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 any technology on either portion 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 for the free response portion of the test, 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 1 8 6a 6 2a 3 6b 6 Points Earned 2b 3 7a 2 3 5 7b 3 4a 3 8 1 4b 5 Free Response 52 5 7 Multiple Choice 48 Test Total 100 Page 1 of 12
Multiple Choice. There are 18 multiple choice questions. Each question is worth 2 3 points and has one correct answer. The multiple choice problems will count as 48% of the total grade. Use a number 2 pencil and bubble in the letter of your response on the scantron sheet for problems 1 18. 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 a function f x (2 pts.) lim f x x 3 lim f x x ( ) such that: ( ) = 3 lim 5 f ( x ) = x 2 + ( ) = 1 f ( 2) and f ( 3) are both undefined. How many of the following statements are TRUE? f x ( ) has a removable discontinuity at x = 3. ( ) has a removable discontinuity at x = 1. ( ) has a removable discontinuity at x = 2. f x f x a) 1 b) 3 c) 2 d) 0 2. Find the average rate of change of f (x) = 3 x 2 over 4,7. a) 1 3 b) 3 10 c) 7 d) 9 10 3. Let k ( x) = x3 + 4x 2. Find dk 2x dx k=2 a) 16 b) 14 c) 4 d) 6 Page 2 of 12
4. Identify the coordinates for any point(s) where y = x 2 + 2x 8 has a horizontal tangent. a) ( 1,0 ) b) ( 1, 9 ) c) ( 4,0) and ( 2,0) d) no horizontal tangents 5. A man throws a heavy rock straight up with an initial velocity of 16 ft/sec. The rock 2 reaches a height of st () = 16t 16t ft after t seconds. How high does the rock go? a) 16 ft b) ½ ft c) 32 ft d) 4 ft 6. Evaluate lim h 0 ( x + h) 2 x 2. h a) h b) 2x c) 0 d) 2xh + h 2 Use the following table of values to answer questions 7-8. 7. Let h x = f g x. Find h! 2. x f x f! x g x g! x 1 3 7 2 4 2 1 8 1 5 a) 32 b) 40 c) 35 d) 28 8. Let kx ( ) = x f( x). Find k '(2). a) 18 b) 8 c) 17 d) 16 Page 3 of 12
9. 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) Page 4 of 12
10. 4 Find the derivative of y = (3x + 1). 5 a) c) dy dx = 60 b) (3x + 1) 6 dy dx = 60 d) (3x + 1) 6 dy dx = dy dx = 20 (3x +1) 4 20 (3x + 1) 6 The graph of the derivative, f '( x ), is shown to the right. Use this graph to answer the next two questions (11-12). 11. On what interval is the graph of f( x ) increasing? (2 pts.) a) (, ) b) ( 3, ) c) ( 1, 5 ) d) (,3) 12. On what interval is the graph of f( x ) concave down? (2 pts.) a) (,3) b) ( 1, 5 ) c) (, ) d) ( 3, ) Page 5 of 12
Use the following graph of f x to answer questions 13 16. 13. For what values of x is f x not differentiable? (2 pts.) a) 8, 5, 2, 0 b) 8, 5, 0 only c) 8, 2, 0 only d) x = 8, 0 only ( ) 14. lim f x x 0 (2 pts.) a) 6 b) 5 c) 4 d) dne 15. Determine from the graph whether f x has any absolute extrema on 10,10. (2 pts.) a) Absolute minimum and maximum b) Absolute maximum only c) Absolute minimum only d) No absolute extrema 16. How many of the following statements are TRUE? lim f( x) = 2 x 1 f '(1) < 0 f ''(1) < 0 a) 3 b) 0 c) 1 d) 2 Page 6 of 12
17. Determine the derivative of y = cos3x + 1 4cos x. a) y = sin3x + 4tan 2 x b) y = 3sin3x + 1 sec x tan x 4 c) y = sin3x + 1 4 tan2 x d) y = 3sin3x + 4sec x tan x 18. Given f x = 4x + 2a, x 3 3x!, what value of a will make f continuous at x = 3? 1, x > 3 a) 26 b) 19 c) 7 d) 14 Page 7 of 12
Free Response. The Free Response questions will count as 52% 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. (8 pts.) A small balloon is released at a point 80 feet away from an observer, who is on level ground. The balloon goes straight up at a rate of 8 feet per second. How fast is the distance from the observer to the balloon increasing when the balloon is 60 feet high? Your answer must include a well labeled picture, a written description of the variables including units (use t for time), a statement of what rate(s) you are given, what rate you want, a general equation, the associated calculus and algebra, and a clearly stated solution with units. x 80 h x: true distance between observer and balloon (ft) h: height of balloon (ft) t: time (sec) Given: dh = 8 ft/sec General equation: h 2 + 80 2 = x 2 Derivative: 2h dh h dh = 2x dx = x dx Specific instant: 100 60 80 dx Want: =? when h = 60 60(8) = 100 dx 480 = 100 dx dx = 4.8 ft/sec General picture (the only numbers labeled should be constants) 0.5 variables described 1 Units on variable 0.5 what rates are given 0.25 What rate wanted 0.25 general equation 1 BELOW HERE FOLLOW THEIR WORK UNLESS TRIVIAL Derivative of general equation 1.5 Specific instant 1 Substitutions 0.5 Solved 1 Sentence conclusion (mathematical or English) 0.5 Only earned if correct Page 8 of 12
