Math 2300 – Calculus II – University Of Colorado

Math 2300 – Calculus II – University of ColoradoSpring 2011 – Final exam review problems: ANSWER KEY21. Find fx (1, 0) for f (x, y) xesin(x y).(x2 y 2 )3/222. Consider the solid region W situated above the region 0 x 2, 0 y x, and bounded above by2the surface z ex .An equivalent description for the region 0 x 2, 0 y x is: 0 y 2, y x 2.(a) Write an integral that evaluates the area of each cross-section of W with vertical planes y a,where a is a constant in [0, 2]. Can you evaluate this integral (as an expression depending on a)?In each cross-sectionalplane y Za, the interval for x is given by a x 2, so the area of theZ 222cross-sections isf (x, a) dx ex dx, which can’t be evaluated directly.aa(b) Write an integral that evaluates the area of each cross-section of W with vertical planes x b,where b is a constant in [0, 2]. Can you evaluate this integral (as an expression depending on b)?In each cross-sectional plane x b, the interval for y is given by 0 y b, so the area of theZ bZ b22cross-sections isf (b, y) dy eb dy beb .00(c) Write a convenient double integral that evaluates the volume of the region W, and evaluate it.Z ZzdA RZ20Zx2ex dy dx 0Z20Z22ex dx dyyThe previous remarks indicate that the first option is the only convenient one, so the volume ofthe solid region can be calculated as:Z02Zxx2edy dx 0Z20x2yey xy 0(d) Letf (x, y) p(a) Find fx (x, y). x/ x2 y 2 .dx Z22x2xe0exdx 22 01e42px2 y 2 .(b) Let g(x) fx (x, 0).pDoes limx!0 g(x) exist? If so, evaluate it. If not, explain why not. Hint:Note that f (x, 0) x2 x . No.xxlim g(x) lim p lim.2x!0x!0 x xx!0The limit does not exist because3. Evaluate:8 x x (x x) x : 1x11 if x 0,if x 0.

(a)Z218714(b)y2(x2 y) dx dy.yZ1014e3 9e18ZZeey1 exdx dy.ln x164. (a) Set up a double integral that corresponds to the mass of the semidiskpD {(x, y) : 0 y 4 x2 },if the density at each of its points is given by the function (x, y) x y.264(b) Find the mass of the above semidisk.15X5. The power seriesCn xn diverges at x 7 and converges at x Z3. At x 22Zp4 x2x2 y dy dx09, the series is(a) Conditionally convergent(b) Absolutely convergent(c) Divergent(d) Cannot be determined.6. The power seriesisXCn (x5)n converges at x 5 and diverges at x 10. At x 11, the series(a) Conditionally convergent(b) Absolutely convergent(c) Divergent(d) Cannot be determined.7. The power seriesXCn xn diverges at x 7 and converges at x 3. At x 4, the series is(a) Conditionally convergent(b) Absolutely convergent(c) Divergent(d) Cannot be determined.8. In order to determine if the series1Xk 1pk 1k2 1converges or diverges, the limit comparison test can be used. Decide which series provides the bestcomparison.2

1X1(a)k(b)(c)k 11Xk 11Xk 11k3/21k29. Decide whether the following statements are true or false. Give a brief justification for your answer.Z 1Z 1(a) If f is continuous for all x andf (x)dx converges, then so doesf (x)dx for all positive a.0TRUEa(b) if f (x) is continuous and positive for x 0 and if lim f (x) 0, thenx!1Z1f (x)dx converges.0FALSEZ 1Z 1R1(c) Iff (x)dx and 0 g(x)dx both converge then(f (x) g(x))dx converges. TRUE00Z 1Z 1Z 1(d) Iff (x)dx andg(x)dx both diverge then(f (x) g(x))dx diverges. FALSE00010. For this problem, state which of the integration techniques you would use to evaluate the integral, butDO NOT evaluate the integrals. If your answer is substitution, also list u and du; if your answeris integration by parts, also list u, dv, du and v; if your answer is partial fractions, set up thepartial fraction decomposition, but do not solve for the numerators; if your answer is trigonometricsubstitution, write which substitution you would use.Z(a)tan xdx Use substitution w cos x, so that dw sin xZdx(b)Factor out the denominator as x2 9 (x 3)(x 3), and use partial fractions tox2 91ABrewrite the integrand as 2 .x9x 3 x 3Z(c)ex cos xdx Start by using an integration by parts, with u ex and dv cos xdx, so thatdu ex dx and v sin x. (Then use another integration by parts with U ex and dV sin xdx,and solve the resulting equation for the integral.)Z p9 x2(d)dx Start by using a trig substitution x 3 sin t, so that dx 3 cos tdt.x2Zsin(ln x)1(e)dx Substitute w ln x, so that dw dx.xxZ1(f)x3/2 ln xdx Use integration by parts, with u ln x and dv x3/2 dx, so that du andx2 5/2v 5x .Z1p(g)dx Trig substitution x 2 tan t, so that dx 2 sec2 t.2x 4Z xe 1(h)dx The numerator is the derivative of the denominator. Use substitution w ex x, soex xthat dw (ex 1)dx.3

