# 7.6: Taylor's Theorem Revisited - Mathematics

The following is a version of Taylor's Theorem with an alternative form of the remainder term.

Theorem (PageIndex{1})

(Taylor's Theorem)

Suppose (f in C^{(n+1)}(a, b), alpha in(a, b),) and

[P_{n}(x)=sum_{k=0}^{n} frac{f^{(k)}(alpha)}{k !}(x-alpha)^{k}.]

Then, for any (x in(a, b)),

[f(x)=P_{n}(x)+int_{alpha}^{x} frac{f^{(n+1)}(t)}{n !}(x-t)^{n} d t.]

Proof

By the Fundamental Theorem of Calculus, we have

[int_{alpha}^{x} f^{prime}(t) d t=f(x)-f(alpha),]

which implies that

[f(x)=f(alpha)+int_{alpha}^{x} f^{prime}(t) d t.]

Hence the theorem holds for (n=0 .) Suppose the result holds for (n=k-1,) that is,

[f(x)=P_{k-1}(x)+int_{alpha}^{x} frac{f^{(k)}(t)}{(k-1) !}(x-t)^{k-1} d t.]

Let

[F(t)=f^{(k)}(t),]

[g(t)=frac{(x-t)^{k-1}}{(k-1) !},]

and

[G(t)=-frac{(x-t)^{k}}{k !}.]

Then

[egin{aligned} int_{alpha}^{x} frac{f^{(k)}(t)}{(k-1) !}(x-t)^{k-1} d t &=int_{alpha}^{x} F(t) g(t) d t &=F(x) G(x)-F(alpha) G(alpha)-int_{alpha}^{x} F^{prime}(t) G(t) d t &=frac{f^{(k)}(alpha)(x-alpha)^{k}}{k !}+int_{alpha}^{x} frac{f^{(k+1)}(t)}{k !}(x-t)^{k} d t, end{aligned}]

Hence

[f(x)=P_{k}(x)+int_{alpha}^{x} frac{f^{(k+1)}(t)}{k !}(x-t)^{k} d t,]

and so the theorem holds for (n=k). (quad) Q.E.D.

Exercise (PageIndex{1})

(Cauchy form of the remainder)

Under the conditions of Taylor's Theorem as just stated, show that

[int_{alpha}^{x} frac{f^{(n+1)}(t)}{n !}(x-t)^{n} d t=frac{f^{(n+1)}(gamma)}{n !}(x-gamma)^{n}(x-alpha)]

for some (gamma) between (alpha) and (x .)

Exercise (PageIndex{2})

(Lagrange form of the remainder)

Under the conditions of Taylor's Theorem as just stated, show that

[int_{alpha}^{x} frac{f^{(n+1)}(t)}{n !}(x-t)^{n} d t=frac{f^{(n+1)}(gamma)}{(n+1) !}(x-alpha)^{n+1}]

for some (gamma) between (alpha) and (x .) Note that this is the form of the remainder in Theorem (6.6 .1,) although under slightly more restrictive assumptions.

## 7.6: Taylor's Theorem Revisited - Mathematics

Notes for my MSRI lectures on 3/13/18  pdf

My mathematical work is supported by NSF grant DMS-1100784. I am an editor for Journal of the AMS, Algebra and Number Theory, and IMRN. Please consider submitting appropriate papers to these journals. All such submissions must go through the journal website papers cannot be submitted through editors, and any submissions on famous open problems are subject to the strict rules as described at the lower half of this page.

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## H.-Kaletha-Weinstein study guide and FAQ

Tasho Kaletha, Jared Weinstein and I have just posted a joint paper, On the Kottwitz conjecture for local shtuka spaces. This is a heavily revised and rewritten version of a preprint by Tasho and Jared which has been available since 2017 (henceforth “KW”).

Is the main result here the same as in KW?
More or less, yes. The main theorem confirms the Kottwitz conjecture for the cohomology of moduli spaces of local shtukas, ignoring the Weil group action and allowing for an “error term” consisting of a non-elliptic virtual representation.

Is the rough idea of the proof still the same as in KW?
Yes: the idea is still to compute the cohomology by applying a suitable Lefschetz-Verdier trace formula, and explicitly computing the “local terms” at all of the elliptic fixed points.

OK, what’s new, then?
Well, basically all of the details are different. First and foremost, the discussion of the relevant Lefschetz-Verdier trace formula for v-stacks has been completely rewritten, using Lu-Zheng’s magical formalism of symmetric monoidal 2-categories of cohomological correspondences. This leads to huge conceptual simplifications, both in the statements and the proofs. Actually, we go beyond the formalism of Lu-Zheng, introducing certain 2-categories of “based cohomological correspondences”. This formalism play a crucial role in the proof of Theorem 4.5.3, which is a key ingredient in the proof of the main theorem. It’s also hard to imagine proving Theorem 4.6.1 without this formalism.

Secondly, a key calculation of local terms on the -affine Grassmannian, Theorem 5.1.3, now has a completely different and correct (!) proof, via a degeneration to a similar calculation on the Witt vector affine Grassmannian. This degeneration argument relies on the observation that the local terms appearing in the Lefschetz-Verdier trace formula are compatible with any base change, Proposition 5.3.1 this seems to be a new observation even for schemes. (My first proof of Proposition 5.3.1 involved checking the commutativity of 500 diagrams, but eventually I realized that it follows immediately from the Lu-Zheng formalism!) Once we’ve degenerated to the Witt vector affine Grassmannian, we make an intriguing global-to-local argument using the trace formula and known properties of the weight functors in geometric Satake, and relying crucially on a recent result of Varshavsky.

There are some other notable improvements:

• All assumptions in KW of the form “assume that some representation admits an invariant -lattice” have now been eliminated. This is far from formal, since we definitely don’t have a 6-functor formalism with -coefficients at hand. Instead, the idea is to reduce “by a continuity argument” to the case where things do admit lattices. The key ingredient here is Theorem 6.5.4, formerly a conjecture of Taylor, which should have lots of other applications.
• The final part of the main theorem, giving practical criteria for the error term to vanish, is new.
• Theorem 1.0.3 is new.

On a practical note, the Fargues-Scholze paper is now finished and available, so we can properly build on the powerful machinery and results in that paper.

Finally, the margins are no longer gigantic.

