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, smart watch 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 a 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 9 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 5 4 6 1b 4 5 6 1c 4 6 6 2a 5 7 6 2b 6 8 1 Points Earned 2c 4 Free Response 6 3 7 Multiple Choice 4 Test Total 1 Page 1 of 13
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 4% 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. Which of the following represent the equation of the tangent line of the curve h(x) = e ' at the point x = ln (3)? a) y = (3 + ln(3))x 3 b) y = ln(3) x + 3 c) y = 3x 3ln (3) d) y = 3x + 3 3ln (3) 2. Graphed below is the tangent line at the point P(x, y ) for some differentiable function y = f(x). Also marked is a second point on the curve Q(x 5, y 5 ). Which of the following choices best represents the slope of this line? a) 7(' 8)97(' : ) ' 8 9' : b) f ; (x ) c) f ;; (x ) d) f(x ) 3. Determine the interval(s) on which g(x) = '= >?'>@ is continuous. ' 8 9A a) (, 3) ( 3,3) (3, ) b) (, 3) (3, ) c) ( 3,3) d) (3, ) Page 2 of 13
4. Which conclusion can be drawn from the following two its? f(x) = ' 5 G ' H f(x) = 5 3 a) f has a removable discontinuity at x = J K b) f has a horizontal asymptote of y = J and a vertical asymptote of x = 2 K c) f has a slant asymptote and a vertical asymptote of x = J K d) f has a horizontal asymptote of y = 2 and a vertical asymptote of x = J K 5. Find M M' sino x. a) MP = 4(sin x cos x )K M' b) MP M' = 4 sink x c) MP M' = 4 sink x cos x d) MP = 4 sin x cos x M' 6. Where does f(x) = cos 2x 2x have horizontal tangent line(s) on [,2π]? a) x = KX O b) x = π c) x = X O d) x = JX O Page 3 of 13
7 Consider the table of data below. x 1.99 1.999 2.1 2.1 x 5 4 x 2 3.99 3.999 4.1 4.1 Which of the following is true? I. 9O does not exist ' 5 '95 II. 9O ' 5 '95 III. There is a removable discontinuity at x = 2 a) I and III only b) II and III only c) II only d) none of I, II, or III 8. Evaluate the following it. sec(x + h) sec x Y Z h a) sec 5 x b) csc 5 x c) csc x cot x d) sec x tan x Page 4 of 13
9. Below is the graph of a differentiable function. Which of the following choices best represents the graph of its first derivative? a) b) c) d) Page 5 of 13
1. Find the average rate of change of f(x) = cos x on the interval ^, X K _. (3 pts.) a) K 5X b) K 5 c) K X d) 5 11. Consider the graph of the following function. Considering only the visible region of the graph, at how many points is the function not continuous and not differentiable? a) not continuous at five points, not differentiable at five points b) not continuous at five points, not differentiable at two points c) not continuous at two points, not differentiable at five points d) not continuous at three points, not differentiable at three points Page 6 of 13
12. Let the displacement of a Yo-Yo at time t be given by d(t) = cos(πt) + 1 feet. The Yo-Yo s velocity is maximal when its acceleration is zero. Find where the Yo- Yo s acceleration is zero. a) t = O + k, k: non-negative integer b) t = X + k, k: non-negative integer X 8 5 c) t = 5 + k, k: non-negative integer d) t = k, k: non-negative integer Use the following table to answer the next two questions: x 1 2 3 4 5 f(x) 3 5 1 f (x) 5 2-5 -8-1 g(x) 4 5 1 3 2 g (x) 2 1 2 15 2 13. Evaluate M M' (3 pts.) 7(') k(') l 'm5. a)? J b) O J c) 5 J d) K J 14. Let h(x) = fng(x)o, find h ; (3). (3 pts.) a) 1 b) 2 c) 5 d) 1 Page 7 of 13
