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1 Student s Printed Name: Instructor: CUID: Section # : You are not permitted to use a calculator on any part of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or any technology on any part of this test. All devices must be turned off while you are in the testing room. During this test, any communication with any person (other than the instructor or his designated proctor) in any form, including written, signed, verbal, or digital, is understood to be a violation of academic integrity. No part of this test may be removed from the testing room. Read each question very carefully. In order to receive full credit, you must: 1. Show legible and logical (relevant) justification which supports your final answer. 2. Use complete and correct mathematical notation. 3. Include proper units, if necessary. 4. Give exact numerical values whenever possible. You have 90 minutes to complete the entire test. On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test. Student s Signature: Do not write below this line. Free Response Problem Possible Points Points Earned Free Response Problem Possible Points 1a 6 5b 4 1b ab cd e 3 Free Response Multiple Choice 35 Points Earned 5a 5 Test Total 100 Page 1 of 12

2 Multiple Choice. There are 12 multiple choice questions. Each question is worth 2 3 points and has one correct answer. The multiple choice problems will count as 35% of the total grade. Use a number 2 pencil and bubble in the letter of your response on the scantron sheet for problems For your own record, also circle your choice on your test since the scantron will not be returned to you. Only the responses recorded on your scantron sheet will be graded. You are NOT permitted to use a calculator on any portion of this test. 1. Consider the function f x ( ) shown below: For which x -value(s) is the function is NOT differentiable? a) x = 1, 2 only b) x = 1, 1, 2, 3 only c) x = 1, 2, 3 only d) x = 1, 1, 3only e) x = 1 only 2. Find the instantaneous rate of change of y = sec x at x = π 4. a) 0 b) 1 c) 2 d) e) Page 2 of 12

3 3. Evaluate lim x 1 x 2 1 sin x 1 ( ). a) 1 b) 0 c) 1 2 d) 2 e) 1 4. For which x-value(s) does f (x) = ln( x) x 2 have a horizontal tangent line? (2 pts.) a) 1 2 b) 1 2 c) 1 2 d) 1 2 e) 1 2, 1 2 Page 3 of 12

4 5. The graph of f (x) is given below. Choose the answer that represents the graph of the derivative of f (x). a) b) c) d) e) There is intentionally no choice (e). Page 4 of 12

5 6. A particle moves according to a law of motion s = f (t) = t 3 12t t, t > 0, where ( ) = 3t 2 24t + 36 = 3( t 6) ( t 2) m/s. s in meters and t in seconds. Clearly, v t Choose the expression that would help find the total distance traveled during the first 9 seconds. a) s(2) s(0) + s(6) s(2) + s(9) s(6) b) s(2) s(0) + s(6) s(2) + s(9) s(6) c) s(9) s(0) d) s(9) s(0) e) ( s(2) s(0) ) + ( s(6) s(2) ) + ( s(9) s(6) ) 7. Strontium-90 has a half-life of 28 days. A sample has a mass of 22 mg initially. Find a formula for the mass remaining after t days. a) m(t) = 11 7 t b) m(t) = 22e ln2t c) m(t) = 22e ln2 28 t ln1 14 d) m(t) = 22e t e) m(t) = t 8. Find the average rate of change of f (x) = 4x 2 5 over the interval 1 2,1. a) 2 b) 3 c) 5 d) 8 e) 4 Page 5 of 12

6 9. A cylindrical pool with radius 20 ft is being filled at a rate of 3 ft 3 / min. Use the fact that the volume of a cylinder is V = πr 2 h, where r is the radius and h is the height, to find how quickly the height of water is increasing. a) dh dt = 400π 3 b) dh dt = 3 20π c) dh dt = 3 20π d) dh dt = 3 40πh 400π e) dh dt = 3 400π 10. A bacteria culture initially contains 50 cells and exponentially grows. After 1 hour, the population contains 150 cells. Based on this information, the number of cells after t hours is B(t) = 50 3 t ( ) cells. Find when the population reaches 300 cells. a) t = ln2 hours b) t = hours c) t = ln6 ln 3 hours ln6 d) t = 3 hours e) t = 2 hours 11. If the tangent line to f (x) at the point 3,4 ( )? f 3 a) 3 b) 2 c) 1 d) 4 e) 1 ( ) also contains the point 2,1 ( ), what is Page 6 of 12

