'(c_{1})}{G'(c_{1})} = \dots = \frac{F^{(n)}(c_{n})}{G^{(n)}(c_{n})} .$$

Set $$c = c_{n}$$:

$$\frac{F(x)}{G(x)} = \frac{F^{(n)}(c)}{G^{(n)}(c)}$$

Step 4: Substitute back

$$\begin{align} \frac{F(x)}{G(x)} = \frac{f(x) - \sum^{n-1}_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^{k}}{(x-a)^{n}} = \frac{F^{(n)}(c)}{G^{(n)}(c)} \end{align}$$

By the Power Rule, repeated derivatives of $$(x - a)^{n}$$, $$G^{(n)}(c) = n(n-1)...1$$, so:

$$\frac{F^{(n)}(c)}{G^{(n)}(c)} = \frac{f^{(n)}(c)}{n(n-1)\cdots1} = \frac{f^{(n)}(c)}{n!}.$$

This leads to:

$$\begin{align} f(x) - \sum^{n-1}_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^{k} = \frac{f^{(n)}(c)}{n!}(x-a)^{n} \end{align}.$$

By rearranging, we get:

$$\begin{align} f(x) = \sum^{n-1}_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^{k} + \frac{f^{(n)}(c)}{n!}(x-a)^{n} \end{align},$$

or because $$c_{n} = a$$ eventually:

$$f(x) = \sum^{n}_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^{k}.$$

Derivation for the mean value forms of the remainder
Let G be any real-valued function, continuous on the closed interval between $a$ and $x$  and differentiable with a non-vanishing derivative on the open interval between $a$  and $x$, and define

$$ F(t) = f(t) + f'(t)(x-t) + \frac{f''(t)}{2!}(x-t)^2 + \cdots + \frac{f^{(k)}(t)}{k!}(x-t)^k. $$

For $$ t \in [a,x] $$. Then, by Cauchy's mean value theorem,

for some $\xi$ on the open interval between $a$  and $x$. Note that here the numerator $F(x)-F(a)=R_k(x)$ is exactly the remainder of the Taylor polynomial for $y=f(x)$. Compute

$$\begin{align} F'(t) = {} & f'(t) + \big(f''(t)(x-t) - f'(t)\big) + \left(\frac{f^{(3)}(t)}{2!}(x-t)^2 - \frac{f^{(2)}(t)}{1!}(x-t)\right) +  \cdots \\ & \cdots + \left( \frac{f^{(k+1)}(t)}{k!}(x-t)^k - \frac{f^{(k)}(t)}{(k-1)!}(x-t)^{k-1}\right) = \frac{f^{(k+1)}(t)}{k!}(x-t)^k, \end{align}$$

plug it into ($$) and rearrange terms to find that

$$ R_k(x) = \frac{f^{(k+1)}(\xi)}{k!}(x-\xi)^k \frac{G(x)-G(a)}{G'(\xi)}.$$

This is the form of the remainder term mentioned after the actual statement of Taylor's theorem with remainder in the mean value form. The Lagrange form of the remainder is found by choosing $$ G(t) = (x-t)^{k+1} $$ and the Cauchy form by choosing $$ G(t) = t-a$$.

Remark. Using this method one can also recover the integral form of the remainder by choosing

$$ G(t) = \int_a^t \frac{f^{(k+1)}(s)}{k!} (x-s)^k \, ds,$$

but the requirements for f needed for the use of mean value theorem are too strong, if one aims to prove the claim in the case that f is only absolutely continuous. However, if one uses Riemann integral instead of Lebesgue integral, the assumptions cannot be weakened.

Derivation for the integral form of the remainder
Due to absolute continuity of $$f^{(k)}$$ on the closed interval between $a$ and $x$  its derivative $$f^{(k+1)}$$ exists as an $$L^1$$-function, and we can use fundamental theorem of calculus and integration by parts. This same proof applies for the Riemann integral assuming that $$f^{(k)}$$ is continuous on the closed interval and differentiable on the open interval between $a$ and $x$, and this leads to the same result than using the mean value theorem.

The fundamental theorem of calculus states that

$$ f(x)=f(a)+ \int_a^x \, f'(t) \, dt.$$

Now we can integrate by parts and use the fundamental theorem of calculus again to see that

$$ \begin{align} f(x) &= f(a)+\Big(xf'(x)-af'(a)\Big)-\int_a^x tf''(t) \, dt \\ &= f(a) + x\left(f'(a) + \int_a^x f(t) \,dt \right) -af'(a)-\int_a^x tf(t) \, dt \\ &= f(a)+(x-a)f'(a)+\int_a^x \, (x-t)f''(t) \, dt, \end{align} $$

which is exactly Taylor's theorem with remainder in the integral form in the case $$k=1$$. The general statement is proved using induction. Suppose that

Integrating the remainder term by parts we arrive at

$$\begin{align} \int_a^x \frac{f^{(k+1)} (t)}{k!} (x - t)^k \, dt = & - \left[ \frac{f^{(k+1)} (t)}{(k+1)k!} (x - t)^{k+1} \right]_a^x + \int_a^x \frac{f^{(k+2)} (t)}{(k+1)k!} (x - t)^{k+1} \, dt \\ = & \ \frac{f^{(k+1)} (a)}{(k+1)!} (x - a)^{k+1} + \int_a^x \frac{f^{(k+2)} (t)}{(k+1)!} (x - t)^{k+1} \, dt. \end{align}$$

Substituting this into the formula in ($$) shows that if it holds for the value $$k$$, it must also hold for the value $$k+1$$. Therefore, since it holds for $$k=1$$, it must hold for every positive integer $$k$$.

