APPM 1360 Final Exam Fall 2013

Transcription

1 APPM 6 Final Exam Fall On the front of our bluebook, please write: a grading ke, our name, student ID, and section and instructor. This exam is worth 5 points and has 9 questions. Show all work! Answers with no justification will receive no points. Please begin each problem on a new page. No notes, calculators, or electronic devices are permitted.. points) Evaluate the following: a) x )x + ) b) Solve for in the differential equation: d x sin x with initial value π). a) Use partial fractions. x )x + ) A x + Bx + C x + ) Ax + ) + Bx + C)x ) A + B)x + B + C)x + A C) x x + ) x + x A + B, B + C, A C A, B, C x x + }{{} ux + ) x + ln x ln x + arctan x ) + C b) This is a separable equation. d x sin x d }{{} x sin }{{ x } u dv

2 Use integration b parts with u x, du, and dv sin x, v cos x. Plug in the initial value π, ). ln x cos x + x cos x+sin x+c e e π+c C π x cos x+sin x π e cos x x cos x + sin x + C. 4 points) Consider the region shown at right. Set up, but do not evaluate, the integrals to calculate: a) The volume generated b rotating this region about the line x. b) Consider the upper curve x /, defined on the interval shown on the graph. Find the surface area of the solid generated b rotating the curve about the -axis. The lower curve is not relevant for this part.) x 4 a) The curves intersect at, ) and, ). B the Shell Method, V πrh )) x x 4 π x) 4 x b) S πr + d/) πx + x. 4 points) Do the following converge or diverge? Explain our work and name an test or theorem ou use. a) + e x b) { ln m) m } m5 c) n n + n a) Since < + e x < e x and also is convergent. is convergent, b the Comparison Theorem ex + e x

3 b) Alternate solution: + e x t + e x Let u + e x, du e x. +et [ ln du uu ) u u The integral is convergent. n ] +e t a ln m) m m m m The sequence converges to. n + n c) Since > n n and n n + also is divergent. n +e t ln ) [ ] +e t du ln u ln u u u e t + e t ln ) ln + ln ln L H ln m L H m m m m n is a divergent p-series, b the Comparison Test Alternate solution: Use the Limit Comparison Test with b n / n. a n n b n n + n n n n + n n n n + 4 n + Since is a divergent p-series, also is divergent. n n n n 4. 6 points) Find the values of x for which the following series converge absolutel, converge conditionall, or diverge: a) x + ) n n n b) n! xn n a) Use the Ratio Test. a n+ n a n x + ) n+ n n + ) n x + ) n x + < The series is absolutel convergent on 4 < x <. Now check the endpoints. When x, the series n converges b the Limit Comparison Test with the convergent n

4 p-series n. When x 4, the series ) n n converges b the Alternating Series Test n since n + ) < n and n n. Therefore the series n x + ) n n is absolutel convergent for 4 x, is not conditionall convergent for an x, and is divergent for x < 4, x >. b) Use the Ratio Test. a n+ n a n n+ x n+ n n + )! n! n x n x n n + Since the ratio is, the series is absolutel convergent for all x, and not conditionall convergent or divergent for an x points) Let a, a + /4 + /, a + /4 + /4 + / + /, a 4 + /4 + /4 + /4 + / + / + /,.... a) Does the sequence {a n } converge? If es, find its it. Explain. b) Does a n converge? Explain our work and state an test or theorem ou use. a) a n is the quotient of geometric series. The numerator has ratio /4 and the denominator has ratio /. The sum of an infinite geometric series is S a/ r). a + 4 n n n ) 4 n n n + n ) / 4 ) n n b) B the Test for Divergence, since n a n, the series diverges. 6. points) / ) a) Find a Maclaurin series for x sinx ). b) Use our answer from part a) to find a series for x sinx ). c) Use our answer from part b) to find the 8th order Talor polnomial T 8 x) for the integral. d) Use T 8 x) to approximate the value of x sinx ). e) Is our approximation an overestimate or underestimate? Explain our reasoning.

5 a) sin x b) n ) n x n+ n + )! x sinx ) C + n x sinx ) n ) n x 4n+4 n + )!4n + 4) ) n xx ) n+ n + )! n ) n x 4n+ n + )! c) T 8 x) C + x4 4 x8 48 d) x sinx ) T 8 ) T 8 ) e) Because the series is alternating, the approximation is an underestimate points) a) Sketch the graphs of r cos θ and r. b) Find the area of the region inside r cos θ and outside r. a) 68 6-exams-fall.nb r@t_d : - Cos@tD; s@t_d : ; Show@ParametricPlot@88r Cos@tD r@td, r Sin@tD r@td<, 8r Cos@tD s@td, r Sin@tD s@td<<, 8t,, Pi<, 8r,, <, PlotRange Ø 88-.5,.5<, 8-.5,.5<<, AspectRatio Ø ê 4, PlotStle Ø 88Opacit@.D, Blue<, 8Opacit@D, White<<, Mesh Ø None, Ticks Ø 88-, -, <, 8-, <<, Frame Ø None, AxesLabel Ø 8x, <, AxesStle Ø 88Arrowheads@AutomaticD, Thickness@.D<, 8Arrowheads@AutomaticD, Thickness@.D<<, LabelStle Ø 4, PlotStle Ø 88Thickness@.D, Black<, 8Thickness@.D, Black<<D, PolarPlot@8- Cos@tD, <, 8t,, Pi<, PlotStle Ø 88Thickness@.6D, Black<, 8Thickness@.6D, Black<<, Mesh Ø NoneD D b) We wish to find the area of the shaded region. The curve r cos θ for θ to π) and curve r for θ to π) intersect at cos θ θ π/. We double the area of the shaded region above the x-axis. π A r r) dθ π/ π π/ [ θ + sin θ cos θ) ) π dθ + cos θ) ) dθ ] π π/ π π π/ π π/ 4 cos θ ) dθ + cos θ) dθ )) π +

6 8. 6 points) Write the word TRUE if alwas true) or FALSE. For this problem, no explanation is necessar. a) If a n diverges, then a n diverges. b) The length of the curve r cos θ is given b L π/ 4 cos θ + sin θ cos θ dθ. c) If the parametric equations x ft) and gt) are twice differentiable, then d d /dt d x/dt. d) The second order Talor polnomial T x) centered at a is used to approximate fx) x on the interval [,.]. An estimate for the error of the approximation using Talor s Formula is 6.). a) True because if a n converges, then so does a n. b) True. Note that r is not defined for cos θ <. If r cos θ, then dr dθ sin θ. The length cos θ of the entire curve is four times the length for θ to π/. L r + c) False. d d dt d/) /dt ) dr π/ dθ 4 dθ cos θ + sin θ cos θ dθ. d) True. R x) f z) x ) for z between x and. On the interval [,.], f z)! 8z 5/ 8 and x ).), so R x) 8!.) 6.).

7 9. points) Match the graphs shown below to the following equations. No explanation is necessar. a) x + / d) r 4/θ b) x /4 e) r cos θ)/4 c) x cos t, sin t f) r / + cosθ) 4 ) d ) f ) c 4) a