Spring /07/2017

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1 MA 38 Calculus II with Life Science Applications FIRST MIDTERM Spring /07/207 Name: Sect. #: Do not remove this answer page you will return the whole exam. No books or notes may be used. Use the backs of the question papers for scratch paper. You may use a graphing calculator during the exam, but NO calculator with a Computer Algebra System (CAS) or a QWERTY keyboard is permitted. Absolutely no cell phone use during the exam is allowed. The first part of the exam consists of 0 multiple choice questions, each worth 5 points. Record your answers on this page by filling in the box corresponding to the correct answer. For example, if is correct, you must write a b c d e Do not circle answers on this page, but please do circle the letter of each correct response in the body of the exam. It is your responsibility to make it CLEAR which response has been chosen. You will not get credit unless the correct answer has been marked on both this page and in the body of the exam. The second part of the exam consists of five open-response questions and one bonus question. When answering these questions, check your answers when possible. Clearly indicate your answer and the reasoning used to arrive at that answer. Unsupported answers may receive NO credit.. a b c d e GOOD LUCK! 2. a b c d e 3. a b c d e QUESTION SCORE OUT OF 4. a b c d e Multiple Choice 50 pts 5. a b c d e 6. a b c d e 7. a b c d e 8. a b c d e. 0 pts 2. 0 pts 3. 0 pts 4. 0 pts 5. 0 pts 9. a b c d e 0. a b c d e Bonus. TOTAL 0 pts 00 pts

2 Please make sure to list the correct section number on the front page of your exam. In case you forgot your section number, consult the following table: Sections # Lecturer Time/Location Alberto Corso MWF 0:00 am - 0:50 am, CB 0 Section # Recitation Instructor Time/Location 00 Dustin Hedmark TR 0:00 am - 0:50 am, CB Dustin Hedmark TR :00 am - :50 am, CB Rafael Eduardo Rojas TR 03:00 pm - 03:50 pm, CB 34

3 . Consider the region enclosed by the curves and the x-axis as indicated below. y = x y = 2 x I. II. III ( ) 2 y + y dy ( ) 2 x x dx x dx ( ) 2 x dx Which of the above integrals measures the area of this region? I. only II. only III. only I. and III. only II. and III. only 2. Compute the average of the function f(x) = 3x over the interval [, 4]

4 3. Find sin(x) cos(cos(x)) dx cos(sin(x)) + C sin(cos(x)) + C cos(x) sin(sin(x)) + C sin(x) + cos(x) + C sin 2 (x) + cos 2 (x) + C 4. Consider the following ln(ln(x)) x ln(x) dx = What substitution did we make? u du = 2 u2 + C u = ln(x) u = x ln(x) u = ln(ln(x)) u = / ln(x) u = e x

5 5. Let f be a function defined on the interval [, 4] with continuous derivatives. Suppose in addition that we are given the following information about f and f f() = 4 f(4) = 6 f () = 5 f (4) = 5. Find the value of 4 xf (x) dx Suppose that f is a differentiable function. Which of the following is equal to the integral x 2 f (x) dx x 3 3 f(x) + C x 2 f(x) 2xf(x) dx 2xf (x) x 2 f(x) dx 2xf (x) + x 2 f (x) + C x 3 3 f(x) x 3 3 f (x) dx

6 7. Which of the following is the correct form of the partial fraction expansion of x 2 + 3x 0 (x 2 + 4x + 6)(x 2 )(x + )? A x + B (x ) 2 + C x + + A (x + ) 2 + B x + Cx + D x 2 + 4x + 6 A x + + B (x + ) 2 + C x + A x + + Bx + C x 2 + Dx + E x 2 + 4x + 6 A x + + B (x + ) 2 + C x + Dx + E x 2 + 4x + 6 D x 2 + 4x + 6 Dx + E x 2 + 4x Which of the following represents 4 x 2 + 8x + 5 dx? 2(ln x ln x + 5 ) + C 2(ln x + 5 ln x + 3 ) + C 2(ln x + 3 ln x + 5 ) + C 4 ln(x 2 + 8x + 5) + C 4( x + 4x 2 + 5x) + C

7 9. Evaluate the following integral 2 x 3 dx /4 3/8 0 Divergent 0. Let a > 3 be a fixed number. Evaluate the improper integral a (x 3) 2 dx a 3 0 Divergent (a + 3) 2 (a 3) 3

8 . Suppose that a change in biomass B(t) at time t during the interval [0, 2] follows the equation ( ) db π dt = cos 6 t for 0 t 2. Suppose that B(0) = B 0. Expresss the cumulative change in biomass during the interval [0, t] as an integral. Using your previous answer, find an explicit formula for B(t). What is the value of the biomass at the end of the interval [0, 2] compared with the value at time 0? pts: /0

9 2. Use an appropriate substitution to evaluate the integral 2x 3 x 2 + dx. [Clearly indicate your answer and the steps used to arrive at that answer. Unsupported answers will receive no credit.] pts: /0

10 3. Evaluate the following integral 6 ln(x) dx x3 [Clearly indicate your answer and the steps used to arrive at that answer. Unsupported answers will receive no credit.] pts: /0

11 4. Evaluate the following integral 4x 2 x (x + ) 2 (x 3) dx [Clearly indicate your answer and the steps used to arrive at that answer. Unsupported answers will receive no credit.] pts: /0

12 5. Integrate the following improper integrals: [Clearly indicate your answer and the steps used to arrive at that answer. Unsupported answers will receive no credit.] 5 2 x 2 dx 3x 2 e x3 dx pts: /0

13 Bonus. Use the Comparison Test for Infinite Integrals to determine whether or not the following integral converges. x 2 x 0 + dx. Be sure to clearly indicate what you are using as your comparison function. pts: /0

Spring /06/2018

Spring /06/2018 MA 38 Calculus II with Life Science Applications FIRST MIDTERM Spring 28 2/6/28 Name: Sect. #: Do not remove this answer page you will return the whole exam. No books or notes may be used. Use the backs