Derivation for the remainder of multivariate Taylor polynomials
We prove the special case, where $$f:\mathbb R^n\to\mathbb R$$ has continuous partial derivatives up to the order $$k+1$$ in some closed ball $$B$$ with center $$\boldsymbol{a}$$. The strategy of the proof is to apply the one-variable case of Taylor's theorem to the restriction of $$f$$ to the line segment adjoining $$\boldsymbol{x}$$ and $$\boldsymbol{a}$$. Parametrize the line segment between $$\boldsymbol{a}$$ and $$\boldsymbol{x}$$ by $$\boldsymbol{u}(t)=\boldsymbol{a}+t(\boldsymbol{x}-\boldsymbol{a})$$ We apply the one-variable version of Taylor's theorem to the function $$g(t) = f(\boldsymbol{u}(t))$$:

$$ f(\boldsymbol{x})=g(1)=g(0)+\sum_{j=1}^k\frac{1}{j!}g^{(j)}(0)\ +\ \int_0^1 \frac{(1-t)^k }{k!} g^{(k+1)}(t)\, dt.$$

Applying the chain rule for several variables gives

$$\begin{align} g^{(j)}(t)&=\frac{d^j}{dt^j}f(\boldsymbol{u}(t))\\ &= \frac{d^j}{dt^j} f(\boldsymbol{a}+t(\boldsymbol{x}-\boldsymbol{a}))\\ &= \sum_{|\alpha| =j} \left(\begin{matrix} j\\ \alpha\end{matrix} \right) (D^\alpha f) (\boldsymbol{a}+t(\boldsymbol{x}-\boldsymbol{a})) (\boldsymbol{x}-\boldsymbol{a})^\alpha \end{align}$$

where $$\tbinom j \alpha$$ is the multinomial coefficient. Since $$\tfrac{1}{j!}\tbinom j \alpha=\tfrac{1}{\alpha!}$$, we get:

$$ f(\boldsymbol{x})= f(\boldsymbol{a}) + \sum_{1 \leq |\alpha| \leq k}\frac{1}{\alpha!} (D^\alpha f) (\boldsymbol{a})(\boldsymbol{x}-\boldsymbol{a})^\alpha+\sum_{|\alpha|=k+1}\frac{k+1}{\alpha!} (\boldsymbol{x}-\boldsymbol{a})^\alpha \int_0^1 (1-t)^k (D^\alpha f)(\boldsymbol{a}+t(\boldsymbol{x}-\boldsymbol{a}))\,dt.$$

title: Taylor's theorem


text:
United States Atomic Energy Commission

The United States Atomic Energy Commission (AEC) was an agency of the United States government established after World War II by the U.S. Congress to foster and control the peacetime development of atomic science and technology. President Harry S. Truman signed the McMahon/Atomic Energy Act on August 1, 1946, transferring the control of atomic energy from military to civilian hands, effective on January 1, 1947. This shift gave the members of the AEC complete control of the plants, laboratories, equipment, and personnel assembled during the war to produce the atomic bomb.

An increasing number of critics during the 1960s charged that the AEC's regulations were insufficiently rigorous in several important areas, including radiation protection standards, nuclear reactor safety, plant siting, and environmental protection.

By 1974, the AEC's regulatory programs had come under such strong attack that the U.S. Congress decided to abolish the AEC. The AEC was abolished by the Energy Reorganization Act of 1974, which assigned its functions to two new agencies: the Energy Research and Development Administration and the Nuclear Regulatory Commission. On August 4, 1977, President Jimmy Carter signed into law the Department of Energy Organization Act, which created the Department of Energy. The new agency assumed the responsibilities of the Federal Energy Administration (FEA), the Energy Research and Development Administration (ERDA), the Federal Power Commission (FPC), and various other federal agencies.

History
In creating the AEC, Congress declared that atomic energy should be employed not only in the form of nuclear weapons for the nation's defense, but also to promote world peace, improve the public welfare and strengthen free competition in private enterprise. At the same time, the McMahon Act which created the AEC also gave it unprecedented powers of regulation over the entire field of nuclear science and technology. It furthermore explicitly prevented technology transfer between the United States and other countries, and required FBI investigations for all scientists or industrial contractors who wished to have access to any AEC controlled nuclear information. The signing was the culmination of long months of intensive debate among politicians, military planners and atomic scientists over the fate of this new energy source and the means by which it would be regulated. President Truman appointed David Lilienthal as the first Chairman of the AEC. Congress gave the new civilian AEC extraordinary power and considerable independence to carry out its mission. To provide the AEC exceptional freedom in hiring its scientists and engineers, AEC employees were exempt from the civil service system. The AEC's first order of business was to inspect the scattered empire of atomic plants and laboratories to be inherited from the U.S. Army.

Because of the great need for security, all production facilities and nuclear reactors would be government-owned, while all technical information and research results would be under AEC control. The National Laboratory system was established from the facilities created under the Manhattan Project. Argonne National Laboratory was one of the first laboratories authorized under this legislation as a contractor-operated facility dedicated to fulfilling the new AEC's missions. Argonne was the first of the regional laboratories to involve universities in the Chicago area. Others were the Clinton (CEW) labs and the Brookhaven National Laboratory in the Northeast, although a similar lab in Southern California did not eventuate.

