Math 132 Exam 1 Fall 2016 16 multiple choice questions worth 4 5 points each 2 hand graded questions worth 14 points each Exam covers sections 4 9 through 6 2 No calculators For the multiple choice questions mark your answer on the answer card Show all your work for the written problems Your ability to make your solution clear will be part of the grade Useful Formulas n X i i 1 n X i2 i 1 n X i 1 3 n n 1 2 n n 1 2n 1 6 i n n 1 2 2 Area Circle Radius r r2 Area Ellipse With Semi Major Axis a Semi Minor Axis b ab Math 132 Exam 1 Page 2 of 19 1 Let f x be the function satisfying f 00 x sin x ex f 0 0 2 and f 0 0 Find f 2 A e 2 2 1 B e 2 C 2e 2 2 D e 2 2 E e 2 1 2 F e 2 G e 2 1 Solution Integrate twice f 0 x cos x ex C1 Use f 0 0 2 to get C1 2 and f 0 x cos x ex 2 f x sin x ex 2x C2 Use f 0 0 to get C2 1 and f x sin x ex 2x 1 Plugging in gives f 2 e 2 2 2 Suppose you know the following about a function f x Z 4 f x dx 4 1 8 Z f x dx 3 2 8 Z f x dx 4 1 Z Find 2 2f x 6 dx 1 A 5 B 3 C 0 D 2 E 4 Math 132 Exam 1 Page 3 of 19 F 6 G 8 H None of the above Solution Z 2 2f x 6 dx 2 1 Z 2 Z f x dx 6 1 Z 2 dx 1 2 Z 2 f x dx 6 1 6 2 f x dx 1 Z 8 Z 8 6 2 f x dx f x dx 2 1 1 6 2 4 3 8 2 Math 132 Exam 1 Page 4 of 19 3 Suppose f x f 0 x and f 00 x are all continuous functions and you know the following additional information about f x f 2 4 f 6 12 f 0 2 5 f 0 6 11 f 00 2 6 f 00 6 10 Z If possible find 6 f 00 x dx 2 A 0 B 2 C 4 D 6 E 8 F 10 G 12 Z H We do not have enough information to determine 2 Solution Use the FTC Part 2 Z 6 f 00 x dx f 0 6 f 0 2 11 5 6 2 Z 4 Compute 2 A 0 B 2 C 4 D 6 6 1 x 4 dx 6 f 00 x dx Math 132 Exam 1 Page 5 of 19 E 8 F 10 G 12 H None of the above Solution If you draw the graph you can just compute the area Looking at the graph you should be able to see that the area is 8 You could also split the integral up that might be slightly easier 5 y y 1 x 4 4 3 2 1 x 2 3 4 5 6 7 Math 132 Exam 1 Z Page 6 of 19 2 sin5 x cos x dx 5 Find 0 A 0 B 16 C 16 1 3 D E 13 F 1 G 1 H None of the above Solution Z 2 5 Z sin x cos x dx x 2 u5 du x 0 6 x 2 0 Let u sin x du cos x dx x 2 u sin6 x 6 x 0 6 x 0 6 sin 2 sin6 0 1 0 6 6 6 Z 6 Let u ln x and rewrite the following integral in the variable u Z A u e3u du Z B u du Z C Z D Z E u e2u du u2 ln u du u2 eu du x2 ln x dx Math 132 Exam 1 Z F Z G Z H eu ln u du u3 ln u du u2 e2u du Solution If u ln x then du x1 dx dx x du and x eu Z Z 2 x ln x dx x2 u x du Z ux3 du Z ue3u du Page 7 of 19 Math 132 Exam 1 Z Page 8 of 19 36 4x2 dx 3 7 Compute 0 A 4 5 B 4 5 C 9 D 9 E 18 F 18 G 36 H None of the above Solution Z 3 Z 36 0 4x2 dx 2 3 9 x2 dx 0 Graphing y 9 x2 is a quarter of a circle of radius 3 and thus has area 14 32 94 Multiply this by two to get the answer 8 A function f x and a number b satisfy the question Z x f t 3 dt 24x 3 4 t b What is b A 0 B 1 C 2 D 3 E 4 F ln 3 G ln 8 H It is not possible to determine b Math 132 Exam 1 Page 9 of 19 Solution Plugging in x b makes the integral equal to 0 and gives the equation 3 0 24b 3 Solving gives b 2 Math 132 Exam 1 Page 10 of 19 9 Let f x x2 1 Compute L4 over the interval 1 3 L4 is the Riemann sum using left endpoints as sample points with 4 subdivisions A 0 B 2 75 C 3 D 5 E 9 F 10 G 12 5 H 18 Solution With 4 subdivisions x0 1 x1 0 x2 1 x3 2 and x4 3 3 1 1 x b a 4 4 L4 4 X f xi 1 x i 1 f x0 1 f x1 1 f x2 1 f x3 1 f 1 1 f 0 1 f 1 1 f 2 1 1 1 0 1 1 1 4 1 10 10 Identify the definite integral that is equal to the limit of Riemann Sums lim n X n i 1 Z 4i 1 n 8 5 x 1 7 dx A 0 Z 8 x 4 7 dx B 4 Z C 2 6 x8 dx 4 n Math 132 Exam 1 Z Page 11 of 19 4 x8 dx D 0 Z E 5 x 1 8 dx 2 Z F 5 x8 dx 1 G None of the above Solution You should be able to see that x n4 which means b a 4 Then you 8 While there are several possibilities one is that f x x8 can see f xi 1 4i n and xi 1 4i n Since xi a i x this means that a 1 Putting all these Z 5 x8 dx together gives 1 Math 132 Exam 1 Page 12 of 19 3 Z 3x2 2 dx lim Rn where Rn is the right hand Riemann sum Find Rn …
View Full Document
Unlocking...