MATH 132: CALCULUS II SYLLABUS


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1 MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early Transcendentals, vol., Customized for University of Massachusetts Amherst, by James Stewart, Brooks/ColeThompson Learning, General Course Description: Math 32 continues the study of singlevariable calculus. It deals with definite and indefinite integrals; infinite sequences and series; and plane curves whose x and y coordinates are functions of another variable such as time. The central concepts are: accumulated (net) change, as realized by the notion of definite integral; successively better approximations of functions by polynomials, as represented by the concept of power series. The emphasis is on on problemsolving and understanding concepts rather than on proving theorems. Learning goals for Calculus II (Math 32): () Continue to become a competent user of calculus. (2) Continue to develop problemsolving skills, especially in formulating verbal descriptions as mathematical problems and in constructing long, multistep solution. (3) Continue to develop ability to write wellorganized, coherent solutions to problems. (4) Become adept at computing indefinite integrals symbolically through use of basic methods. (5) Be able to formulate as a definite integral a problem about net change in a varying quantity. (6) Become familiar with the process of successive approximations to a quantity or a function. Required topics: A. The definite integral (Note: Items & 2 are now covered in Math 3, but are included here for completeness. This material should be quickly reviewed.). Areas and distances: approximation by sums, leading to The definite integral as a limit of Riemann sums. Leftendpoint, rightendpoint, and midpoint Riemann sums. Definition of definite integral as limit of Riemann sums. Applying the definition for a linear function. Calculating integrals of special functions by using geometry, e.g., b (m x + k) dx and a r r2 x r 2 dx. Linearity, endpointadditivity, and comparison properties of definite integrals. 3. The Fundamental Theorem of Calculus (FTC). Area and other functions of the form F (x) = x f(t) dt. a Statement of the FTC. At least an intuitive justification, or a plausibility argument, for the FTC. Using FTC to evaluate definite integrals. 4. Indefinite integrals. The f(x) dx notation. Indefinite integrals corresponding to derivatives of powers (including nonintegral powers) and of basic elementary transcendental functions. 5. The Net Change Theorem: statement and uses. 6. Integration by substitution (in both indefinite and definite integrals). B. Applications of integration. Applications to geometry. Area between curves. Volumes by slicing perpendicular to a line and, as a special case, volume of solid of revolution. (Not volume by shells.)
2 2 2. (If time permits.) One or two nongeometric applications, to be chosen by the course chair or individual instructors from topics such as those listed below. No more than one week should be devoted to this; this could even be done as individual or group projects. Owing to the conceptual physical understanding required, the fluid pressure and center of mass are not suggested here. Work (physics). Average value of a function (with concrete scientific instances). Consumer surplus (economics). Blood flow or cardiac output (biological science). Probability density functions (sciences and engineering). C. Methods of integration. Techniques of symbolic integration. Integration by parts, including repeated integration by parts and examples leading to equations of the form f(x) dx = g(x) + c f(x) dx. (If time permits. Trigonometric integrals and their application in trigonometric substitutions. Integration by partial fractions of a rational function whose denominator factors into two distinct linear factors. Use of recursion formulas for integrals. Note: It is suggested that calculator and/or computer technology capable of doing symbolic integration be demonstrated in order to show how integration is often done in practice and to indicate that named, nonelementary functions arise as antiderivatives. 2. Approximate integration. Midpoint, Trapezoidal, and Simpson s Rules. Qualitative comparison of the methods accuracy (but not bounds on the error). 3. Improper integrals: infinite endpoints and discontinuous integrands. D. Series and power series Note: This topic is placed here, before parametric equations, to ensure that the essential topics of power series expansions and approximation by Taylor polynomials are reached.. Sequences and limits of sequences: meaning of sequential limit; algebraic Limit Laws; the Squeeze Theorem and the Monotonic Convergence Theorem (statements and use). 2. Series Notions of convergence and sum of an infinite series. Geometric series and application to rational values of repeating decimals. The nth Term Test for divergence. 3. Testing series of constants for convergence Note: In this course, the important thing is power series and approximation by Taylor polynomials and not series of constants. So testing for convergence should introduce the notion of bounding the error in approximating the sum of a series by a partial sum and should emphasize those methods that are: (i) needed to establish convergence or divergence of standard examples such as harmonic, alternating harmonic, and pseries; (ii) most relevant to finding the radius of convergence of power series; and/or (iii) needed in order to justify the Ratio Test. The emphasis should be on examples that are simple rather than artificial or technically complicated. The Integral Test and bounds on the error. (Note: It is not clear that the Integral Test in its full generality is actually appropriate. It could suffice to use the argument behind that test to test pseries.) The Comparison Test (but not the Limit Comparison Test). The Alternating Series Test and bounding the error of the nth partial sum. Absolute convergence implies convergence (but omit terminology of absolute convergence and conditional convergence). The Ratio Test (but not the Root Test). 4. Power series. 5. Representation of functions as power series.