On 11 March 1948 Lilienthal and Kenneth Nichols were summoned to the White House where Truman told them "I know you two hate each other’s guts". He directed that "the primary objective of the AEC was to develop and produce atomic weapons", Nichols was appointed a major general and replaced Leslie Groves as chief of the Armed Forces Special Weapons Project (AFSWP); previously Lilienthal had opposed his appointment. Lilienthal was told to "forgo your desire to place a bottle of milk on every doorstop and get down to the business of producing atomic weapons." Nichols became General Manager of the AEC on 2 November 1953.

The AEC was in charge of developing the U.S. nuclear arsenal, taking over these responsibilities from the wartime Manhattan Project. In its first decade, the AEC oversaw the operation of Los Alamos Scientific Laboratory, devoted primarily to weapons development, and in 1952, the creation of new second weapons laboratory in California, the Lawrence Livermore National Laboratory. The AEC also carried out the "crash program" to develop the hydrogen bomb (H-bomb), and the AEC played a key role in the prosecution of the Rosenbergs for espionage.

The AEC also began a program of regular nuclear weapons testing, both in the faraway Pacific Proving Grounds and at the Nevada Test Site in the western United States. While the AEC also supported much basic research, the vast majority of its early budget was devoted to nuclear weapons development and production.

After serving as director of the Manhattan Project's Los Alamos Laboratory, physicist J. Robert Oppenheimer voiced strong opinions to the AEC, as chairman of its general advisory board of nuclear scientists, against development of the "super" or hydrogen bomb along with Lilienthal. Subsequently, Lilienthal left the AEC at the White House's request in 1950 and Oppenheimer's appointment to the board was not renewed in 1952. With them removed, President Truman announced his decision to develop and produce the hydrogen bomb. The first test firing of an experimental H-bomb ("Ivy Mike") was carried out in the Central Pacific on November 1, 1952, under President Truman. Furthermore, U.S. Navy Admiral Lewis W. Strauss was appointed in 1953 by the new President Eisenhower as the Chairman of the AEC, to carry out the military development and production of the H-bomb.

Lilienthal wanted to give high priority to peaceful uses, especially with nuclear power plants. However, coal was still cheap, and the electric power industry was not interested. The first experimental nuclear power plant was started in Pennsylvania under President Eisenhower in 1954.

Domestic uranium procurement program
The AEC developed a program for sourcing uranium domestically. Before 1947, the main sources for the mineral had been Canada and (what was then) the Belgian Congo, though the Manhattan Project also secretly processed uranium from the tailings of vanadium plants in the US West during World War II. The Colorado Plateau was known to contain veins of carnotite ore, which contains both vanadium and uranium. The AEC developed its program in accordance with the principle of free enterprise. Rather than discovering, mining, and processing the ore itself, the federal government provided geological information, built roads, and set a fixed rate for purchasing ore through one of the mills in the area. This prompted individuals to discover and produce the ore, which the government would then buy. The AEC was the only legal buyer of uranium from the beginning of the program in 1947 through 1966. From 1966 to the end of the program in 1970, the AEC continued to buy uranium to support the market until private industry could develop sufficiently.

Because the government itself was not producing ore, it claimed that it had no obligation to regulate miner safety. A congressional report published in 1995 concluded that, "The government failed to act to require the reduction of the hazard by ventilating the mines, and it failed to adequately warn the miners of the hazard to which they were being exposed." The Radiation Exposure Compensation Act of 1990 sought to compensate miners and families who developed cancer as a result of exposure to radon gas in uranium mines.

Regulations and experiments
The AEC was connected with the U.S. Department of Defense by a "Military Liaison Committee"'. The Joint Committee on Atomic Energy exercised congressional oversight over the AEC and had considerable power in influencing AEC decisions and policy.

The AEC's far-reaching powers and control over a subject matter which had far-reaching social, public health, and military implications made it an extremely controversial organization. One of the drafters of the McMahon Act, James R. Newman, famously concluded that the bill made "the field of atomic energy [an] island of socialism in the midst of a free-enterprise economy".

Before the Nuclear Regulatory Commission (NRC) was created, nuclear regulation was the responsibility of the AEC, which Congress first established in the Atomic Energy Act of 1946. Eight years later, Congress replaced that law with the Atomic Energy Act of 1954, which for the first time made the development of commercial nuclear power possible, and resolved a number of other outstanding problems in implementing the first Atomic Energy Act. The act assigned the AEC the functions of both encouraging the use of nuclear power and regulating its safety. The AEC's regulatory programs sought to ensure public health and safety from the hazards of nuclear power without imposing excessive requirements that would inhibit the growth of the industry. This was a difficult goal to achieve, especially in a new industry, and within a short time the AEC's programs stirred considerable controversy. Stephanie Cooke has written that:

"the AEC had become an oligarchy controlling all facets of the military and civilian sides of nuclear energy, promoting them and at the same time attempting to regulate them, and it had fallen down on the regulatory side ... a growing legion of critics saw too many inbuilt conflicts of interest."