Sure. The main Theorem 1.0.2/6.5.1 in the paper is essentially a combination of two separate results, Theorem 6.5.2 and Theorem 3.2.9. Theorem 3.2.9 is conditional on the refined local Langlands correspondence, and the material needed to formulate and prove this result is developed in Chapter 3 and Appendix A. The bulk of the paper (sections 4-6 and Appendices B-C) is devoted to building up enough material to prove Theorem 6.5.2.

Each of sections 3, 4, and 5 is fairly self-contained, although section 5 depends slightly on section 4. Section 6 builds heavily on sections 4 and 5, and also on [FS21]. Section 5 is actually devoted to proving a single result, Theorem 5.1.3 (and its reinterpretation Theorem 5.1.4), and can be treated as a black box.

One thing you could do is read the introduction, then jump to Proposition 6.4.7 and work your way backwards as needed until all the notation makes sense. This proposition is the technical heart of the paper. The key inputs into the proof of Proposition 6.4.7 are Theorem 5.1.4, Theorem 4.5.3, and Corollary 4.3.8. The use of “the trace formula” in this whole argument is actually somewhat hidden: it is entirely encoded in the equality sign labelled “Cor. 4.3.8” at the bottom of p. 66.

From Proposition 6.4.7, Theorem 6.4.9 (a weak form of Theorem 6.5.2) follows quickly, and then Theorem 6.5.2 follows from Theorem 6.4.9 by a continuity argument which relies crucially on Theorem 6.5.4.

This sounds cool, but you didn’t punt any key ingredients into a separate and currently nonexistent paper, did you?
I’m glad you asked. In working out the trace formula formalism in enough generality to give a conceptual proof of the main theorem, it became clear at some point that we needed the existence and expected properties of the functors and in etale cohomology for “smooth-locally nice” maps between Artin v-stacks. For maps which are actually “nice”, and in particular non-stacky, these functors were constructed by Scholze, but extending them to the setting of stacky maps is not formal. Already in the analogous setting of schemes versus Artin stacks, these functors for Artin stacks were only constructed recently by Liu-Zheng, making heavy use of -categories. In a forthcoming paper, Jared and I together with my postdoc Dan Gulotta will explain how to construct and for smooth-locally nice maps of Artin v-stacks. The arguments will very closely follow the arguments of Liu-Zheng, with the exception of one key non-formal piece of input (which took me three years of intermittent work).

Why do you only prove an explicit formula for the virtual character of restricted to elliptic elements of ? What’s so special about elliptic elements in this context?
The fixed points of elliptic elements lie in the “admissible locus” inside the relevant closed Schubert cell. This is discussed in detail in the introduction.

Fine, but maybe there’s still an obvious explicit formula for the virtual character of at any strongly regular semisimple element.
Here’s an example to illustrate why I think this is a hard problem. I’ll be slightly heuristic about the details if you want more precision, please leave a comment.

Let’s take and , so we’re in the Lubin-Tate/Drinfeld setting with the units in the quaternion algebra over . Let be the trivial representation of . Then as a virtual representation of , by an old calculation of Schneider-Stuhler. Note that is a principal series representation, hence non-elliptic, so the virtual character of on elliptic elements of is the constant function . This matches perfectly with the fact that any elliptic has two fixed points in , both contained in , and the “naive” local terms of the relevant sheaf at both these points are . Here is the evident open immersion.

On the other hand, if is regular semisimple and nonelliptic, then it’s conjugate to some with . In this case there are still two fixed points, but they both lie in the “boundary” . Since restricted to the boundary is identically zero, its naive local terms at both fixed points are zero. Nevertheless, the trace formula still works, so the contribution of the true local terms at these two points must add up to the virtual character of evaluated at . This character value can be computed explicitly by van Dijk’s formula, and turns out to be . So this slightly strange expression needs to emerge from the sum of these two local terms.

Putting these observations together, we see that in this example, the true local terms do not equal the naive local terms at the non-elliptic fixed points. Morally, the analysis in Section 3 of the paper can and should be interpreted as an unravelling of naive local terms (at elliptic fixed points). Hopefully this makes the difficulty clear.

Can you give an example where the error term in Theorem 1.0.2 is nonzero?
Sure, look at the two-dimensional Lubin-Tate local Shimura datum and take trivial again, so with the Steinberg parameter. Then as discussed in the previous answer, while the sum on the right-hand side in Theorem 1.0.2 is just . So in this case is, up to sign, a reducible principal series representation.

It would be very interesting to formulate an extension of the Kottwitz conjecture which covered all discrete L-parameters. Since the Kottwitz conjecture is best regarded as a reflection of the Hecke eigensheaf property in Fargues’s conjecture, this is closely related to understanding Fargues’s conjecture for general discrete parameters.

It seems like the Kottwitz conjecture should also make sense for local shtukas over Laurent series fields. Why doesn’t the paper cover this case too?
Good question! The short answer is that all of the p-adic geometry should work out uniformly over Laurent series fields and over finite extensions of , and indeed Fargues-Scholze handles these two situations uniformly. However, the literature on local Langlands and harmonic analysis for reductive groups over positive characteristic local fields case is pretty thin. This is the main reason we stuck to the mixed characteristic case. For more on the state of the literature, take a look at section 1.6 in Kal16a.

Do the methods of this paper give any information about for non-basic b?
Yes: If is non-basic, or is basic and is parabolically induced, our methods can be applied to prove that is always a virtual combination of representations induced from proper parabolic subgroups of . This is a soft form of the Harris-Viehmann conjecture. The details of this argument will be given elsewhere (in the sense of Deligne-Mumford).

Any surprising subtleties?
Lemma 6.5.5 was a bracing experience. My first “proof” was absolute nonsense, due to some serious misconceptions I had about harmonic analysis on p-adic groups – big thanks to JT for setting me straight. My second proof didn’t work because I cited a result in a famous article (>100 mathscinet citations) whose proof turns out to be fallacious – big thanks to JFD for setting me straight. Tasho and I devised the current proof, which in hindsight is actually a very natural argument. (I was hoping, apparently unreasonably, for a shortcut.)

