MATH 132 SOLUTION FOR FINAL EXAM FALL 2009 1 Evaluate 0 38 B cos B dB 1 E F G 1 H I J K L 1 M Solutions u 38 B 9 B B l H 2 Evaluate E F 1 B sin 2x B G 1 H 1 I 1 J 1 K 1 L M 1 N 1 Solutions B 38 B B B 9 B B B sin 2x B B 9 B 9 B B B 9 B 38 B 1 1 1 W9 B sin 2x B B 9 B 38 B l H 2 3 Use partial fractions to evaluate 1 E 1 24 F 1 38 G 1 46 B B B 1 B D 1 56 E 1 69 F 1 75 G 1 80 H 1 91 I 2 14 E F Solutions BB B 1 EB B E F B E 8 F B B 2 B X 2 8 B B 1 B B B B 68 B 68 B l 1 68 68 68 68 68 H Find what becomes of the integral 4 x B B when you make the substitution x 2 tan 1 1 E 8 F sec G ec 2 H cot I sc J K t 8 L 8 M cot N csc Solution X 2 8 dx 2 sec 4 x B B sec tan2 sec 8 E tan tan 2 sec 5 Find the area of the region enclosed by 0 B B 8 0 B B E 1 F G 1 H I J 16 K L M N Solutions B B 0 9 B 8 B S8 638 3 9 W9 E B B B B B B B B l J 6 Find the volume of the solid you get by revolving the region enclosed by the curve B C the lines B and C about the B axis A 1 F 1 G 1 H 1 I 2 1 J 1 K 4 1 L 1 M 1 N 1 Solution Region of revolution in x y plane is enclosed by y B 8 C A3 2 B FC 3 7 29 Z 1 B B 1 B B l 1 I 7 Suppose S is the solid whose base in the region bounded by the parabola C 4 B and the line y 0 and whose cross sections perpendicular to the y axis are squares For each y between 0 and 4 find a formula for the cross section A y A y B y G C H y I C J y K C L y M C N C Solution X 2 1398 2 3 9 2 B 90 C 4 B 8 3 9 97 98 2 B B3 9 2 38 6 3 C J 9 2 C 2 3 90 2 2 6 81 2 B A2 B 3 2 B 99 38 90 938 B C 98 C B X 2 0 9 E C B C C N Calculate the arc length of the curve y A F G H I J x K x L M N C C Solution B B D B 9 6 81 2 3 6 B B B B l M 8 Determine whether the improper integral 0 1 B determine its value A 0 B 2 C H 4 I J M K L 8 8 Solution B B B 0 B converges and if so N 3 1 1 B B B l K A force of 20 N is required to hold a spring stretched 5 m beyond its natural length How much work is done in stretching it from 5 m to 10 m beyond its natural length A 50 J B 90 J C 110 J I 230 J J 250 J D 130 J E 150 J F 170 J G 90 J H 210 J Solution K3 8 5 5 8 J B B X 2 0 9 B B B l N I C 11 Find the solution to the initial value differential equation E C F C C C G C H C I C J C K C L C 68 I C 38 N C 9 Solution C C G X 2 8 C G C G W9 A 1 C E 12 Find the sum of the E 1eometric series F G H I J K L M N Solution 3 1 2 1 97 3 3 A3 2 8 X 2 0 9 3 98 1 9 F 4 13 If cos 2x a8 B8 3 Q 6 38 3 E F 8 G H I J find K L a M N B Solution 9 B B x B x so cos 2x B x x X 23 13 x G 14 Find the function whose Maclaurin series is B B B B B E 38 B B 9 B G 68 B H B I B K B L B M 8 B N 8 B J B B B B x B By substitution we get B B B B B I Solution a8 x 1 8 6 15 If 8 at c A 0 B 1 is the Taylor series of the function f x cos x then find the value of a the coeficient of the term a x 16 G H I 0 1 J K L M N Solution a x 0 B B 9 B 38 B W9 0 1 X 2 8 A 1 H 16 Find the interval of convergence of the power series E B E 4 B I 0 B 4 4 F B G B F 4 B 4 G 0 B J B 8 8 4 B 8 8 H B H 0 B 4 l B l Solution We use Ratio Test l B l l B l J 9 98 1 8 A 8 8 8 l B l A23 2 7 8 B B B J 9 B A 1 8 8 8 8 8 8 8 3 1 8 0 9 B A 1 3 1 W9 38 6 90 98 1 8 3 0 B 4 K 17 Find the 4 2 term in the binomial series for the function f x 1 E B F B G B H B I K B L B M B N B J B B B Solution B B x B x B B B B FC 3 398 A 1 1 B B B B X 2 2 7 3 2 8 B L Using the error formula for alternating series what error do you make by approximating S 8 E B C 8 8 8 8 D E by S i e the sum of the first three terms F G H I J Solution The error is no worse than the absolute value of the fourth term i e n 4 which is H Find the term a in the Maclaurin series of A 0 B 1 C 1 D E F K B 38 B B f B …
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