Math 132 Final Examination May 4 2012 6 multiple choice 4 long answer 100 points General Instructions Please answer the following without use of calculators You may refer to up to four 3x5 cards but no other notes Part I of the exam is multiple choice while Part II is long answer Part I Instructions If you do not have a pencil to fill out your answer card please ask to borrow one from your proctor Write your Student ID number on the six blank lines on the top of your answer card and shade in the corresponding bubbles to the right of each digit Fill in the bubble corresponding to each of the following 6 questions Each is worth 4 points On Part I no partial credit will be given 1 The Taylor series for f x X xk a k 0 k b c d e X x3k k 0 k X xk k 0 3k X k 0 X x3k xk k 3 f g X xk k 3 k X x3k k 3 h None of the above 1 around 0 is 1 x3 2 Evaluate X 16 i 2 3 i 4 a 0 4 b 7 c 1 d 2 e 3 f 4 g 5 36 h 7 i Does not converge oscillates j Does not converge diverges to n X 8i3 by recognizing it as a Riemann sum 4 n i 1 n 3 Evaluate lim a 0 1 b 2 c 1 3 d 2 e 2 5 f 2 g 3 7 h 2 i 4 9 j 2 k Diverges to 4 Which of the following series can the Alternating Series test be used on X 1 i I 2 i 0 i 1 II X 1 i 2 i 0 i cos i III X 1 i i i 1 ie a None of them b I only c II only d III only e I and II only f I and III only g II and III only h All of I II and III 2i 1 i2 i 2i 3i3 5 Evaluate lim a 0 1 b 3 2 c 5 1 d 2 2 e 3 f 1 4 g 3 3 h 2 i 2 j 3 k Diverges to 6 Evaluate Z 0 x x sin dx 2 a 2 b 4 c d 2 e 0 f 1 g 2 h i 4 j 2 k Diverges Name Id Math 132 Part II Instructions Answer the following on the exam sheet showing all your work Correct answers without correct supporting work may not receive full credit You may use the back of each page for additional answer space please clearly indicate if you have done so or scratch work Please put your name and student id number on each page of Part II now 1 Calculations a 7 points Find all solutions to the differential equation y 0 y assume x 1 1 x b 6 points Find an upper bound for 2e x 3ex 4 sin x 5 cos x on the interval 1 3 2 c 6 points Using a power series find f 100 0 and f 101 0 for f x ex Name Id Math 132 2 Integrals a 5 points Set up an integral for the area of the surface obtained by rotating the curve y e 2x around the x axis for x between 0 and You need not evaluate the integral in question b 6 points Show that the improper integral Z 0 1 dx converges 1 x4 c 8 points Find the volume of the solid obtained by rotating the region below the 1 about the x axis for 0 x curve y 3x 1 Name Id Math 132 3 Series and power series Use the back if you need additional space a 7 points Using partial fractions find a power series representation for 4 x 1 x 3 b 8 points Find the radius and interval of convergence for the power series c 6 points Does the series or diverge X X 2i xi 2 i 0 i 2 1 i converge absolutely converge conditionally 3 2 1 i 0 i Name Id Math 132 4 Integrating cos x2 a 1 point We have discussed that cos x2 has no closed form antiderivative In 1 2 sentences explain what this means b 6 points Using the Fundamental Theorem of Calculus and an appropriate definite integral give an antiderivative of cos x2 c 6 points Using a Taylor series expansion give a power series representation of an antiderivative of cos x2 d 2 points Using part c find a series representing Z 1 0 cos x2 dx
View Full Document
Unlocking...