3 Examples derived from geometric series. Termbyterm differentiation and integration. 6. Taylor series (and Maclaurin series). Using the definition to find Taylor series. Uniqueness of Taylor series expansion: A power series expansion of a function is its Taylor series. Standard examples: Maclaurin series for e x, sin x, cos x, arctan x [and perhaps also for ln( + x)]. Approximating functions by Taylor polynomials; Taylor s Inequality for error bounds. Approximating values of functions using Taylor polynomials, with error bounds in some cases. E. Parametric equations and polar equations.. Curves defined by parametric equations. Graphs of parametric equations. Elimination of the parameter. Arc length of parametric curves. (But not tangents to parametric curves or area and surface area calculations involving parametric curves.) 2. Using polar coordinates. Polar equations of graphs. Conversion between polar and rectangular equations. Arc length in polar coordinates. (But not area in terms of polar coordinates.) 3
4 4 Representative problems to solve: These problems are intended strictly to suggest the level and coverage of the course; they are not meant as a template for exam questions. () Evaluate without technology or a Table of Integrals: (a) (cos x + 5 sin x) dx (f) ( ) x 2 + x dx (g) 2 (c) x + 4 dx (h) (d) x e 2 x dx (i) sin x (e) + cos 2 x dx (j) x + 3 x 2 3 x + 2 dx x 2 sin x dx 2 x 4 x 2 dx tan 3 x sec 5 x dx e 2 x e x 2 dx (2) (a) Approximate 0 e x2 dx by the Riemann sum with n = 4 subintervals and left endpoints as sample points. Is this Riemann sum an overestimate or an underestimate of the exact value? Approximate the same integral by using Simpson s Rule with n = 4 subintervals. (3) Find the area of the bounded region enclosed by the curves y = x 2 9 and y = 9 x 2. (4) Find the volume of the solid obtained by rotating around the xaxis the region bounded by the curve y = tan x and the lines y = 0 and x = π/4. (5) Determine the following derivatives. ( (a) sin ( 3 e t + t ) ) 32 d t d d t ( π d d t sin ( 3 e x + x ) ) 32 d x t 3 (6) The rate r at which people become ill with the flu at time t days after an epidemic begins is given by r = 000 t e t/20 people per day. How many people become ill with the flu during the first 0 days of this epidemic? (7) Does the improper integral converge or diverge, and why? If it converges, find its value. (a) 2 0 x 4 x 2 dx 0 arctan x x 2 + dx (8) Does the series converge? Why or why not? (a) n 3 n + n 4 (c) 4 n (d) (e) n = 2 ( ) n n 2 n (ln n) 2 (9) The sequence {S n } n= of partial sums of the series n= a n is given by S n = n/(5 + n) (for, 2, 3,... ). (a) Does the series n= a n converge and, if so, to what sum? Use the formula for S n to find an explicit formula for a n in terms of n. (0) Find the largest interval on which the power series n = 0 (x ) n (n + ) 3 n converges. () Find a power series representation for f(x) = 4/( + 2 x) around a = 0; write the power series using sigma ( ) notation. State for which x the power series actually has sum f(x).
5 5 (2) Calculate the degree 3 Taylor polynomial T 3 (x) of g(x) = / x around a =. (3) (a) Starting with the Maclaurin series for e x (which you may just write down), obtain a power series representation of e x2. Use your answer to (a) to express 0 e x2 dx as the sum of an infinite series. (c) Find a bound on the error if the first three (nonzero) terms of the series you obtained in were used to approximate 0 e x2 dx. (d) How many (nonzero) terms of the series obtained in would you need to use in order to approximate 0 e x2 dx with an error at most 0 5? (Do not actually obtain that approximation!) (4) Find all points at which the curve with parametric equations x = t 2 +, y = t 3 t has a horizontal tangent. (5) As the parameter t increases forever, starting at t = 0, the curve with parametric equations { x = e t cos t, y = e t sin t spirals inward toward the origin, getting ever closer to the origin (but never actually reaching it) as t. Find the length of this entire spiral curve. (6) (a) Find a Cartesiancoordinate equation for the curve having polar equation r = 2 cos θ. Sketch together the curves with polar equations r = 2 cos θ and r =. (c) Find some polar coordinates of each point where the curves intersect.
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