Closing thoughts:
-At least for me, the proof of Theorem 6.5.4 is a very striking illustration of the raw Wagnerian power of the machinery developed in [FS21]. This result was totally out of reach a few years ago, but with the machinery of [FS21] in hand, the proof is less than a page! Note that this argument crucially uses the and Hecke operator formalism of [FS21] with general coefficient rings.
-Let me again emphasize how heavily the paper depends on the recent preprints [LZ20], [Var20], and (of course) [FS21].
-The paper depends, more significantly than most, on a choice of isomorphism . It’s naturally of interest to wonder how everything depends on this choice. Imai has formulated a conjectural answer to this question. Of course, “independence of ” results in the etale cohomology of diamonds are probably very hard.

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GenEd Learning Objective: Key Literacies

This course is the first in a sequence of three calculus courses designed for students in the earth and mineral sciences and related fields. Topics include limits of functions, continuity the definition of the derivative, various rules for computing derivatives (such as the product rule, quotient rule, and chain rule), implicit differentiation, higher-order derivatives, solving related rate problems, and applications of differentiation such as curve sketching, optimization problems, and Newton's method the definition of the definite integral, computation of areas, the Fundamental Theorem of Calculus, integration by substitution, and various applications of integration such as computation of areas between two curves, volumes of solids, and work.

Enforced Prerequisite at Enrollment: Math 22 and Math 26 or Math 26 and satisfactory performance on the mathematics placement examination or Math 40 or Math 41 or satisfactory performance on the mathematics placement examination.

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

Calculus is an important building block in the education of any professional who uses quantitative analysis. This course introduces and develops the mathematical skills required for analyzing change and creating mathematical models that replicate real-life phenomena. The goals of our calculus courses include to develop the students' knowledge of calculus techniques and to use the calculus environment to develop critical thinking and problem solving skills. The concept of limit is central to calculus MATH 140 begins with a study of this concept. Differential calculus topics include derivatives and their applications to rates of change, related rates, linearization, optimization, and graphing techniques. The Fundamental Theorem of Calculus, relating differential and integral calculus begins the study of Integral Calculus. Antidifferentiation and the technique of substitution is used in integration applications of finding areas of plane figures and volumes of solids of revolution. Trigonometric functions are included in every topic. Students may only take one course for credit from MATH 110, 140, 140A, 140B, and 140H.

Enforced Prerequisite at Enrollment: Math 22 and Math 26 or Math 26 and satisfactory performance on the mathematics placement examination or Math 40 or Math 41 or satisfactory performance on the mathematics placement examination.

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

MATH 141 is the second course in a two- or three-course calculus sequence for students in science, engineering and related fields. Calculus is an important building block in the education of any professional who uses quantitative analysis. This course further introduces and develops the mathematical skills required for analyzing growth and change and creating mathematical models that replicate reallife phenomena. The goals of our calculus courses include to develop the students' knowledge of calculus techniques and to use the calculus environment to develop critical thinking and problem solving skills. This course covers the following topics: logarithms, exponentials, and inverse trigonometric functions applications of the definite integral and techniques of integration sequences and series power series and Taylor polynomials parametric equations and polar functions. Students may take only one course for credit from MATH 141, 141B, and 141H.

Enforced Prerequisite at Enrollment: MATH 140 or MATH 140A or MATH 140B or MATH 140E or MATH 140G or MATH 140H.

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

Techniques of integration and applications to biology elementary matrix theory, limits of matrices, Markov chains, applications to biology and the natural sciences elementary and separable differential equations, linear rst-order differential equations, linear systems of differential equations, the Lotka-Volterra equations. Students may take only one course for credit from MATH 141, 141B, and 141H.

Enforced Prerequisite at Enrollment: MATH 140 or MATH 140A or MATH 140B or MATH 140E or MATH 140G or MATH 140H.

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

MATH 141E is the second course in a two- or three-course calculus sequence for students in science, engineering and related fields. Calculus is an important building block in the education of any professional who uses quantitative analysis. This course further introduces and develops the mathematical skills required for analyzing growth and change and creating mathematical models that replicate reallife phenomena. The goals of our calculus courses include to develop the students' knowledge of calculus techniques and to use the calculus environment to develop critical thinking and problem solving skills. This course covers the following topics: logarithms, exponentials, and inverse trigonometric functions applications of the definite integral and techniques of integration sequences and series power series and Taylor polynomials parametric equations and polar functions.

Enforced Prerequisite at Enrollment: MATH 140 or MATH 140A or MATH 140B or MATH 140E or MATH 140G or MATH 140H.

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

This course is the second in a sequence of three calculus courses designed for students in the earth and mineral sciences and related fields. Topics include inverse functions of exponential, logarithmic, and trigonometric functions indeterminate forms and L'Hopital's rule various techniques of integration, including integration by parts, trigonometric integrals, trigonometric substitution, and partial fractions improper integration infinite sequences and series, tests for convergence and divergence of infinite series, including the integral test, comparison tests, ratio test, root test power series, Taylor and MacLaurin Series.

Enforced Prerequisite at Enrollment: MATH 140 or MATH 140A or MATH 140B or MATH 140E or MATH 140G or MATH 140H.

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

MATH 141 is the second course in a two- or three-course calculus sequence for students in science, engineering and related fields. Calculus is an important building block in the education of any professional who uses quantitative analysis. This course further introduces and develops the mathematical skills required for analyzing growth and change and creating mathematical models that replicate reallife phenomena. The goals of our calculus courses include to develop the students' knowledge of calculus techniques and to use the calculus environment to develop critical thinking and problem solving skills. This course covers the following topics: logarithms, exponentials, and inverse trigonometric functions applications of the definite integral and techniques of integration sequences and series power series and Taylor polynomials parametric equations and polar functions. Students may take only one course for credit from MATH 141, 141B, and 141H.

Enforced Prerequisite at Enrollment: MATH 140 or MATH 140A or MATH 140B or MATH 140E or MATH 140G or MATH 140H.

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

Formal courses given infrequently to explore, in depth, a comparatively narrow subject which may be topical or of special interest.

Bachelor of Arts: Quantification

Courses offered in foreign countries by individual or group instruction.