The AEC had a history of involvement in experiments involving radioactive iodine. In a 1949 operation called the "Green Run", the AEC released iodine-131 and xenon-133 to the atmosphere which contaminated a 500000 acre area containing three small towns near the Hanford site in Washington. In 1953, the AEC ran several studies on the health effects of radioactive iodine in newborns and pregnant women at the University of Iowa. Also in 1953, the AEC sponsored a study to discover if radioactive iodine affected premature babies differently from full-term babies. In the experiment, researchers from Harper Hospital in Detroit orally administered iodine-131 to 65 premature and full-term infants who weighed from 2.1 to 5.5 lbs. In another AEC study, researchers at the University of Nebraska College of Medicine fed iodine-131 to 28 healthy infants through a gastric tube to test the concentration of iodine in the infants' thyroid glands.

Public opinion and abolition of the AEC
During the 1960s and early 1970s, the Atomic Energy Commission came under fire from opposition concerned with more fundamental ecological problems such as the pollution of air and water. Under the Nixon Administration, environmental consciousness grew exponentially and the first Earth Day was held on April 22, 1970. Along with rising environmental awareness came a growing suspicion of the AEC and public hostility for their projects increased. In the public eye, there was a strong association between nuclear power and nuclear weapons, and even though the AEC had made a push in the late 1960s, to portray their efforts as being geared toward peaceful uses of atomic energy, criticism of the agency grew. The AEC was chiefly held responsible for the health problems of people living near atmospheric test sites from the early 1960s, and there was a strong association of nuclear energy with the radioactive fallout from these tests. Around the same time, the AEC was also struggling with opposition to nuclear power plant siting as well as nuclear testing. An organized push was finally made to curb the power held by the AEC, and in 1970 the AEC was forced to prepare an Environmental impact statement (EIS) for a nuclear test in northwestern Colorado as part of the initial preparation for Project Rio Blanco.

In 1973, the AEC predicted that, by the turn of the century, one thousand reactors would be needed producing electricity for homes and businesses across the United States. However, after 1973, orders for nuclear reactors declined sharply as electricity demand fell and construction costs rose. Some partially completed nuclear power plants in the U.S. were stricken, and many planned nuclear plants were canceled.

By 1974, the AEC's regulatory programs had come under such strong attack that Congress decided to abolish the agency. Supporters and critics of nuclear power agreed that the promotional and regulatory duties of the AEC should be assigned to different agencies. The Energy Reorganization Act of 1974 transferred the regulatory functions of the AEC to the new Nuclear Regulatory Commission (NRC), which began operations on January 19, 1975. Promotional functions went to the Energy Research and Development Administration which was later incorporated into the United States Department of Energy.

Lasting through the mid-1970s, the AEC, along with other entities including the Department of Defense, National Institutes of Health, the American Cancer Society, the Manhattan Project, and various universities funded or conducted human radiation experiments. The government covered up most of these radiation mishaps until 1993, when President Bill Clinton ordered a change of policy. Nuclear radiation was known to be dangerous and deadly (from the atomic bombings of Hiroshima and Nagasaki in 1945), and the experiments were designed to ascertain the detailed effect of radiation on human health. In Oregon, 67 prisoners with inadequate consent to vasectomies had their testicles exposed to irradiation. In Chicago, 102 volunteers with unclear consent received injections of strontium and cesium solutions to simulate radioactive fallout.

AEC Chair
Atomic Energy Commission Commissioners
 * Sumner Pike : October 31, 1946 – December 15, 1951
 * David E. Lilienthal, Chairman : November 1, 1946 – February 15, 1950
 * Robert F. Bacher : November 1, 1946 – May 10, 1949
 * William W. Waymack : November 5, 1946 – December 21, 1948
 * Lewis L. Strauss : November 12, 1946 – April 15, 1950; Chairman : July 2, 1953 – June 30, 1958
 * Gordon Dean : May 24, 1949 – June 30, 1953; Chairman : July 11, 1950 – June 30, 1953
 * Henry DeWolf Smyth : May 30, 1949 – September 30, 1954
 * Thomas E. Murray : May 9, 1950 – June 30, 1957
 * Thomas Keith Glennan : October 2, 1950 – November 1, 1952
 * Eugene M. Zuckert : February 25, 1952 – June 30, 1954
 * Joseph Campbell : July 27, 1953 – November 30, 1954
 * Willard F. Libby : October 5, 1954 – June 30, 1959
 * John von Neumann : March 15, 1955 – February 8, 1957
 * Harold S. Vance : October 31, 1955 – August 31, 1959
 * John Stephens Graham : September 12, 1957 – June 30, 1962
 * John Forrest Floberg : October 1, 1957 – June 23, 1960
 * John A. McCone, Chairman : July 14, 1958 – January 20, 1961
 * John H. Williams : August 13, 1959 – June 30, 1960
 * Robert E. Wilson : March 22, 1960 – January 31, 1964
 * Loren K. Olson : June 23, 1960 – June 30, 1962
 * Glenn T. Seaborg, Chairman : March 1, 1961 – August 16, 1971
 * Leland J. Haworth : April 17, 1961 – June 30, 1963
 * John G. Palfrey : August 31, 1962 – June 30, 1966
 * James T. Ramey : August 31, 1962 – June 30, 1973
 * Gerald F. Tape : July 15, 1963 – April 30, 1969
 * Mary I. Bunting : June 29, 1964 – June 30, 1965
 * Wilfrid E. Johnson : August 1, 1966 – June 30, 1972
 * Samuel M. Nabrit : August 1, 1966 – August 1, 1967
 * Francesco Costagliola : October 1, 1968 – June 30, 1969
 * Theos J. Thompson : June 12, 1969 – November 25, 1970
 * Clarence E. Larson : September 2, 1969 – June 30, 1974
 * James R. Schlesinger, Chairman : August 17, 1971 – January 26, 1973
 * William O. Doub : August 17, 1971 – August 17, 1974
 * Dixy Lee Ray : August 8, 1972; Chairman : February 6, 1973 – January 18, 1975
 * William E. Kriegsman : June 12, 1973 – January 18, 1975
 * William A. Anders : August 6, 1973 – January 18, 1975

