Please note that, except for question 1, you are expected to provide supporting calculations or otherwise explain how you arrived at your answers.

Transcription

1 Fnal Exam Practce Problems Sprng 2008 WARNING: THIS IS SIMPLY A SMALL SET OF PROBLEMS OF TYPES THAT WOULD BE APPROPRIATE FOR THE FINAL EXAM. BE SURE TO REVIEW THE IN-CLASS TESTS, HOMEWORK, AND LECTURE NOTES Instructons: Ths examnaton conssts of xxx (xx) questons on xx (xx) pages. Each problem s worth 0 ponts. Please make sure you have a complete exam. Enter your name, secton, and TA nformaton at upper rght on ths page and your name or ntals on each of the other pages. Please note that, except for queston, you are expected to provde supportng calculatons or otherwse explan how you arrved at your answers. Expressons such as 3 ln 76 2, e 5, 5 ln( 2) + arctan π 4 need not be evaluated on your calculator., etc. that are part of fnal answers If you do prefer to work wth floatng pont values then be sure the grader can determne what they represent. Problem Score Homework Total Page

2 . (0 pts - 2 pts each) Respond as ndcated to each of the followng. You do not need to show work for ths problem. a a. By defnton = lm n = b. Accordng to the rato test a n s absolutely convergent f n = lm n c. Suppose f(x) s contnuous on ( 0, ) and f( ) = a for,2,... Suppose further that we are gven that the ntegral test apples to a and that d < f( x) x. If R n = a s = n + the remander then we have the estmate n + f( x) dx R n d f( x) x d. For whch values of r does the seres r = 0 e. For whch values of p does the seres t q converge absolutely? converge absolutely? f. The alternatng seres n = a n converges f lm = 0 and a g. Suppose f(x) s a contnuous functon such that { a } s a sequence such that a = f( ) for =,2,... and d < f( x) x. The ntegral test can be used to determne whether converges provded f(x) has two addtonal propertes: () 0 f( x ) for all x and (b) a Page 2

3 2. SHOW YOUR WORK ON BOTH PARTS OF THIS PROBLEM. (a) (5 ponts) Determne whether the seres n = 2 n 2 ln( n) s convergent or not. (b) (5 ponts) For whch values of z s the followng seres absolutely convergent 7( 2 5z) n? n 2 n = 0 2 n 3. (5 ponts) Use the ntegral test to determne whether the seres n = n 2 + dvergent. You must use the ntegral test and must show that t apples. s convergent or 4. (0 ponts) Determne whether the seres n = n 3 + n s convergent or dvergent by usng the comparson test. You must use the comparson test, and must verfy that t apples, and must explan or demonstrate why the seres beng used for comparson s convergent or dvergent. 5. Calculate the exact values for each of the followng a. (5 ponts) = 3 sn π 6 b. (5 ponts) n = n ( n + 3) = 6. a. (5 ponts) What s the least n for whch you can be certan by the alternatng seres test that n ( ) s wthn Page 3

4 .05 of ( )? b. (5 ponts) The partal sum, S 8 = 8 j = a j of the seres ( ) j j s j = 7. (0 ponts) A car travels counterclockwse around a crcular track at a constant speed, crclng the track sx (6) tmes per hour. The track has a radus of 2 mles. If the track s assumed to be the graph of x 2 + y 2 = 4 and at tme t=0 the car s at (0,2), then the poston of the car at tme t hours s gven parametrcally by x(t) = y(t) = 8. The curve C s descrbed parametrcally by x( t ) = t t +, y( t ) = 3+ 4t + t 3 a. (3 ponts) Calculate a parametrc form for the tangent lne to C at the pont correspondng to t=. b. ( 3 ponts) Calculate a parametrc form for the normal lne to C at the pont correspondng to t=. dy b. (2 ponts) Calculate at the pont where t =. dx c. (2 ponts) Gve a formula ( n terms of the parameter t ) for dy2. DO NOT SIMPLIFY 2 dx YOUR ANSWER. 9. ( 0 ponts) A ladder 0 feet rests aganst the rght wall of a narrow passageway wth ts base aganst the left wall. However the rght wall s movng to the rght at a constant rate of 3 feet per mnute. At tme t=0 the passageway s 4 feet wde. Gve parametrc equatons descrbng P(t), the pont of contact of the ladder wth the rght wall for tmes between t = 0 and t = 3 mnutes. The dagram represents the tme t = 0. Page 4

5 0. Calculate the followng. SHOW ALL WORK + tan( 2 x ) 2 a. (5 ponts) dx = + tan( 2 x ) sn( x ) b. (5 ponts) d 4 cos( x ) 2 + x = 0 π 4. Suppose f( x ) = 2 x + 3 x + 5 for x>5/3 a. (5 ponts) Calculate f ( - ) (x) b. (5 ponts) For your functon f ( - ) (x) n part (a) show f ( - ) ( f( x ) ) = x 2. Calculate the followng: a. d x cos( x) x = b. d x ln( x) x = Page 5

6 c. d sn( x ) e x x = d. d x3 e x x = - 3. ( 0 ponts) The table at rght gves the values of a functon f(x) and ts frst four dervatves at selected values of x. x f(x) f '(x) f ( 2 ) f ( 3 ) f ( 4 ) a. ( 3 ponts) Calculate the Taylor polynomal of order 3 for f(x) at x = 0. Do not smplfy your answer. b. ( 3 ponts) Calculate the Taylor polynomal of order3 for f(x) at x = Do not smplfy your answer. c. ( 3 ponts) Calculate the Taylor polynomal of order 3 for f(x) at x = Do not smplfy your answer. d. (2 ponts) Let h( x ) = 3 x f( x ). Calculate the Taylor polynomal of degree 3 for h(x) at x = Do not smplfy your answer. e. (2 ponts) Calculate the Taylor polynomal of degree 3 for f ' (x) ( the dervatve of f(x) ) at x =0 Do not smplfy your answer a. (5 ponts) Calculate the trapezod rule estmate for d x 3 + x x wth two sub-ntervals. Use the approprate error formula to calculate the maxmum error n ths estmate. Page 6

7 b. (5 ponts) Calculate the Smpson's rule estmate for d x 3 + x x for 4 sub-ntervals a. (5 pts) Use partal fractons to evaluate dx ( x 5 )( x 2) b. (5 ponts) Calculate the partal fracton expanson of f(x) = ntegrate the result. x 3 + x + x ( x 2 + ) DO NOT (6) Evaluate each of the followng ntegrals a. (5 ponts) d sn ( x + 2) 3 x b. (5 ponts) d sn( 3 x ) cos( 7 x) x (7) 0 pts Evaluate each of the followng a. ( 5 ponts) dx sn( π x ) 3 b. (5 ponts) dx 9x 2 3 (8) Gve an example of a seres a such that lm a = converge). You must explan why t does not converge 0 and yet a does not 9. SHOW YOUR WORK. A car travels counterclockwse around a crcular track at a constant speed, crclng the track sx (6) tmes per hour. The track has a radus of 2 mles. If the track s assumed to be the graph of x 2 + y 2 = 4 and at tme t=0 the car s at (0,2), then the poston of the car at tme t hours s gven parametrcally by ( 4 ponts) x(t) = (4 ponts) y(t) = Page 7

8 ( 2 ponts) The poston of the car at tme t = π 4 s (, ) Calculate the followng lmts c. lm x 0 ( 3x ) 3 x = d. lm n e. lm x n n 2 = x arctan( x ) = f. lm x 7 x 2 4 x + 0 ( 2 x )( 3x ) = Page 8