Bachelor of Arts: Quantification

International Cultures (IL)

Fundamental concepts of arithmetic and geometry, including problem solving, number systems, and elementary number theory. For elementary and special education teacher certification candidates only. A student who has passed EDMTH 444 may not take MATH 200 for credit. MATH 200 Problem Solving in Mathematics (3) (GQ) This is a course in mathematics content for prospective elementary school teachers. Students are assumed to have successfully completed two years of high school algebra and one year of high school geometry. Students are expected to have reasonable arithmetic skills. The content and processes of mathematics are presented in this course to develop mathematical knowledge and skills and to develop positive attitudes toward mathematics. Problem solving is incorporated throughout the topics of number systems, number theory, probability, and geometry, giving future elementary school teachers tools to further explore mathematical content required to convey the usefulness, beauty and power of mathematics to their own students.

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

Mathematical ways of thinking, number sequences, numeracy, symmetry, regular polygons, plane curves, methods of counting, probability and data analysis. For elementary and special education teacher certification candidates only.

General Education: Quantification (GQ)

This course studies the foundations of elementary school mathematics with an emphasis on problem solving. MATH 201 Problem Solving in Mathematics II (3) (GQ) Problem Solving in Mathematics II studies the foundations of elementary school mathematics with an emphasis on problem solving. Mathematical ways of thinking are integrated throughout the study of probability, statistics, graphing, geometric shapes, and measurement. This course is designed for prospective teachers not only to gain the ability to explain the mathematics in elementary school courses, but also to help them comprehend the underlying mathematical concepts. Gaining a deeper understanding will enable them to assist their young students in the classroom since effective mathematical teaching requires understanding what students know, what they need to learn, and then helping them to learn it well.

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

Topics in calculus with an emphasis on applications in engineering technology. MATH 210 Calculus with Engineering Technology Applications (3) is a three-credit course to be taken either after the MATH 81, MATH 82, MATH 83 sequence or after a semester of college-level calculus. The content of the course is geared toward the needs of engineering technology majors and places a large emphasis on technology and applications. The course provides mathematical tools required in the upper division engineering technology courses. A primary goal is to have students use technology to solve more realistic problems than the standard simplistic ones that can be solved by "pencil and paper." Student evaluation will be performed through exams, quizzes, graded assignments, and a cumulative final exam. It is expected that MTHBD 210 will be offered every semester with an enrollment of 44-80 students.

Enforced Prerequisite at Enrollment: MATH 83 or MATH 140

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Integrative Thinking

GenEd Learning Objective: Key Literacies

Topics in ordinary differential equations, linear algebra, complex numbers, Eigenvalue solutions and Laplace transform methods. MATH 211 Intermediate Calculus and Differential Equations with Applications (4) MATH 211 is a three-credit course to be taken after MATH 210. The content of the course is geared toward the needs of engineering technology majors and places a large emphasis on technology and applications. The course provides mathematical tools required in the engineering technology courses at the sixth semester and above. A primary goal is to have students use technology to solve more realistic problems than the standard simplistic ones that can be solved by "pencil and paper." Student evaluation will be performed through exams, quizzes, graded assignments, and a cumulative final exam.

Enforced Prerequisite at Enrollment: MATH 210

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

Systems of linear equations matrix algebra eigenvalues and eigenvectors linear systems of differential equations. MATH 220 Matrices (2-3) (GQ) (BA) This course meets the Bachelor of Arts degree requirements.Systems of linear equations appear everywhere in mathematics and its applications. MATH 220 will give students the basic tools necessary to analyze and understand such systems. The initial portion of the course teaches the fundamentals of solving linear systems. This requires the language and notation of matrices and fundamental techniques for working with matrices such as row and column operations, echelon form, and invertibility. The determinant of a matrix is also introduced it gives a test for invertibility. In the second part of the course the key ideas of eigenvector and eigenvalue are developed. These allow one to analyze a complicated matrix problem into simpler components and appear in many disguises in physical problems. The course also introduces the concept of a vector space, a crucial element in future linear algebra courses. This course is completed by a wide variety of students across the university, including students majoring in engineering programs, the sciences, and mathematics. (In case of many of these students, MATH 220 is a required course in their degree program.)

Enforced Prerequisite at Enrollment: MATH 110 or MATH 140 or MATH 140B or MATH 140E or MATH 140G or MATH 140H

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

Honors course in systems of linear equations matrix algebra eigenvalues and eigenvectors linear systems of differential equations. MATH 220H Honors Matrices (2) (GQ)(BA) This course meets the Bachelor of Arts degree requirements. This course is intended as an introduction to linear algebra with a focus on solving systems for linear equations. Topics include systems of linear equations, row reduction and echelon forms, linear independence, introduction to linear transformations, matrix operations, inverse matrices, dimension and rank, determinants, eigenvalues, eigenvectors, diagonalization, and orthogonality.The typical delivery format for the course is two 50-minute lectures per week, with typical assessment tools including examinations, quizzes, homework, and writing assignments.In contrast to the non-honors version of this course, the honors version is typically more theoretical and will often include more sophisticated problems. Moreover, certain topics are often discussed in more depth and are sometimes expanded to include applications which are not visited in the non-honors version of the course.

Enforced Prerequisite at Enrollment: MATH 110 or MATH 140 or MATH 140B or MATH 140E or MATH 140G or MATH 140H

Bachelor of Arts: Quantification

General Education: Quantification (GQ)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

Three-dimensional analytic geometry vectors in space partial differentiation double and triple integrals integral vector calculus. Students who have passed either Math 231 or MATH 232 may not schedule Math 230 or MATH 230H for credit.

Enforced Prerequisite at Enrollment: MATH 141 or MATH 141B or MATH 141E or MATH 141G or MATH 141H

Bachelor of Arts: Quantification

Honors course in three-dimensional analytic geometry vectors in space partial differentiation double and triple integrals integral vector calculus. Students who have passed either MATH 231 or MATH 232 may not schedule MATH 230 or MATH 230H for credit. MATH 230H Honors Calculus and Vector Analysis (4) This course is the third in a sequence of three calculus courses designed for students in engineering, science, and related fields. Topics include vectors in space, dot products, cross products vector-valued functions, modeling motion, arc length, curvature functions of several variables, limits, continuity, partial derivatives, directional derivatives, gradient vectors, Lagrange multipliers double integrals, triple integrals line integrals, Green's Theorem, Stokes' Theorem, the Divergence Theorem.The typical delivery format for the course is four 50-minute lectures per week, with typical assessment tools including examinations, quizzes, homework, and writing assignments.In contrast to the non-honors version of this course, the honors version is typically more theoretical and will often include more sophisticated problems. Moreover, certain topics are often discussed in more depth and are sometimes expanded to include applications which are not visited in the non-honors version of the course.