Ecology
For many years, the AEC provided the most conspicuous example of the benefit of atomic age technologies to biology and medicine. Shortly after the Atomic Energy Commission was established, its Division of Biology and Medicine began supporting diverse programs of research in the life sciences, mainly the fields of genetics, physiology, and ecology. Specifically concerning the AEC's relationship with the field of ecology, one of the first approved funding grants went to Eugene Odum in 1951. This grant sought to observe and document the effects of radiation emission on the environment from a recently built nuclear facility on the Savannah River in South Carolina. Odum, a professor at the University of Georgia, initially submitted a proposal requesting annual funding of $267,000, but the AEC rejected the proposal and instead offered to fund a $10,000 project to observe local animal populations and the effects of secondary succession on abandoned farmland around the nuclear plant.

In 1961, AEC chairman Glenn T. Seaborg established the Technical Analysis Branch (to be directed by Hal Hollister) to study the long-term biological and ecological effects of nuclear war. Throughout the early 1960s, this group of scientists conducted several studies to determine nuclear weapons' ecological consequences and their implications for human life. As a result, during the 1950s and 1960s, the U.S. government placed emphasis on the development and potential use of "clean" nuclear weapons to mitigate these effects.

In later years, the AEC began providing increased research opportunities to scientists by approving funding for ecological studies at various nuclear testing sites, most notably at Eniwetok, which was part of the Marshall Islands. Through their support of nuclear testing, the AEC gave ecologists a unique opportunity to study the effects of radiation on whole populations and entire ecological systems in the field. Prior to 1954, no one had investigated a complete ecosystem with the intent to measure its overall metabolism, but the AEC provided the means as well as the funding to do so. Ecological development was further spurred by environmental concerns about radioactive waste from nuclear energy and postwar atomic weapons production. In the 1950s, such concerns led the AEC to build a large ecology research group at their Oak Ridge National Laboratory, which was instrumental in the development of radioecology. A wide variety of research efforts in biology and medicine took place under the umbrella of the AEC at national laboratories and at some universities with agency sponsorship and funding. As a result of increased funding as well as the increased opportunities given to scientists and the field of ecology in general, a plethora of new techniques were developed which led to rapid growth and expansion of the field as a whole. One of these techniques afforded to ecologists involved the use of radiation, namely in ecological dating and to study the effects of stresses on the environment.

In 1969, the AEC's relationship with science and the environment was brought to the forefront of a growing public controversy that had been building since 1965. In search for an ideal location for a large-yield nuclear test, the AEC settled upon the island of Amchitka, part of the Aleutian Islands National Wildlife Refuge in Alaska. The main public concern was about their location choice, as there was a large colony of endangered sea otters in close proximity. To help defuse the issue, the AEC sought a formal agreement with the Department of the Interior and the U.S. state of Alaska to help transplant the colony of sea otters to other former habitats along the West Coast.

Arctic ecology
The AEC played a role in expanding the field of arctic ecology. From 1959 to 1962, the Commission's interest in this type of research peaked. For the first time, extensive effort was placed by a national agency on funding bio-environmental research in the Arctic. Research took place at Cape Thompson on the northwest coast of Alaska, and was tied to an excavation proposal named Project Chariot. The excavation project was to involve a series of underground nuclear detonations that would create an artificial harbor, consisting of a channel and circular terminal basin, which would fill with water. This would have allowed for enhanced ecological research of the area in conjunction with any nuclear testing that might occur, as it essentially would have created a controlled environment where levels and patterns of radioactive fallout resulting from weapons testing could be measured. The proposal never went through, but it evidenced the AEC's interest in Arctic research and development.

The simplicity of biotic compositions and ecological processes in the arctic regions of the globe made ideal locations in which to pursue ecological research, especially since at the time there was minimal human modification of the landscape. All investigations conducted by the AEC produced new data from the Arctic, but few or none of them were supported solely on that basis. While the development of ecology and other sciences was not always the primary objective of the AEC, support was often given to research in these fields indirectly as an extension of their efforts for peaceful applications of nuclear energy.