7 12. Choose the graph of a function f satisfying all of the given conditions. The dashed lines are asymptotes. lim fx ( ) =, lim fx ( ) =, lim fx ( ) = 2, lim fx ( ) = 2, f (0) = 2, lim fx ( ) DNE + x 3 x 3 x x 0 x 0 a) b) c) d) e) The Free Response section follows. PLEASE TURN OVER YOUR SCANTRON while you work on the Free Response questions. You are welcome to return to the Multiple Choice section at any time. Page 7 of 12

8 Free Response. The Free Response questions will count as 65% of the total grade. Read each question carefully. In order to receive full credit you must show legible and logical (relevant) justification which supports your final answer. Give answers as exact answers. You are NOT permitted to use a calculator on any portion of this test. 1. a. (6 pts.) Use the limit definition of the derivative to show that the derivative of the function f (x) = 3 5x is f (x) = 3 5x. 2 b. (4 pts.) Find the equation of the normal line to f (x) = 3 5x at x = (6 pts.) Find dy dx for sin4 y = y 2 + 4e 3x Page 8 of 12

9 3. Consider f ( x) = 5x3 + 9x 2 17x + 3 = 4x 3 4x ( ) a. Evaluate lim x 1 f x (x 1)(5x 1)(x + 3). 4x(x 1)(x +1) b. (2 pts.) Is f (x) continuous at x = 1? Support your answer by commenting on the three conditions for continuity. c. Calculate lim f ( x). x 0 + d. (2 pts.) What does part (c) tell us about f (x)? Choose one: There is a vertical asymptote at x = 0. There is a removable discontinuity at x = 0. fxis () continuous at x = 0. e. Does f (x) have any horizontal asymptotes? Support your answer with limit(s). If yes, state the equation of the horizontal asymptote. Page 9 of 12

10 4. (9 pts.) A kite, blown by the wind, moves horizontally (parallel to the ground) at a speed of 8 ft/s. During this time, the kite remains at 50 feet above the ground. At what rate is the angle between the string and the ground decreasing when 100 feet of string have been let out? Your answer must include a written description of the variables including units, a statement of what rate you are given, a general equation, the associated calculus and algebra, and a clearly stated solution with units. Work must be clear and logical to receive full points. θ s x 50 s: x: θ : t: Info you may find helpful while working this problem: ( ) 2 = Page 10 of 12

11 5. Compute the derivative of each of the following functions. Use appropriate notation to denote the derivative. DO NOT SIMPLIFY the derivatives. a. (5 pts.) f (x) = π 5 + log 3 ( x 2 +1) + 3 tan x b. (4 pts.) g(x) = tan 1 4x 5x 6. (5 pts.) Evaluate the following limit: lim x x x 2 7. (6 pts.) If f (x) = ( csc x) ex, compute f (x). Page 11 of 12

12 8. (6 pts.) If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t seconds is s(t) = 80t 16t 2 ft. What is the velocity of the ball when it is 96 feet above the ground on its way down? Be sure to include units with your answer. 9. (1 pt.) Check to make sure your Scantron form meets the following criteria. If any of the items are NOT satisfied when your Scantron is handed in and/or when your Scantron is processed one point will be subtracted from your test total. My scantron: is bubbled with firm marks so that the form can be machine read; is not damaged and has no stray marks (the form can be machine read); has 12 bubbled in answers; has MATH 1070 and my Section number written at the top; has my Instructor s name written at the top; has Test No. 1 written at the top; has Test Version A both written at the top and bubbled in below my CUID; and shows my correct CUID both written and bubbled in (bubble in a 0 in place of the C). Page 12 of 12

MthSc 107 Test 1 Spring 2013 Version A Student s Printed Name: CUID:

MthSc 107 Test 1 Spring 2013 Version A Student s Printed Name: CUID: Student s Printed Name: CUID: Instructor: Section # : You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or