15. Given f(x) = 4x + 2 we know that f(x) = 6. Thus, given ε >, there exists ' δ > such that x 1 < δ implies f(x) 6 < ε. Find δ. a) 4ε b) ε c) 5 ε d) O ε 16. Find the derivative of y = lnne wxyz: ' o. a) ' 8 9 c) e y (wxyz: ') ' 8 9 b) d) { }~z: >' 8 >' 8 17. Find the slope of the tangent to f(x) = cos 9 ' O ƒ at 2, X K ƒ. (3 pts.) a) O 5 b) 5 c) 5 d) O 5 18. Consider the following it. 3x 3 ' K x 3 Which of the following represents the it after one step (using a conjugate)? a) K'9K ' K ('9K)( K'>K) c) K'9A ' K ('9K)( K'>K) b) K'>A ' K ('9K)( K'9K) K'9A d) ' K '9K 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 8 of 13
Free Response. The Free Response questions will count as 6% 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. All work to be graded must be on this test paper. 1. Find the it of each of the following function, if it exists. Show all work. You will not be given any credit for using L Hopital s Rule to find the its. a. (5 pts.) ' 9 ' ' 8 9 x K 1 ' x 5 1 = ' x 5 + x + 1 = = 3 ' x + 1 2 b. (4 pts.) 8 (X') ' 5 z ('95) 8 (x 1)(x 5 + x + 1) (x 1)(x + 1) cos 5 (πx) ' 5 z (x 2) 5 = 1 small pos = + OR 8 (X') = (X') ' 5 z ('95) 8 ' 5 z '95 ƒ5 = ( ) 5 = + c. (4 pts.) 3x 5 2x ' H 2x K 3 K'8 95' ' H 5' 9K 1 x K = 1 ' H x K 3 x 2 x 5 2 3 = + x K x yš ƒ5 = 2 = Factoring numerator 2 Factoring denominator 1 Simplification 1 Correctly finds it (award only if 1 correct) -1 including / in work -1 missing equals (-.5 for only one missing equals) -1 missing its (-.5 for only one missing) Excused 1 missing = sign ONLY if it was at a line break -.5 it notation carried too far Shows evidence of understanding that 2 denominator gets small Recognizes it goes to positive infinity 2 (correct work must be shown; no credit for correct answer with no proper work) -.5 to -1 poor notation (-1 if is ever written in denominator; -2 number/=dne -2 wrong sign -3 wrong sign and small not talked about -2 mixing x with small neg in denominator -2 never plugs in x=value to numerator (and has x even after it is dropped) -1 specific x-values plugged in -4 small/small -2 incorrect sign on infinity (no other mistakes) -1 it notation used more than appropriate proper WORK for it to infinity 2 Finds the overall it (award only if 2 correct, correct work must be shown; no credit for correct answers with no proper work) -.5 to -1 poor notation (-.5 missing it in one step) -1 no it notation -.5 if not multiplying by a form of 1-1 if one sign or term is incorrect -1 if one step is incorrect equated to another step -4 for small in work Page 9 of 13
2. Find the first derivative of the following functions. You do NOT need to simplify. a. (5 pts.) f(t) = 6 t 4t K csc(2t) + 9t + π J. f ; (t) = 6 1 2 t9 5 4t K ( 2 csc 2t cot 2t) 4 3t 5 csc(2t) + 9 + = K + 8tK csc 2t cot 2t 12t 5 csc(2t) + 9 Derivative of first term 1 Product rule for second term 2 Derivative of third term 1 Zero derivative of constant term 1-1 incorrectly labeling derivative or not labeling derivative -.5 notational errors -1/2 each deducted for egregious simplification errors b. (6 pts.) y = π?' + log? ne '8 > o + ln(4sin 9 x) y ; = ln π π?' 6 + 1 ln 6 e '8 > e'8 > 2x + = ln π π?' 6 + 2x ln 6 + 1 sin 9 x 1 x 5 1 4sin 9 x 4 1 1 x 5 Derivative of first term 2 Derivative of second term 2 Derivative of third term 2-1 incorrectly labeling derivative or not labeling derivative -.5 notational errors -1/2 each deducted for egregious simplification errors c. (4 pts.) y = 2 log J (3ln x ) y ; = 2 1 ln 5 1 3 ln x 3 x = 2 x ln 5 ln x Hold multiplier 1 Derivative of outside log function, keep 1 Derivative of first inside, keep... 1 Derivative of second inside 1-1 incorrectly labeling derivative or not labeling derivative -.5 notational errors -1/2 each deducted for egregious simplification errors Page 1 of 13