Reports
The AEC issued a large number of technical reports through their technical information service and other channels. These had many numbering schemes, often associated with the lab from which the report was issued. AEC report numbers included AEC-AECU (unclassified), AEC-AECD (declassified), AEC-BNL (Brookhaven National Lab), AEC-HASL (Health and Safety Laboratory), AEC-HW (Hanford Works), AEC-IDO (Idaho Operations Office), AEC-LA (Los Alamos), AEC-MDCC (Manhattan District), AEC-TID (Technical Information Division), and others. Today, these reports can be found in library collections that received government documents, through the National Technical Information Service (NTIS), and through public domain digitization projects such as the Technical Report Archive & Image Library, which are available via HathiTrust.

title: United States Atomic Energy Commission


text:
Robert Moon (postal inspector)

Robert Aurand Moon (April 15, 1917, Williamsport, Pennsylvania, US – April 10, 2001, Leesburg, Florida, US), sometimes called "Mr. ZIP", is considered the father of the ZIP Code or Zone Improvement Plan, a mechanism to route mail in the United States.

Biography
The U.S. Postal Service had been providing a premium level "air mail" service since the introduction of regular air transportation. As air transportation reliability improved and cost per ounce of mail decreased, it began to make financial sense to transport larger quantities of mail via intercontinental airlines rather than ship, and interstate via the old airmail routes rather than via train. But this made the sorting of mail by hand the new "bottleneck" for mail service. Rather than leisurely hand sorting mail bound for various destinations while passenger trains traveled between town centers, there was a need for machine sorting speed that could match the increased volume traveling between communities by jet and direct trucking.

Rather than being sorted within a city, destined for another city, the structure of delivery had by necessity become sorting at regional sorting centers that were connected by various modes of transport (rail, air and road), with the regional centers responsible for routing mail and packages to each local post office in their region.

In 1944, Moon submitted his idea for the "ZIP Code" while working as a postal inspector in Philadelphia, but no action was taken. Following his third submission of the concept, in 1963 it received the approval of a top-level postal service committee which shares credit for further development of the Zoning Improvement Plan, "ZIP". Mr. Moon's system described the routing to general regions of the country using the first three digits of what would eventually become a five-digit and later a nine-digit system. The next two digits of the first five-digit numeric code would be for smaller delivery areas, and were the work of others. The first Directory of Post Offices using five-digit ZIP code numbers would be published in 1963.

The "Mr. ZIP" cartoon character that was seen on postal delivery vehicles across the United States was originally developed by ad man Harold Wilcox, whose father was a postman. Wilcox's agency, Cunningham & Walsh, originally displayed the image for a Chase Manhattan bank-by-mail campaign.

ZIP codes have been proven to reducing delays and errors in processing mail while allowing increased postal service volume. The impact of the introduction of ZIP codes has led to decreased marketing costs per prospective customer, incrementally lowering costs of goods and services to every person in the country, because businesses can target their advertising and marketing to specific neighborhood demographics.

title: Robert Moon (postal inspector)


text:
Microsoft Bob

Microsoft Bob was a Microsoft software product intended to provide a more user-friendly interface for the Windows 3.1x, Windows 95 and Windows NT operating systems, supplanting the Windows Program Manager. The program was released on March 11, 1995 and discontinued in early 1996. Microsoft Bob presented screens showing a house, with rooms that the user could visit containing familiar objects corresponding to computer applications, such as a desk with pen and paper and a checkbook. Clicking on the pen and paper would open the system's word processor.

A cartoon dog named Rover and other cartoon characters provided guidance using speech balloons.

Upon release, Microsoft Bob was criticized in the media and did not gain wide acceptance with users, which resulted in its discontinuation. Its legacy would be observed in future Microsoft products, notably the use of virtual assistants. The Rover character later reappeared as a Windows XP search companion.

History
Microsoft Bob was released in March 1995 (before Windows 95 was released), although it had been widely publicized under the codename Utopia. The project leader for Bob was Karen Fries, a Microsoft researcher. The design was based on research by professors Clifford Nass and Byron Reeves of Stanford University. Melinda Gates, wife of Bill Gates, was the marketing manager for the product. Microsoft originally purchased the domain name bob.com from Boston-area techie Bob Antia, but later traded it to Bob Kerstein for the windows2000.com domain name.

Applications

Microsoft Bob includes various office-suite programs such as a finance application and a word processor. The user interface was designed to simplify the navigational experience for novice computer users.

Similar to early graphical shells like Jane, the main interface is portrayed as the inside of a house, with different rooms corresponding to common real-world room styles such as a kitchen or family room. Each room contains decorations and furniture, as well as icons that represent applications. The user may also fully customize the entire house and has full control over each room's decoration. The user may add, remove or reposition all objects. The user can also add or remove rooms and change the destinations of each door. The program offers multiple themes for room designs and decorations, such as contemporary and postmodern.

The applications built into Microsoft Bob are represented by matching decorations. For example, clicking on a clock opens the calendar, while a pen and paper represent the word processor. The user can also add shortcuts to applications on the computer. These shortcuts display the icon inside various styles of decorations, such as boxes and picture frames.

Bob includes the ability to install new applications, but because of the failure of the product, only a single add-on application package, Microsoft Great Greetings, was released.

Released just as the Internet was beginning to become popular, Bob offered an email client with which a user could subscribe to MCI Mail, a dial-up email account. The price was $5.00 per month to send up to 15 emails. Each email was limited to 5,000 characters, and each additional email after the limit was reached was an additional 45 cents. Customers were required to call a toll-free phone number to set up the account.

Bob features "Assistants", cartoon characters intended to help the user navigate the virtual house or perform tasks in the main interface or within the built-in applications.

