APPM 1360 Final Exam Spring 2016

Transcription

1 APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan n) /n d) n a) The sequence converges to lim a n lim ln 8n LH ln 8. n n n + b) The integral is convergent : Note that ln d lim t + lim t + t [ }{{} dv d v / ln ln }{{} uln dud/ ] lim ln t + ln t lim t t LH ln t lim + t + t t d ) t d t lim t + n [ e n )) ln t t ln. 4 /t lim lim t + t 3 t t +. ln 4 ] t ) c) The series diverges by the Test for Divergence: lim n arctan n) /n /). d) Apply the Ratio Test. lim a n+ n a n lim n + )!) e n n e n+) n!) lim LH The series absolutely converges. n n + ) e n+ n + ) LH lim lim n e n+ <. n 4en+

2 . 8 points) Consider the circles shown. The largest circle has a radius of m, the net largest circle has a radius of m, and the net has a radius of m, etc. Each circle has half the radius of the previous one. Suppose there is an infinite number of circles and S represents the total area of all the circles. a) Epress S as an infinite series of the n th circle. a n by finding an epression for a n, where a n is the area n b) Is a n bounded? If so, find the bounds. If not, eplain why not. c) Let s n equal the nth partial sum of the series. Find s 3. Simplify your answer. d) Evaluate lim n s n+ s n ). e) Find the total area S or eplain why it does not eist. a) The series is geometric with a 4 and ratio r /4. ) n S ar n n n n n b) Since a n is a positive, decreasing sequence, it is bounded above by a 4 and bounded below by, and therefore a n is bounded. c) s 3 a + a + a d) lim s n+ s n ) lim a n+ lim n n n 4 n. Alternate solution: This is a convergent geometric series r < ). lim s n lim s n+ S n n e) The sum of the geometric series is S 3. 8 points) Let y 3 +. lim s n+ s n ) S S. n 4 / a) Consider the shaded region shown bounded by the curve and the -ais,. Find the volume of the solid generated by rotating the region about the y-ais. Simplify your answer. - + b) Set up but do not evaluate an integral to find the area of the surface generated when the curve,, is rotated about the line.

3 a) By the shell method and partial fraction decomposition / V 3 + d [ ln ln ln 9 ) 8 / ] / ) d ln 3 ln ) ln + Partial fraction decomposition: 3 + Let. Then B. Let. Then A. A + A ) + B ) B A ) + B ) 3 + Alternatively: Match terms: A + B) and A B, so A and B. / ) 3) b) S ) ) d 4. points) The function g) equals the Taylor series g) a) Use the fourth degree Taylor polynomial T 4 ) to approimate the value of g). Simplify your answer. b) Find an error bound for the approimation in part a). Justify your answer. c) What is the eact value of g)? d) Find a series representation for g ). Epress your answer in sigma notation starting with n. Simplify your answer. a) T 4 ) g) T 4 )

4 b) The series g)! 3! + 4! ) n Series Test with b n /n!: n n! satisfies the conditions of the Alternating n + )! < n! b n+ < b n and lim b n lim n n n!. By the Alternating Series Estimation Theorem, an error bound is g) T 4 ) < b 5 5!. Alternate solution: Note that the given series ) n n n! is the Maclaurin series for g) e +. Using n Taylor s Formula the error is R 4 ) g5) z) 5 for and < z <. 5! The derivative g 5) z) e z < e, so R 4) < 5!. c) The given series g) e. ) n n n! is the Maclaurin series for g) e +. At, n d) Differentiating term by term we get g )! + 3 3! 4 4! + Alternate g ) d ) ) n n d n! n n n n ) n!. ) n n n n n )! n ) n n n 5. 6 points) Consider the curve C given by the parametric equations a sint), y b cost), where t and a, b are positive constants with a > b. a) Eliminate the parameter to obtain an equation in rectangular coordinates that describes C. b) Sketch a graph of the curve C. In your graph: Indicate the starting and ending positions of the parametric equation. Use an arrow to show the direction the curve is traversed. Label all intercepts. c) Using the given parametric equations set up an integral to find the area of the region enclosed by the curve C. Evaluate this integral. d) Set up but do not evaluate an integral that equals the arc length of the curve C. e) Find dy d and d y in terms of the parameter t. d a) Using the identity sin t + cos t, a Cartesian equation for the ellipse is a + y b. n!.

5 b) + / - c) By symmetry d) L e) A 4 / 4ab a L a b cos ta cos t) dt 4ab [t + sint) ] / / / cos t dt 4ab + cost)) dt ) ab ab. a cos t) + b sin t) dt or using the Cartesian form y b a ) b + a d. a dy d dy/dt b sin t d/dt a cos t b a tan t d d y d dt dy/d) b/a) sec t b d/dt a cos t a sec3 t 6. 8 points, 6 points each) Unrelated short answer questions. a) Always True or False? If f) g) for all and f) d converges. b) Find the 5 term of the Taylor series for sin centered at a. b a cos 3 t c) How large should n be to guarantee that the Midpoint Rule approimation for a) is accurate to within? Simplify your answer. 4!) 3 a, g) d converges, then ln d False. For eample, let f) and g). Then f) g) for and d converges, but ) d diverges. Note that the integral Comparison Theorem applies to functions f and g that are both positive.

6 b) The 5 term of a Taylor series centered at a is f 5) a) a) 5. The derivatives for 5! f) sin occur in a cycle of 4: n f n) ) f n) /) sin cos sin 3 cos 4 sin It follows that f 5) /) f ) /) and the desired term is 5! ) 5. c) Let f) ln, f ) + ln, and f ) /. In the interval [, ], f. Solve E M /4!) 3 for n, letting the upper bound K. E M 7. points) Consider the rose curve r sin3θ). Kb a)3 4n 4!) 3 4n n 4 subintervals 43 a) Sketch a graph of the curve in the y-plane and the corresponding graph in the rθ-plane. For both graphs, label all intercepts. b) Find an equation of the tangent line at the tip of the leaf in the first quadrant. [Note: the tip of the leaf occurs when r is a maimum]. c) Show that the tangent line in part b) is perpendicular to the line that connects the origin to the tip of the leaf. d) Set up, but do not evaluate, an integral to find the area inside one leaf of the given curve. e) Set up but do not evaluate an integral that equals the eact arc length of one leaf of the given curve. a) θ) θ) θ - -

7 b) First find the tangent slope. At θ /6, r attains a maimum value of. d dy d dθ sin3θ) sin θ) sin3θ) cos θ) d dθ dy d ) 3 + θ/6 ) + 3 The point of tangency is r cos/6), r sin/6)) The tangent line is y ) 3 3. c) The line connecting the origin to the point sin3θ) cos θ + 3 sin θ cos3θ) sin3θ) sin θ + 3 cos θ cos3θ) ) 3,. ) 3, has slope /, which is 3/ 3 the negative reciprocal of the tangent slope 3. The lines are therefore perpendicular. d) A e) L /3 /3 sin 3θ) dθ sin 3θ) + 3 cos3θ)) dθ