Enforced Prerequisite at Enrollment: MATH 141 or MATH 141B or MATH 141E or MATH 141G or MATH 141H

Bachelor of Arts: Quantification

Analytic geometry in space partial differentiation and applications. Students who have passed MATH 230 or MATH 230H may not schedule this course.

Enforced Prerequisite at Enrollment: MATH 141 or MATH 141B or MATH 141E or MATH 141G or MATH 141H

Bachelor of Arts: Quantification

Honors course in analytic geometry in space partial differentiation and applications. Students who have passed MATH 230 or MATH 230H may not schedule this course. MATH 231H Honors Calculus of Several Variables (2) This course covers a subset of the material found in MATH 230. Topics include vectors in space, dot products, cross products vector-valued functions, modeling motion, arc length, curvature functions of several variables, limits, continuity, partial derivatives, directional derivatives, gradient vectors, Lagrange multipliers.The typical delivery format for the course is two 50-minute lectures per week, with typical assessment tools including examinations, quizzes, homework, and writing assignments.In contrast to the non-honors version of this course, the honors version is typically more theoretical and will often include more sophisticated problems. Moreover, certain topics are often discussed in more depth and are sometimes expanded to include applications which are not visited in the non-honors version of the course.

Enforced Prerequisite at Enrollment: MATH 141 or MATH 141B or MATH 141E or MATH 141G or MATH 141H

Bachelor of Arts: Quantification

Multidimensional analytic geometry, double and triple integrals potential fields flux Green's, divergence and Stokes' theorems. Students who have passed MATH 230 may not schedule this course for credit.

Enforced Prerequisite at Enrollment: MATH 231 or MATH 231H

Bachelor of Arts: Quantification

This course will cover systems of differential equations, multivariable calculus, and applications to biology and the life sciences. Students will learn about complex numbers, and their relation to oscillations. Analysis of biologically relevant mathematical models will include the linear stability of couples nonlinear systems, and the method of separation of timescales. The course will also introduce probability theory in a biological context, including conditional probability, Bayes Theorem, probability distributions, and stochastic modeling in the life sciences.

Enforced Prerequisites at Enrollment: MATH141B or instructor approval

First- and second-order equations special functions Laplace transform solutions higher order equations. Students who have passed MATH 251 may not schedule this course for credit.

Enforced Prerequisite at Enrollment: MATH 141 or MATH 141B or MATH 141E or MATH 141G or MATH 141H

Bachelor of Arts: Quantification

First- and second-order equations special functions Laplace transform solutions higher order equations Fourier series partial differential equations.

Enforced Prerequisite at Enrollment: MATH 141 or MATH 141B or MATH 141E or MATH 141G or MATH 141H

Bachelor of Arts: Quantification

Honors course in first- and second-order equations special functions Laplace transform solutions higher order equations Fourier series partial differential equations. MATH 251H Honors Ordinary and Partial Differential Equations (4) This course serves as an introduction to ordinary and partial differential equations. Topics include various techniques for solving first and second order ordinary differential equations, an introduction to numerical methods, solving systems of two ordinary differential equations, nonlinear differential equations and stability, Laplace transforms, Fourier series, and partial differential equations.The typical delivery format for the course is four 50-minute lectures per week, with typical assessment tools including examinations, quizzes, homework, and writing assignments.In contrast to the non-honors version of this course, the honors version is typically more theoretical and will often include more sophisticated problems. Moreover, certain topics are often discussed in more depth and are sometimes expanded to include applications which are not visited in the non-honors version of the course.

Enforced Prerequisite at Enrollment: MATH 141 or MATH 141B or MATH 141E or MATH 141G or MATH 141H

Bachelor of Arts: Quantification

Fourier series partial differential equations. Students who have passed MATH 251 may not schedule this course for credit. This course serves as the continuation of MATH 250 (Ordinary Differential Equations) and provides an elementary treatment of partial differential equations and Fourier series. Once a student completes both MATH 250 (3 credits) and MATH 252 (1 credit), the student will have completed all of the material in MATH 251 (4 credits). In particular, the student will be able to find solutions to given partial differential equations and will be able to utilize the tools from the field of Fourier series in the process.

Creative projects, including nonthesis research, which are supervised on an individual basis and which fall outside the scope of formal courses.

Bachelor of Arts: Quantification

Formal courses given infrequently to explore, in depth, a comparatively narrow subject which may be topical or of special interest.

Bachelor of Arts: Quantification

Fundamental techniques of enumeration and construction of combinatorial structures, permutations, recurrences, inclusion-exclusion, permanents, 0, 1- matrices, Latin squares, combinatorial designs.

Enforced Prerequisite at Enrollment: MATH 220 or MATH 220H

Bachelor of Arts: Quantification

Recitation for MATH 310H - Concepts in Combinatorics.

Enforced Prerequisite at Enrollment: MATH 220 or Concurrent: MATH 310H

Bachelor of Arts: Quantification

Honors version of elementary and enumerative combinatorics.

Prerequisite: MATH 220

Bachelor of Arts: Quantification

Basic methods of mathematical thinking and fundamental mathematical structures, primarily in the context of numbers, groups, and symmetries.

Bachelor of Arts: Quantification

Writing Across the Curriculum

Introduction to mathematical proofs elementary number theory and group theory. Students who have passed CMPSC 360 may not schedule this course for credit.

Enforced Prerequisite at Enrollment: MATH 141 or MATH 141B or MATH 141E or MATH 141G or MATH 141H

Bachelor of Arts: Quantification

Writing Across the Curriculum

An introduction to rigorous analytic proofs involving properties of real numbers, continuity, differentiation, integration, and infinite sequences and series.

Enforced Prerequisite at Enrollment: MATH 141 or MATH 141B or MATH 141E or MATH 141G or MATH 141H

Bachelor of Arts: Quantification

A recitation component to MATH 312H, practice in problem solving.

Enforced Prerequisite at Enrollment: MATH 140H and MATH 311M or Concurrent: MATH 312H

Bachelor of Arts: Quantification

Basic methods of mathematical thinking and fundamental structures, primarily in the context of infinite sets, real numbers, and metric spaces.