Gateway 2000 edition
A special Gateway 2000-branded edition of Microsoft Bob 1.00a was bundled with the Gateway 2000 computer around 1995. This edition contains Gateway branding on the login screen along with additional rooms and backgrounds not seen in the retail version. One additional room is the attic, which contains the image of a Gateway 2000 computer box. In addition to the extra rooms, more icons appear by default in the new rooms.

Reception and legacy
Although a Consumer Electronics Show demonstration was met with generally positive reactions, reviewers generally derided the software, and Microsoft Bob became one of Microsoft's more visible product failures. The New York Times found the characters irritating and the home design apparently the work of an "esthetically challenged sixth-grader," criticized the hardware requirements and storage file formats and concluded that the program was not as simple to use as Microsoft had advertised. The Washington Post called the home environment "sterile" and "lifeless," wrote that the characters' cuteness wore thin quickly and criticized the scarce customization and access to Windows components. According to PC Data, the real sales from Bob's release until its discontinuation amounted to only about 58,000 copies—far short of Microsoft's estimate that it would sell millions as had Microsoft Works and Encarta. Despite being discontinued just one year after launch, Microsoft Bob continued to be sharply criticized in reviews and popular media. In 2017, Melinda Gates acknowledged that the software "needed a more powerful computer than most people had back then."

Microsoft Bob was listed seventh in PC World magazine's list of the 25 worst tech products of all time and as the number-one worst product of the decade by CNET.com. It was also listed among the 50 worst inventions in Time magazine, which called Bob "overly cutesy" and an "operating system designed around Clippy." Microsoft's Steve Ballmer mentioned Bob as an example in which "we decided that we have not succeeded and let's stop [now]."

Microsoft employee Raymond Chen disclosed that an encrypted copy of Bob was included on Windows XP installation CDs in order to consume space to prevent piracy. It was thought that by consuming an additional 30 megabytes on the disc (in the era of dial-up internet access), users with 56 kbit/s modems would be dissuaded from attempting to download the software illegally. Retired Microsoft engineer David Plummer has identified himself as the employee responsible for the encrypted copy. The installer routine would check for the "blob of Bob" and if an "OEM blob" was detected, only an OEM product key would be accepted. Tech journalist Harry McCracken called the story "a delightfully urban legend-y tale" and noted its similarities to an April Fools' Day joke claiming that Bob was hidden in Windows Vista.

The use of virtual assistants in Microsoft Bob later inspired Clippit, nicknamed "Clippy" the paperclip, the default Office Assistant in Microsoft Office. Rover, the dog mascot character introduced with Bob, became a "search companion" for Windows XP's file-search function.

Microsoft graphic designer Vincent Connare designed the typeface Comic Sans when he noticed that Rover's speech was displayed in Times New Roman, which he felt was inappropriate for a cartoon dog. Although Connare's font did not appear in the final release of Microsoft Bob because its characters did not fit within any of the typographical grids, it later debuted in Microsoft's 3D Movie Maker and became an additional typeface for Windows 95.

title: Microsoft Bob


text:
Melinda French Gates

Melinda French Gates (born Melinda Ann French; August 15, 1964) is an American philanthropist, former multimedia product developer and manager at Microsoft, and the ex-wife of its co-founder and billionaire Bill Gates. French Gates has consistently been ranked as one of the world's most powerful women by Forbes magazine. In 2000, she and her then-husband Bill Gates co-founded the Bill & Melinda Gates Foundation, the world's largest private charitable organization. She and her ex-husband have been awarded the US Presidential Medal of Freedom and the French Legion of Honour. In early May 2021, Bill and Melinda Gates announced they were getting divorced. She was recognized as one of the BBC's 100 women of 2021.

Early life and education
Melinda Ann French was born on August 15, 1964, in Dallas, Texas. She is the second of four children born to Raymond Joseph French Jr., an aerospace engineer, and Elaine Agnes Amerland, a homemaker. She has an older sister and two younger brothers.

French, a Catholic, attended St. Monica Catholic School, where she was the valedictorian of her class. At age 14, French was introduced to the Apple II by her father and Mrs. Bauer, a school teacher who advocated teaching computer science at the all-girls school. It was from this experience she developed her interest in computer games and the BASIC programming language.

French graduated as valedictorian from Ursuline Academy of Dallas in 1982. She earned a bachelor's degree in computer science and economics from Duke University in 1986 and an MBA from Duke's Fuqua School of Business in 1987. At Duke, French was a member of the Kappa Alpha Theta sorority, Beta Rho Chapter.

Career

French Gates's first job was tutoring children in mathematics and computer programming. After graduation, she became a marketing manager with Microsoft, being responsible for the development of multimedia products. These included Cinemania, Encarta, Publisher, Microsoft Bob, Money, Works (Macintosh) and Word. She worked on Expedia, which became one of the most popular travel booking websites. In the early 1990s, French Gates was appointed as General Manager of Information Products, a position which she held until 1996. She left Microsoft that year, reportedly, to focus on starting a family.

French Gates served as a member of Duke University's Board of Trustees from 1996 to 2003. She attends the annual Bilderberg Group conference and has held a seat on the Board of Directors of Graham Holdings (formerly The Washington Post Company) since 2004. She was also on the board of directors at Drugstore.com but left in August 2006 to focus on philanthropic projects. Since 2000, French Gates has been in the public eye, stating "As I thought about strong women of history, I realized that they stepped out in some way." This has allowed her work shaping and advancing the goals of the Bill & Melinda Gates Foundation to be publicly recognized. By 2022, Bill and Melinda had given US$59.1 billion of their personal wealth to the Foundation. In 2015, French Gates founded Pivotal Ventures as a separate, independent organization to identify and implement innovative solutions to problems affecting U.S. women and families. On May 13, 2024, French Gates resigned as co-chair of the Bill and Melinda Gates Foundation, to be effective June 7.