2. (3 pts. each) Find the indicated limit. Work must be shown on each limit. x 2 + 2x a. lim x 3 + x 2 x 6 = lim ( x + 2 ) x 3 + x 3 x 3 + ( )( x + 2) = lim x ( x 3) = 3 small + = + Factor, and simplify 1 Work showing constant num, small den 1 Correct answer 1-0.5 poor notation b. lim x x 2 + 2x x 2 x 6 1 x 2 1 x 2 2 1+ = lim x x 1 1 x 6 = 1+ 0 1 0 0 = 1 x 2 Fractional work 1.5 Individual limits 0.5 Correct answer 1-0.5 poor notation 3. (5 pts.) Use implicit differentiation to find dy dx if cos y + x2 y = 5x. sin y dy dx + x2 dy dx + 2xy = 5 sin y dy dx + x2 dy = 5 2xy dx dy ( sin y + x2 ) = 5 2xy dx dy dx = 5 2xy sin y + x 2 Points Awarded Implicit derivative cosy 1 Implicit derivative product 1.5 Implicit 2x 1 Solve for dy/dx 1.5 Incorrect implicit derivative should be followed through unless trivial -1.5 f * g -5 no dy/dx and therefore nothing to solve for -1 dy/dx = at the beginning of their implicit line -1 poor notation Page 9 of 12
4. (3, 5 pts. respectively) Find the derivative of the following functions. Use appropriate notation to denote your derivative. DO NOT SIMPLIFY. f b. a. f (x) = tan x x 5 +1 ( )sec 2 x tan x 5x 4 x x 5 +1 ( ) = x5 +1 ( ) 2 ( ) y = sin ( 7 3x 2 5x) + 3 = sin 3x 2 7 ( 5x) 3 4 2 x + y = 7 sin 3x 2 6 ( 5x) cos ( 3x 2 5x) 6x 5 2 x 1 4 ( ) + 3 2 1 4 5 x 4 5. (7 pts.) Use the limit definition of the derivative to show that the derivative of f ( x) = 5 4x is f (x) = f x ( ) = lim h 0 = lim h 0 = lim h 0 = lim h 0 2 5 4x. 5 4( x + h) 5 4x h ( ) ( 5 4x) ( ) + 5 4x 5 4 x + h h( 5 4 x + h ) 4h h( 5 4( x + h) + 5 4x ) 5 4 x + h ( ) + 5 4x ( ) + 5 4x 5 4 x + h 4 5 4x 4h + 5 4x = 4 2 5 4x = 2 5 4x Derivative top 1 Derivative bottom 0.5 Quotient rule used correctly 1.5-2 bad version of quotient rule except -3 f /g -1 lack of derivative notation -0.5 poor notation -1 simplifies and it is incorrect Points Awarded First outside 1 Derivative of sine 1.5 Derivative of second inside 1 Derivative second term 1.5-3 der. outside at der. of inside -2 lack of derivative notation -1.5 faulty algebra changed problem -1 simplifies and it is incorrect Correctly shows implicitly/explicitly limit definition 0.5 Correctly uses this function in limit definition 0.5 Correctly uses conjugate 1 Correctly expands numerator 1.5 Correctly simplifies numerator 1 Correctly simplifies h/h 1 Correctly evaluates limit as h goes to 0 1.5-1 lack of limits more than once, -2 for not saying limit at all -1 lack of equals more than once -1 for leaving "lim" in the last statement. -1 small algebraic errors Page 10 of 12
6. Use the first derivative of f( x ), f '(x) = 4x 3 20x 2, to answer the following: a. (6 pts.) Identify the intervals on which f( x ) is increasing and decreasing. Give the value of x at any local extrema, identifying each as a local maximum or local minimum. Put your answers in the blanks below. (If there is no max or min say none.) f x ( ) = 4x 3 20x 2 f (x) und? never f (x) = 0? 4x 3 20x 2 = 0 ( ) = 0 4x 2 x 5 x = 0, x = 5 + f (x) 0 5 =0 and solved 1 correct sign chart 1.5 Undefined and solve questions help give partial credit but are not required for full credit -6 for entire problem for treating f as f -1 copy error -0.5 notation errors -0.5 combining decreasing intervals Increasing: ( 5, ) 1 pt Decreasing:_ (,0), ( 0,5) 1 pt x-value at local maximum: none 0.5 pts x-value at local minimum: 5 1 pt b. (6 pts.) Identify the intervals on which f( x ) is concave up and concave down. Give the value of x at any points of inflection. Put your answers in the blanks below. (If there is no inflection point say none.) f x ( ) = 12x 2 40x f (x) und? never f (x) = 0? 12x 2 40x = 0 ( ) = 0 4x 3x 10 x = 0, x = 10 3 + + 10 f (x) 0 3 =0 and solved 1.5 correct sign chart 1.5 Undefined and solve questions help give partial credit but are not required for full credit 10 Concave Up: _(,0), 3, _1 pt Concave Down: 0,10 3 1 pt x-value at point(s) of inflection: _0, 10 3 1 pt_ Page 11 of 12
7. Consider the function f x = 2x! 15x! + 7. a. (2 pts.) Find the critical points for the function on, ( ) = 6x 2 30x ( x) und? never ( x) = 0? 6x 2 30x = 0 6x( x 5) = 0 f x f f x = 0, x = 5 ( ). right or wrong (-1/2 for extra information that is incorrect) b. Find the absolute minimum and maximum values of f x = 2x! 15x! + 7 on the interval 1,2. x f x 1 0 7 ( ) 2 15+ 7 = 10 2 16 60 + 7 = 37 Correct x values in table 0.5 Each calculation 0.5 each Correct answer 0.5 each Undefined and solve questions help give partial credit but are not required for full credit -0.5 states point instead of max/min values 5 not included in comparison because not in interval 1,2 Absolute maximum value: 7 Absolute minimum value: 37 8. (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 18 bubbled in answers; has MthSc 107 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. Page 12 of 12