Enforced Prerequisite at Enrollment: MATH 141 or MATH 141B or MATH 141E or MATH 141G or MATH 141H

Bachelor of Arts: Quantification

A recitation component to Math 313H, practice in problem solving.

Enforced Prerequisite at Enrollment: MATH 140H and MATH 311M or Concurrent: MATH 313H

Development thorough understanding and technical mastery of foundations of modern geometry. MATH 313H Concepts of Geometry (3) The central aim of this course is to develop thorough understanding and technical mastery of foundations of modern geometry. Basic high school geometry is assumed axioms are mentioned, but not used to deduce theorems. Approach in development of the Euclidean geometry of the plane and the 3-dimensional space is mostly synthetic with an emphasis on groups of transformations. Linear algebra is invoked to clarify and generalize the results in dimension 2 and 3 to any dimension. It culminates in the last part of the course where six 2-dimensional geometries and their symmetry groups are discussed. This course is a a part of a new "pre-MASS" program (PMASS)aimed at freshman/sophomore level students, which will operate in steady state in the spring semesters. This course is directly linked with a proposed course Math 313R, its 1-credit recitation component. It is highly recommended to all mathematics, physics and natural sciences majors who are graduate school bound, and is a great opportunity for all Schreyer Scholars. The following topics will be covered: Euclidean geometry of the plane (distance, isometries, scalar product of vectors, examples of isometries: rotations, reflections, translations, orientation, symmetries of planar figures, review of basic notions of group theory, cyclic and dihedral groups, classification of isometries of Euclidean plane, discrete groups of isometries and crystallographic restrictions. similarity transformations, selected results from classical Euclidean geometry> Euclidean geometry of the 3-dimensional space and the sphere (distance, isometries, scalar product of vectors, planes and lines in the 3-dimensional space, normal vectors to planes, classification of pairs of lines, isometries with a fixed point: rotations and reflections, orientation, isometries of the sphere, classification of orientation-reversing isometries with a fixed point, finite groups of isometries of the 3-dimensional space, existence of a fixed point, examples: cyclic, dihedral, and groups of symmetries of Platonic solids, classification of isometries without fixed point: translations and screw-motions, intrinsic geometry of the sphere, elliptic plane: a first example of non-Euclidean geometry) Elements of linear algebra and its application to geometry in 2, 3, and n dimension (real and complex vector spaces. linear independence of vectors, basis and dimension, eigenvalues and eigenvectors, diagonalizable matrices, classification of matrices in dimension 2: elliptic, hyperbolic and parabolic matrices, orthogonal matrices and isometries of the n-dimensional space) Six 2-dimensional geometries (Projective geometry, affine geometry, inversions and conformal geometry, Euclidean geometry revisited, geometry of elliptic plane, hyperbolic geometry). The achievement of educational objectives will be assessed through weekly homework, class participation, and midterm and final exams.

Enforced Prerequisite at Enrollment: MATH 140H and MATH 311M or Concurrent: MATH 312H

Group work on challenging problems, discussions and project presentations. MATH 314 PMASS Problem Solving Seminar (1) A 1-credit Problem Solving Seminar will feature group work on challenging problems which require only elementary techniques for their solution. Each student of the PMASS program will be required to participate in two individual or group projects. Unlike those in MASS Program, the projects will not be necessarily closely related to the courses, although the course instructors will be encouraged to offer topics and supervise the work. Some projects will grow out of the work of the problem solving seminar, and the seminar will be a venue for the students to discuss their research projects. This course is a a part of a new "pre-MASS" program (PMASS)aimed at freshman/sophomore level students, which will operate in steady state in the spring semesters. This course is linked with other PMASS courses, and is highly recommended to all mathematics, physics and natural sciences majors who are graduate school bound, and is a great opportunity for all Schreyer Scholars. Each student of the PMASS program will be required to participate in two individual or group projects.The achievement of educational objectives will be assessed through evaluations of the project presentations.

A consideration of selected topics in the foundations of mathematics, with emphasis on development of basic meaning and concepts.

Enforced Prerequisite at Enrollment: MATH 141

Bachelor of Arts: Quantification

Bi-weekly lecture series with multiple invite speakers. MATH 315 PMASS Colloquium (1) This bi-weekly lecture series will feature multiple invited speakers. Unlike MASS colloquia that focus on specific topics, those lectures will be broad in scope and not very technical.We envision that advanced high school students from State College Area High School will attend these lectures that will be properly advertised. This will help to attract talented high school students to undergraduate study of mathematics and related subjects, and will also enhance our existing collaboration with mathematics educators in the area. This course is a a part of a new "pre-MASS" program (PMASS)aimed at freshman/sophomore level students, which will operate in steady state in the spring semesters. This course is highly recommended to all mathematics, physics and natural sciences majors who are graduate school bound, and is a great opportunity for all Schreyer Scholars.

Enforced Prerequisite at Enrollment: MATH 140H and MATH 311M or Concurrent: MATH 312H and MATH 313H and MATH 314H

Combinatorial analysis, axioms of probability, conditional probability and independence, discrete and continuous random variables, expectation, limit theorems, additional topics. Students who have passed either MATH(STAT) 414 or 418 may not schedule this course for credit.

Enforced Prerequisite at Enrollment: MATH 141

Bachelor of Arts: Quantification

Statistical inference: principles and methods, estimation and testing hypotheses, regression and correlation analysis, analysis of variance, computer analysis. Students who have passed STAT (MATH) 415 may not schedule this course for credit.

