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05faex.final - MATH 221 LEC 1 YOUR NAME Instructor...

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Unformatted text preview: MATH 221 LEC 1 YOUR NAME: Instructor: Professor Uhlenbrock CIRCLE your TA’s name: Paul Jenkins Junwu Tu Michael Woodbury Luanlei Zhao FINAL EXAM Thursday, December 22, 2005, 5:05 PM - 7:05 PM 3650 Humanities Building Do all problems and show all your work. Answers without written steps or reasons are worth much less than answers with correct steps and reasons. Unless otherwise indicated you should derive exact solutions to the problems, rather than numerical approximations. You can only get credit for what you write down, not for what you might have meant by a vague answer. Write neatly and display your final answers clearly on the lines provided with the problems. Use the last page for scrap paper. __-| MATH 221 LEC 1 FINAL EXAM THU DEC 22 PROBLEM 1 Evaluate the following limits, showing complete explanations of your work. (Such explanations should include words; explain what you are doing.) Part (a) (6poz'nts) lirn sinh(a:) 12'9—00 ANSWER: ____________________________________ Part (b) (7p0ints) . (a:2 + 523+ 6)(:C + 3) 11m ——————————— z—)2+ 1‘3 — 8 ANSWER: MATH 221 LEC 1 FINAL EXAM THU DEC 22 PROBLEM 2 Part (a) ( 7 points ) Find the second m-derivative of the function f(a:) = 5302/3 — V3332+2 ANSWER: _______________________________.__ Part (b) ( 6 points ) Suppose that the equation es‘y—i—lnac=y3 defines y as a differentiable function of ac. Find 73% in terms of m and y. MATH 221 LEC 1 FINAL EXAM THU DEC 22 PROBLEM 3 ( 14 points ) A closed box with a square base (see the diagram below) is to have a volume of 40.5 cubic feet. , If the material used for the sides costs $10 per square foot, the material used for the bottom ‘costs $12.50 per square foot, and the material used for the top costs $17.50 per square foot, what should the dimensions of the box be so that the cost of the box is minimized? ANSWER: MATH 221 LEC 1 FINAL EXAM THU DEC 22 PROBLEM 4 Part (a) ( 6 points ) Evaluate the integral /4 1 —————-3dt 1 fl(\fl+1) ANSWER: ___________________________________ Part (b) ( 7 points ) Evaluate the integral 1r/2 cosa: f ——-T‘d$ 0 1+sma: MATH 221 LEC 1 FINAL EXAM THU DEC 22 PROBLEM 5 Consider the region R between the m—axis and the graph of the function y = x2 such thatOSmg 1. Part (a) ( 6 points ) Find the volume of the solid obtained when this region R is rotated about the y—axis. ) ANSWER: Part (b) (7120mm) Find the volume of the solid obtained when this region R is rotated about the line y = —1. ANSWER: MATH 221 LEC 1 FINAL EXAM THU DEC 22 PROBLEM 6 ( 15’ points ) Find the area of the region above the m-axis which lies below the curve y = fl and also below the curve y = —a: + 12. ANSWER: MATH 221 LEC 1 FINAL EXAM THU DEC 22 PROBLEM 7 Part (a) ( 6 points ) Compute the derivative Dan (12%“) . ANSWER: Part (b) { 7 points ) Evaluate the integral 9 2V; dm [1 m ANSWER: MATH 221 LEC 1 FINAL EXAM THU DEC 22 PROBLEM 8 (14 points) Part (a) Find the general solution of the differential equation 3/ + 2mg = a: ANSWER: Part (b) Find the particular solution of the above differential equation satisfying m = 0,11 = 2. ANSWER: MATH 221 LEC 1 FINAL EXAM THU DEC 22 PROBLEM 9 Part (a) ( 7 points ) Find the following limit showing all your reasoning: . $3 . sma: — a; + -6- hm ————— arr—)0 1:5 ANSWER: ________________________________ Part (b) (7poz'nts) Find the following limit showing all your reasoning (Hint: Consider logarithms): lirn (1m + 2x)“ 13—)00 ANSWER: ...
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