MATH 122 FINAL EXAM WINTER March 15, 2011
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1 MATH 1 FINAL EXAM WINTER March 15, 011 NAME: SECTION: ONLY THE CORRECT ANSWER AND ALL WORK USED TO REACH IT WILL EARN FULL CREDIT. Simplify all answers as much as possible unless explicitly stated otherwise. This is a closed-book, closed-notes exam. No electronic devices are allowed. IF YOUR SECTION NUMBER IS MISSING OR INCORRECT, 5 POINTS WILL BE DEDUCTED FROM YOUR SCORE. Points PAGE 1 16 points Score PAGE 16 points PAGE 3 16 points PAGE 4 16 points PAGE 5 18 points Points PAGE 6 17 points Score PAGE 7 9 points PAGE 8 16 points PAGE 9 16 points Raw Score (out of 140 ): Final Score (100 * Raw Score / 140 ):
2 FINAL EXAM MATH 1 WINTER (8 points each) Evaluate the integrals by any method. 1. x 3 4 dx 3 x. e 1 e x 4 ln x dx 1
3 FINAL EXAM MATH 1 WINTER (8 points each) Evaluate the integrals by any method x dx 4. 3 dx x 1 0
4 FINAL EXAM MATH 1 WINTER (8 points each) Evaluate the integrals by any method sin 4x cos 4 xdx 6. dx 9 x 3
5 FINAL EXAM MATH 1 WINTER (8 points) Use sigma notation and the appropriate summation formulas to find the net-signed area between the graph of y x 1 0,. Let * x k be the right endpoint of each subinterval. and the x-axis on the interval 8. (8 points) Use the definition of a definite integral to evaluate the following limit on the interval 0,1 : 1 1 n lim xk max xk 0 * k 1 x k 4
6 Final Exam Math 1 Winter (8 points) Use sigma notation and the appropriate summation formulas to find the net-signed area between the graph of y x 1 0,. Let * x k be the right endpoint of each subinterval. and the x-axis on the interval 8. (8 points) Use the definition of a definite integral to evaluate the following limit on the 1 interval 0, : n lim x max k 0 k x k 1 * 1 1 x k 4
7 FINAL EXAM MATH 1 WINTER (9 points) Find the area between the curves y x xe and y x on the interval 0,ln. 10. (9 points) Let R be the region bounded by the curves y 9 x and y 5. Give an integral that represents the volume of the solid by rotating R around the x-axis. Do NOT evaluate the integral. 5
8 FINAL EXAM MATH 1 WINTER (9 points)write down the partial fraction decomposition for the rational function x x1 x 9x Do NOT solve for the unknown constants in the expression. 1. (8 points) Answer True or False to the following statement and justify your answer: d y y 0 The function y cos4xis a solution to the differential equation dx. 6
9 Final Exam Math 1 Winter (9 points)write down the partial fraction decomposition for the rational function x x1 x 3x 16 Do NOT solve for the unknown constants in the expression. 1. (8 points) Answer True or False to the following statement and justify your answer: d y y 0 The function y cos4xis a solution to the differential equation dx. 6
10 FINAL EXAM MATH 1 WINTER (9 points) Suppose that a quantity y yt root of the amount present. a. Write down the differential equation that yt () satisfies. b. Find the general solution to the differential equation. c. Find the specific solution with initial value y 0 3. increases at a rate that is equal to the fifth 7
11 FINAL EXAM MATH 1 WINTER (8 points) Without eliminating the parameter, find dy dx and d y dx parametric curve x e t, y sint. at t 0 for the 15. (8 points) Identify the curve rectangular coordinates. 10 r by transforming the polar equation to 5cos sin 8
12 FINAL EXAM MATH 1 WINTER (8 points) Give an integral that represents the arc length of the inner loop of the limaçon r 1 cos. Do NOT evaluate the integral. 17. (8 points) Give an integral or integrals that represents the area of the region outside r and inside r sin. Do NOT evaluate the integral(s). 9
13 Final Exam Math 1 Winter (8 points) Give an integral that represents the arc length of the inner loop of the limaçon r 1 cos. Do NOT evaluate the integral. 17. (8 points) Give an integral or integrals that represents the area of the region outside r 3 and inside r cos. Do NOT evaluate the integral(s). 9
14 FINAL EXAM MATH 1 WINTER Math 1 Winter Final Exam Formula Sheet Summation Formulas n n n n n n n k, k, k 6 n n n 3 k1 k1 k1 Trigonometry Formulas 1 sin cos sin sin 1 cos cos cos cos 1 sin sin cos cos n1 n sin xcos x n1 n sin xdx sin xdx (for n) n n n1 n cos xsin x n1 n cos xdx cos xdx (for n) n n tan x n 1 n1 n n tan xdx tan xdx (for n ) n n sec xtan x n n sec xdx sec xdx (for n) n1 n1 1 x x sin 1 cos 1 x x cos 1 cos 10
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