Enforced Prerequisite at Enrollment: MATH 318 or STAT 318 or MATH 414 or STAT 414 or STAT 418 or MATH 418

## Advanced Engineering Mathematics 6th Edition - Solutions by Chapter

Advanced Engineering Mathematics | 6th Edition - Solutions by Chapter

• Chapter 1: Introduction to Differential Equations
• Chapter 1.1: Definitions and Terminology
• Chapter 1.2: Initial-Value Problems
• Chapter 1.3: Differential Equations as Mathematical Models
• Chapter 10: Systems of Linear First-Order Differential Equations
• Chapter 10.1: Theory of Linear Systems
• Chapter 10.2: Homogeneous Linear Systems
• Chapter 10.3: Solution by Diagonalization
• Chapter 10.4: Nonhomogeneous Linear Systems
• Chapter 10.5: Matrix Exponential
• Chapter 11: Systems of Nonlinear Differential Equations
• Chapter 11.1: Autonomous Systems
• Chapter 11.2: Stability of Linear Systems
• Chapter 11.3: Linearization and Local Stability
• Chapter 11.4: Autonomous Systems as Mathematical Models
• Chapter 11.5: Periodic Solutions, Limit Cycles, and Global Stability
• Chapter 12: Orthogonal Functions and Fourier Series
• Chapter 12.1: Orthogonal Functions
• Chapter 12.2: Fourier Series
• Chapter 12.3: Fourier Cosine and Sine Series
• Chapter 12.4: Complex Fourier Series
• Chapter 12.5: SturmLiouville Problem
• Chapter 12.6: Bessel and Legendre Series
• Chapter 13: Boundary-Value Problems in Rectangular Coordinates
• Chapter 13.1: Separable Partial Differential Equations
• Chapter 13.2: Classical PDEs and Boundary-Value Problems
• Chapter 13.3: Heat Equation
• Chapter 13.4: Wave Equation
• Chapter 13.5: Laplaces Equation
• Chapter 13.6: Nonhomogeneous Boundary-Value Problems
• Chapter 13.7: Orthogonal Series Expansions
• Chapter 13.8: Fourier Series in Two Variables
• Chapter 14: Boundary-Value Problems in Other Coordinate Systems
• Chapter 14.1: Polar Coordinates
• Chapter 14.2: Cylindrical Coordinates
• Chapter 14.3: Spherical Coordinates
• Chapter 15: Integral Transform Method
• Chapter 15.1: Error Function
• Chapter 15.2: Applications of the Laplace Transform
• Chapter 15.3: Fourier Integral
• Chapter 15.4: Fourier Transforms
• Chapter 15.5: Fast Fourier Transform
• Chapter 16: Numerical Solutions of Partial Differential Equations
• Chapter 16.1: Laplaces Equation
• Chapter 16.2: Heat Equation
• Chapter 16.3: Wave Equation
• Chapter 17: Functions of a Complex Variable
• Chapter 17.1: Complex Numbers
• Chapter 17.2: Powers and Roots
• Chapter 17.3: Sets in the Complex Plane
• Chapter 17.4: Functions of a Complex Variable
• Chapter 17.5: CauchyRiemann Equations
• Chapter 17.6: Exponential and Logarithmic Functions
• Chapter 17.7: Trigonometric and Hyperbolic Functions
• Chapter 17.8: Inverse Trigonometric and Hyperbolic Functions
• Chapter 18: Integration in the Complex Plane
• Chapter 18.1: Contour Integrals
• Chapter 18.2: CauchyGoursat Theorem
• Chapter 18.3: Independence of the Path
• Chapter 18.4: Cauchys Integral Formulas
• Chapter 19: Series and Residues
• Chapter 19.1: Sequences and Series
• Chapter 19.2: Taylor Series
• Chapter 19.3: Laurent Series
• Chapter 19.4: Zeros and Poles
• Chapter 19.5: Residues and Residue Theorem
• Chapter 19.6: Evaluation of Real Integrals
• Chapter 2: First-Order Differential Equations
• Chapter 2.1: Solution Curves Without a Solution
• Chapter 2.2: Separable Equations
• Chapter 2.3: Linear Equations
• Chapter 2.4: Exact Equations
• Chapter 2.5: Solutions by Substitutions
• Chapter 2.6: A Numerical Method
• Chapter 2.7: Linear Models
• Chapter 2.8: Nonlinear Models
• Chapter 2.9: Modeling with Systems of First-Order DEs
• Chapter 20: Conformal Mappings
• Chapter 20.1: Complex Functions as Mappings
• Chapter 20.2: Conformal Mappings
• Chapter 20.3: Linear Fractional Transformations
• Chapter 20.4: SchwarzChristoffel Transformations
• Chapter 20.5: Poisson Integral Formulas
• Chapter 20.6: Applications
• Chapter 3: Higher-Order Differential Equations
• Chapter 3.1: Theory of Linear Equations
• Chapter 3.10: Greens Functions
• Chapter 3.11: Nonlinear Models
• Chapter 3.12: Solving Systems of Linear Equations
• Chapter 3.2: Reduction of Order
• Chapter 3.3: Homogeneous Linear Equations with Constant Coefficients
• Chapter 3.4: Undetermined Coefficients
• Chapter 3.5: Variation of Parameters
• Chapter 3.6: CauchyEuler Equations
• Chapter 3.7: Nonlinear Equations
• Chapter 3.8: Linear Models: Initial-Value Problems
• Chapter 3.9: Linear Models: Boundary-Value Problems
• Chapter 4: The Laplace Transform
• Chapter 4.1: Definition of the Laplace Transform
• Chapter 4.2: The Inverse Transform and Transforms of Derivatives
• Chapter 4.3: Translation Theorems
• Chapter 4.4: Additional Operational Properties
• Chapter 4.5: The Dirac Delta Function
• Chapter 4.6: Systems of Linear Differential Equations
• Chapter 5: Series Solutions of Linear Differential Equations
• Chapter 5.1: Solutions about Ordinary Points
• Chapter 5.2: Solutions about Singular Points
• Chapter 5.3: Special Functions
• Chapter 6: Numerical Solutions of Ordinary Differential Equations
• Chapter 6.1: Euler Methods and Error Analysis
• Chapter 6.2: RungeKutta Methods
• Chapter 6.3: Multistep Methods
• Chapter 6.4: Higher-Order Equations and Systems
• Chapter 6.5: Second-Order Boundary-Value Problems
• Chapter 7: Vectors
• Chapter 7.1: Vectors in 2-Space
• Chapter 7.2: Vectors in 3-Space
• Chapter 7.3: Dot Product
• Chapter 7.4: Cross Product
• Chapter 7.5: Lines and Planes in 3-Space
• Chapter 7.6: Vector Spaces
• Chapter 7.7: GramSchmidt Orthogonalization Process
• Chapter 8: Matrices
• Chapter 8.1: Matrix Algebra
• Chapter 8.10: Orthogonal Matrices
• Chapter 8.11: Approximation of Eigenvalues
• Chapter 8.12: Diagonalization
• Chapter 8.13: LU-Factorization
• Chapter 8.14: Cryptography
• Chapter 8.15: An Error-Correcting Code
• Chapter 8.16: Method of Least Squares
• Chapter 8.17: Discrete Compartmental Models
• Chapter 8.3: Rank of a Matrix
• Chapter 8.4: Determinants
• Chapter 8.5: Properties of Determinants
• Chapter 8.6: Inverse of a Matrix
• Chapter 8.7: Cramers Rule
• Chapter 8.8: The Eigenvalue Problem
• Chapter 8.9: Powers of Matrices
• Chapter 9: Vector Calculus
• Chapter 9.1: Vector Functions
• Chapter 9.10: Double Integrals
• Chapter 9.11: Double Integrals in Polar Coordinates
• Chapter 9.12: Greens Theorem
• Chapter 9.13: Surface Integrals
• Chapter 9.14: Stokes Theorem
• Chapter 9.15: Triple Integrals
• Chapter 9.16: Divergence Theorem
• Chapter 9.17: Change of Variables in Multiple Integrals
• Chapter 9.2: Motion on a Curve
• Chapter 9.3: Curvature and Components of Acceleration
• Chapter 9.4: Partial Derivatives
• Chapter 9.5: Directional Derivative
• Chapter 9.6: Tangent Planes and Normal Lines
• Chapter 9.7: Curl and Divergence
• Chapter 9.8: Line Integrals
• Chapter 9.9: Independence of the Path
• Chapter Appendix I: Derivative and Integral Formulas
• Chapter Appendix II: Gamma Function
• Chapter Appendix III: Table of Laplace Transforms
##### ISBN: 9781284105902

Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This expansive textbook survival guide covers the following chapters: 160. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Since problems from 160 chapters in Advanced Engineering Mathematics have been answered, more than 69369 students have viewed full step-by-step answer. The full step-by-step solution to problem in Advanced Engineering Mathematics were answered by , our top Math solution expert on 03/08/18, 07:27PM.

S. Permutation with S21 = 1, S32 = 1, . , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1 eigenvectors are columns of the Fourier matrix F.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

Invert A by row operations on [A I] to reach [I A-I].

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

Nullspace of AT = "left nullspace" of A because y T A = OT.

Ln = 2,J, 3, 4, . satisfy Ln = L n- l +Ln- 2 = A1 +A

, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' . , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

There are n! orders of 1, . , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

Orthogonal Q times positive (semi)definite H.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

= number of pivots = dimension of column space = dimension of row space.

One free variable is Si = 1, other free variables = o.

Columns of n by n identity matrix (written i ,j ,k in R3).

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

## Engineering Mathematics Questions and Answers – Leibniz Rule – 1

This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Leibniz Rule – 1”.

1. Let f(x) = sin(x)/1+x 2 . Let y (n) denote the n th derivative of f(x) at x = 0 then the value of y (100) + 9900y (98) is
a) 0
b) -1
c) 100
d) 1729

2. Let f(x) = ln(x)/x+1 and let y (n) denote the n th derivative of f(x) at x = 1 then the value of 2y (100) + 100y (99)
a) (99)!
b) (-99)!
c) (100)!
d) (-98)!

3. Let f(x) = (sqrt<1-x^2>) and let y (n) denote the n th derivative of f(x) at x = 0 then the value of 6y (1) y (2) + 2y (3) is
a) -998
b) 0
c) 998
d) -1

4. Let f(x) = tan(x) and let y (n) denote the n th derivative of f(x) then the value of y (9998879879789776) is
a) 908090988
b) 0
c) 989
d) 1729

5. If the first and second derivatives at x = 0 of the function f(x)=(frac) were 2 and 3 then the value of the third derivative is
a) -3
b) 3
c) 2
d) 1

6. For the given function f(x)=(sqrt) the values of first and second derivative at x = 1 are assumed as 0 and 1 respectively. Then the value of the third derivative could be
d) Indeterminate

7. Let f(x)=(frac) and let the n th derivative at x = 0 be given by y (n) Then the value of the expression for y (n) is given by
a) (frac<4>)
b) (sum_^frac<(-1)^i c_<2i>^n><2i+1>)
c) &pin
d) (frac<2n>)

8. Let f(x) = e x sinh(x) / x, let y (n) denote the n th derivative of f(x) at x = 0 then the expression for y (n) is given by
a) (sum_^n frac^n><2i+1>)
b) (sum_^n frac<1><2i+1>)
c) 1
d) Has no general form

Sanfoundry Global Education & Learning Series – Engineering Mathematics.

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## 7.6: Taylor's Theorem Revisited - Mathematics

The proof is completely analogous to that of the Fermat's Theorem except that instead of the set of non-negative remainders <1,2. m-1>we now consider the set of the remainders of division by m coprime to m. In exactly the same manner as before, multiplication by a results in a permutation (but now) of the set Therefore, two products are congruent:

dividing by the left-hand side proves the theorem.

The derivation of the Euler's formula for &phi(m) proceeds in two steps. First, we consider the next simplest case &phi(p a ), where p is prime.

Next, we establish the multiplicative property of &phi:

Since any integer can be (uniquely) represented in the form

with distinct pi's, these two steps combined will lead to a closed form expression for &phi.

### Lemma 1

&phi(p a ) = p a - p a-1 = p a-1 (p - 1) = p a (1 - 1/p).

Indeed, below p a , there are integers divisible by p and hence having common factors with p a . These are . The total of positive integers below p a and coprime to it is then

To prove the second part, assume m = m1m2, where the two factor are coprime. We wish to relate the number &phi(m) of the remainders of division by m coprime to m to the similarly defined quantities &phi(m1) and &phi(m2). Let be coprime to m. Find remainders n1 and n2 of division of n by m1 and m2, respectively:

Obviously, n1 is coprime to m1 and n2 is coprime to m2. Furthermore, n defines n1 and n2 uniquely. Assume now that n1 and n2 (coprime to m1 and m2, respectively) are given. Then the Chinese Remainder Theorem yields a unique n such that (3) holds. This proves (1).

All that remains is to finally derive the Euler's formula for &phi(m). Let m be given by (2). Applying the multiplicative property repeatedly we get