Writing
In 2019, French Gates debuted as an author with the book The Moment of Lift: How Empowering Women Changes the World. Former president Barack Obama starred in a comedy sketch in order to promote it. The book highlights the failure to acknowledge women's unpaid work, drawing on feminist economist Dame Marilyn Waring's book If Women Counted.

Personal life
Melinda began dating Microsoft CEO Bill Gates in 1987, after meeting him at a trade fair in New York. In 1994, she married Gates in a private ceremony held in Lanai, Hawaii. They have three children: daughters Jennifer Gates Nassar and Phoebe Gates and son Rory Gates. The family maintained a home in an earth-sheltered mansion overlooking Lake Washington in Medina, Washington. The family also owned an oceanfront residence in Del Mar, California.

In May 2021, Melinda Gates and her then-husband announced their decision to divorce in a joint statement on social media. This ended 27 years of marriage, and 34 years as a couple. According to The Wall Street Journal, Melinda Gates had been meeting with divorce lawyers since at least October 2019 after Bill's business dealings with convicted sex offender Jeffrey Epstein became public, and she had warned him about associating with Epstein as early as 2013. Although the couple did not have a prenuptial agreement, Melinda Gates (who filed) did not request spousal support. She was allocated over $2 billion worth of shares and stocks from the divorce. The divorce was finalized on August 2, 2021. In March 2022, Melinda said that she and Bill were "friendly" but "not friends".

Awards and recognition
In 1998, Melinda and Bill Gates were each honored with an American Library Association Honorary Membership. In 2002, Melinda and Bill Gates received the Award for Greatest Public Service Benefiting the Disadvantaged, an award given out annually by Jefferson Awards. In December 2005, Melinda and Bill were named by Time as Persons of the Year alongside Bono. Melinda and Bill Gates received the Spanish Prince of Asturias Award for International Cooperation on May 4, 2006, in recognition of their world impact through charitable giving. In November 2006, French Gates was awarded the Insignia of the Order of the Aztec Eagle, together with Bill, who was awarded the Placard of the same order, both for their philanthropic work around the world in the areas of health and education, particularly in Mexico, and specifically in the program "Un país de lectores".

In May 2006, in honor of her work to improve the lives of children locally and around the world, Seattle Children's Hospital dedicated the Melinda French Gates Ambulatory Care building at Seattle Children's (formerly Children's Hospital and Regional Medical Center). She chaired a campaign for the hospital to fundraise $300 million to expand facilities, fund under-compensated and uncompensated care, and grow the hospital's research program to find cures and treatments.

In 2007, French Gates received an honorary doctorate in medicine from the Karolinska Institute in Stockholm, Sweden. In 2009, she and her then-husband received honorary degrees from the University of Cambridge. Their benefaction of $210 million in 2000 set up the Gates Cambridge Trust, which funds postgraduate scholars from outside the UK to study at the university. Lastly, she was awarded an honorary Doctor of Humane Letters by Duke University in 2013 in honor of her philanthropic commitment.

She has been repeatedly recognized by Forbes in its annual list of the 100 Most Powerful Women, ranking #3 in 2013, 2014, 2015 and 2017, #4 in 2012 and 2016, #5 in 2020 and 2021, and #6 in 2011, 2018, 2019 and 2022, #10 in 2023. She was awarded the UCSF medal in 2013. French Gates was appointed an honorary Dame Commander of the Order of the British Empire (DBE) in 2013 for services to philanthropy and international development. In recognition of the foundation's philanthropic activities in India, Bill and Melinda jointly received India's third-highest civilian honor, Padma Bhushan, in 2015. In 2016, President Barack Obama awarded French Gates and her husband with the Presidential Medal of Freedom for their philanthropic efforts.

In 2017, President François Hollande awarded France's highest national honor to French Gates and her husband for their charitable efforts, i.e. as Commander of the Legion of Honour. That year, she was awarded the Otto Hahn Peace Medal 2016 of the United Nations Association of Germany (DGVN), Berlin-Brandenburg, "for outstanding services to peace and international understanding" in the historic Berlin Town Hall. That year, French Gates was listed by UK-based company Richtopia at number 12 in the list of 200 Most Influential Philanthropists and Social Entrepreneurs Worldwide. In June 2024, she delivered the Commencement address at Stanford University.

Women in technology
French Gates's experience of a male-dominated workplace at Microsoft inspired her to encourage more women in the computing field. In September 2016, she announced her desire to increase diversity in the workplace, especially in the technology industry, stating: "Every company needs technology, and yet we're graduating fewer women technologists. That is not good for society. We have to change it." French Gates also spoke about this topic at the 2017 Grace Hopper Celebration of Women in Computing, an annual series of conferences. She was recognized as one of the BBC's 100 women of 2021.

Published works

 * The Moment of Lift: How Empowering Women Changes the World (2019) ISBN 978-1-250-31357-7

title: Melinda French Gates