11. If1Xan (x1)n (x1)(x1)22n 1 (x1)3(x41)48 (x1)516··· ,then the correct formula for an is: Oops, in the original version of this revew sheet, all of the possibleanswers below had a factor of (x 1)n . Please ignore those.( 1)n2n( 1)n 1(b) an 2n( 1)n(c) an 2n( 1)n 1 2(d) an 2n(a) an 12. True or False? If lim an 0 thenn!11Xan converges.n 0(a) True(b) False13. True or False? If lim an 6 0 thenn!1(a) True1Xan diverges.n 0(b) False14. True/False: IfR 1. TrueXan is convergent, then the power seriesPan xn has convergence radius at least1X( 1)n x2n 115. If one uses the Taylor polynomial P3 (x) of degree n 3 to approximate sin x at(2n 1)!0x 0.1, would one get an overestimate or an underestimate? underestimate16. Suppose that x is positive but very small. Arrange the following in increasing order:x, sin x, ln(1 x), 11pcos x x 1cos(x), exp1, x 1x ln(1 x) sin x ex2217. (a) Find the Taylor series about 0 for f (x) x2 ex . x2 ex (b) Is this function even or odd? Justify your answer. even6!(c) Find f (3) (0) and f (6) (0). f (3) (0) 0, f (6) (0) 2!x.11Xx2n 2n!n 018. Suppose that ibuprofen is taken in 200mg doses every six hours, and that all 200mg are delivered tothe patient’s body immediately when the pill is taken. After six hours, 12.5% of the ibuprofen remains.Find expressions for the amount of ibuprofen in the patient immediately before and after the nth pilltaken. Include work; without work, you may receive no credit.Just before the nth dose:4

200(0.125 0.1252 0.1253 · · · 0.125n1) 200nX1k 00.125k 200 ·1 0.125n1 0.125Just after the nth dose:200 200 ·1 0.125n1 0.12519. Find an equation for a sphere if one of its diameters has endpoints (2, 1, 4) and (4, 3, 10). (x(y 2)2 (z 7)2 113)2 20. A cube is located such that its top four corners have the coordinates ( 1, 2, 2), ( 1, 3, 2), (4, 2, 2),and (4, 3, 2). Give the coordinate of the center of the cube. (1.5, 0.5, 0.5)21. Find the equation of the largest sphere with center (5, 4, 9) contained in the first octant. (x(y 4)2 (z 9)2 165)2 22. Evaluate the double integral (using the most convenient method):(a)Z10(b)Z01Zeey1Zxxe2 1dx dyln x2y2ln 2dy dx1 y4423. The following sum of double integrals describes the mass of a thin plate R in the xy-plane, of density(x, y) x y:mass Z04Z2x 8(x, y) dy dx 0Z04Z2x 8(x, y) dy dx0(a) Describe the thin plate (shape, intersections with the coordinate axes). It’s an isosceles trianglewith base along the x axis from 4 to 4, and apex on the y axis at (0, 4).(b) Write an expression for the mass of the plate as only one double integral.Z04Z(8 y)/2(x y) dx dy(y 8)/2(c) Calculate the area and the mass of the plate. 256/324. If one uses the Taylor polynomial P5 (x) of degree n 5 to approximate e xwould one get an overestimate or an underestimate? underestimate1Xxn0n!at x 0.2,25. A slope field for the di erential equation y 0 y e x is shown. Sketch the graphs of the solutions thatsatisfy the given initial conditions. Make sure to label each sketched graph.(a) y(0) 0 RED(b) y(0) 1 GREEN5(c) y(0) 1 BLU E

210!1!2!2!10612

26. For each di erential equation, find the corresponding slope field. (Not all slope fields will be used.)EquationACSlope Fielddy xdxBy 0 xyFy 0 sin(2x)Ddy 2dxBDyAEF227. Let f (x, y) exy xy 2.2y xy 222(a) Find fx (x, y). ex(2xy y 2 )(b) Find fy (x, y). ex y xy (x2 2xy)22@(c) Findfy (x, y). ex y xy (2x3 y 5x2 y 2 2xy 3 2x 2y)@x@22(d) Findfx (x, y). ex y xy (2x3 y 5x2 y 2 2xy 3 2x 2y)What’s curious about the relation of@ythis answer to that of part (c) above? They’re equal.28. Find the equation of the sphere that passes through the origin and whose center is (1, 2, 3).(x1)2 (y2)2 (z73)2 14.

29. You want to estimate sin x using the first 3 nonzero terms in the Taylor series. What formula for theerror bound would you use to get the best estimate for the error? E6 (x) x 7.7!p30. Find the arc length of the curve y x3/2 from (1, 1) to (2, 2 2).827" 134 3/2 134 3/2 #31. Find the arc length of the curve y (x6 8)/(16x2 ) from x 2 to x 3.32. Let f (x, y) x559514410x3 y 2 5xy 4 . Show that@@fx (x, y) fy (x, y) 0.@x@y@fx (x, y) 20x3@x60xy 2 ;@fy (x, y) 60xy 2@y33. Additional problems from your text:(a) Section 8.1 (Areas and Volumes) Exercises 3, 11, 25820x3 .

(b) Section 8.2 (Volumes by Revolution & Cross Sections) Exercises 9, 23, 359

(c) Section 8.3 (Arc Length and Parametric Curves) Exercises 11, 15, 17, 1910

(d) Section 11.4 (Separation of Variables) Exercises 3, 9, 13, 21, 4111

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Math 2300 – Calculus II – University of Colorado Spring 2011 – Final exam review problems: ANSWER KEY 1. Find f x(1,0) for f(x,y) xesin(x2y) (x2 y2)3/2. 2 2. Consider the solid region W situated above the region 0 x 2,