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1 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 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 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 5 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. pts 2. pts 3. pts 4. pts 5. pts 9. a b c d e. a b c d e Bonus. TOTAL pts 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 -4 Alberto Corso MWF : am - :5 am, CB Section # Recitation Instructor Time/Location Julianne Vega TR : am - :5 am, CB Julianne Vega TR : am - :5 am, CB Rachel Petrik TR 2: pm - 2:5 pm, CB 34 4 Rachel Petrik TR : pm - :5 pm, CB 34

3 . The area of the region enclosed by the curves y = x and y = x is described by the following integral: [ x x] dx [x x] dx [x x] dx x dx None of the above 2. Measles is a highly contagious infection of the respiratory 5 system and is caused by a virus. Despite the fact that more than 8% of the world s population is vaccinated for it, measles remains the fifth leading cause of death worldwide. In general, the term pathogenesis refers to the way a disease originates and develops over time. In the case of measles, the virus enters through the respiratory tract and replicates there before spreading into the bloodstream and then the skin. In a person with no immunity to measles the characteristic rash usually appears 5 about two weeks after infection, at which point the virus reaches a peak density in the blood. The virus level then decreases fairly rapidly over the next few days as a result of the immune response. This progression is approximated in the pathogenesis curve graphed on the side. The above measles pathogenesis curve has been modeled by the polynomial 4 2 t (days) f(t) = t(t 2)(t + ), where t is measured in days and f(t) is measured in the number of infected cell per ml of blood plasma. Over the course of the 2-day infection, what is the average level of infection? infected cell per ml of blood plasma infected cell per ml of blood plasma, infected cell per ml of blood plasma, 2.75 infected cell per ml of blood plasma, 5 infected cell per ml of blood plasma

4 3. Find sin 3 (x) cos 2 (x) dx using the substitution u = cos(x). cos 5 (x) 5 cos 4 (x) 4 cos 4 (x) 4 cos 3 (x) 3 cos3 (x) 3 cos6 (x) 6 sin3 (x) 3 + cos2 (x) 2 cos 3 (x) sin 2 (x) 4. Evaluate x e x2 dx using the substitution u = x 2 2e x2 e x2 e x2 2e x2 x 2 2e x3

5 5. Let f be a function defined on the interval [2, 7] with continuous derivatives. Suppose in addition that we are given the following information about f and f f(2) = 5 f(7) = f (7) = 3. Find the value of 7 (2x 4)f (x) dx The value can t be determined without knowing f (2) 6. Which of the following is equal to the integral x cos(2x) dx? x 2 sin(2x) + 4 cos(2x) x 2 sin(2x) 4 cos(2x) x 2 sin(2x) + 4 cos(2x) 2x sin(2x) + cos(2x) 2x sin(2x) 4 cos(2x)

6 7. Which of the following is the correct form of the partial fraction expansion of 84 (x 4) 2 (x 2 + x + 9)? Ax + B (x 4) 2 x + D x 2 + x + 9 A (x 4) 2 + Bx x 2 + x + 9 A x 4 + B (x 4) 2 x 2 + x + 9 A (x 4) 2 + B x 2 + x + 9 A x 4 + B (x 4) 2 x + D x 2 + x Which of the following represents 3x + 4 x 2 + 3x + 2 dx? (3/2x 2 + 4x) ln x 2 + 3x + 2 3/2 ln x 2 + 3x + 2 ln x ln x + 2 ) 2 ln x + + ln x + 2 ln x + + ln x + 2

7 9. Evaluate the following integral 5 x dx ln(5) ln( 2) ln(5) ln(2) ln(5) + ln(2) ln(5) + ln(2) The integral diverges. Complete correctly the statement below The integral x p dx diverges, because it is improper... diverges if p > and converges if < p... converges if p and diverges if < p <... converges if p > and diverges if < p... diverges if p is negative and converges if p is positive

8 . Sketch the region enclosed by the parabolas Then find the area of this region. y = x 2 and y = 4x x 2. pts: /

9 2. Use substitution to evaluate the definite integrals below. [Clearly indicate your answer and the steps used to arrive at that answer. Unsupported answers will receive no credit.] cos x 2 x dx x x dx pts: /

10 3. Use integration by parts to evaluate the integrals below. [Clearly indicate your answer and the steps used to arrive at that answer. Unsupported answers will receive no credit.] (2x + 3)e x dx 4 t ln t dt pts: /

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

12 5. Integrate the improper integrals below. [Clearly indicate your answer and the steps used to arrive at that answer. Unsupported answers will receive no credit.] 5 2 x 2 dx xe x2 dx pts: /

13 Bonus. Evaluate the integral: 4 [Hint: if you set u = x then... ] e x dx Given that n is a positive integer, evaluate the integral: [Hint: if you set u = x then... ] x( x) n dx pts: /

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