3. a. (4 pts.) Apply the it definition of the derivative to find the derivative of f(x) = x 5 + x. f(x + h) f(x) Y Z h (x + h) 5 + x + h (x 5 + x) = Y Z h = Y Z x 5 + 2xh + h 5 + x + h x 5 x h = Y Z 2xh + h 5 + h h You must show your work. Use a derivative rule will result in no credit. h(2x + h + 1) = = (2x + h + 1) = 2x + 1 Y Z h Y Z b. (3 pts.) Find the equation of the normal line to f(x) = x 5 + x at the x = 1. f(1) = 1 + 1 = 2 f ; (x) = 2x + 1 m = f ; (1) = 3 m = 1 3 So the equation of the normal line to f(x) at x = 1 is y 2 = (x 1) K 4. (6 pts.) Suppose that while visiting planet Vogon, Arthur Dent gets the idea to play golf. If he tees off from the edge of a cliff, the height of the golf ball at time t seconds is given by s(t) = 25 + 2t 5t 5 meters. Be sure to include units on your final answers. a. Find the time at which the golf ball hits the ground. An object hits the ground when s(t) = 25 + 2t 5t 5 = 5(t 4t 5) = 5(t 5)(t + 1) = t = 5 and t = 1 Golf ball hits ground at 5 sec. b. What is the velocity of the ball at this point? v(t) = 1t + 2 v(5) = 5 + 2 = 3 m/sec Points Awarded Proper substitution into definition 1 Expands numerator 1 Correctly einates / issue 1 Correct it (award only if correct) 1-3 definition reversed -2 pt for no it notation or incorrect it notation -1 pt for poor it notation -1/2 one or two missing equals -1 more than two missing equals -1/2 pt for it notation carried too far -2 dropping the denominator -1 each skipping a step -1/2 too many steps in one -1 final answer wrong sign work that jumped from a correct point to the correct answer without showing work in between - lose points for whatever steps were not shown y value.5 Slope of tangent.5 Slope of normal 1 Putting line together properly 1-3 answer is not linear ie used derivative function instead of number for slope of tangent -1 simplified normal line incorrectly -2 took derivative again and it is incorrect s(t)= 1 Correctly solves for t 2 Einates solution that is too late.5 Units.5 v(t).5 Evaluates v(t) at time from a.5 Correct answer (award only if.5 correct) Units.5-3 factor = non-zero -2 no velocity function in part b Page 11 of 13
5. (6 pts.) Find the first derivative of e 5'P = cos y + 7. Implicit differentiation: e 5'P ^2x MP MP + 2y_ = sin y + M' M' Solve for dy/dx: 2xe 5'P dy dx + 2ye5'P = sin y dy dx 2xe 5'P dy dy + sin y dx dx = 2ye5'P dy dx [2xe5'P + sin y] = 2ye 5'P dy dx = 2ye5'P 2xe 5'P + sin y Derivative of left side outside.5 Chain rule of left side 1.5 Distribute on left.5 Right side first term 1 Right side second term.5 Group to two sides.5 Factor 1 isolate dy/dx.5 OK to use y -.5 notational error -5 no derivative or implicit not used -.5 to -1.5 algebra errors OR ln e 5'P = ln(cos y + 7) 2xy = ln(cos y + 7) Implicit differentiation: 2x MP + 2y = MP ( sin y) M' P>@ M' Solve for dy/dx: 2x dy dx + sin y cos y + 7 dy dx = 2y dy sin y dx š2x + cos y + 7 = 2y dy dx cos y + 14 + sin y š2x = 2y cos y + 7 dy 2y(cos y + 7) = dx 2x cos y + 14 + sin y Page 12 of 13
6. (6 pts.) Find MP M' for the curve y = (tan x)œ Note: your final answer should be simplified and in terms of x only. y = (tan x) O ' ln y = ln(tan x) O ' ln y = 4 ln tan x x Implicit differentiation MP = O P M' ' wxy ' sec5 x + O ' 8 ln tan x = y ^O Š 8 ' O y wxy ' _ M' ' wxy ' ' 8 MP = (tan M' x)œ ^O Š 8 ' O y wxy ' _ ' wxy ' ' 8 So MP Points Awarded Natural log both sides, 1.5 simplify the right hand side Derivative of lny 1 Derivative of right side 2 Solve for dy/dx 1 Substitute y.5-1 missing dy/dx on left which reappears -.5 notation errors -6 attempting to use the power rule -1 mixed line w derivative = function 7. (6 pts.) For the curve x K + y K = 1, it can be shown that MP M' = '8 P 8. Find M8 P M' 8 for xk + y K = 1. Start with MP = '8 M' P 8 d 5 y dx 5 = 2xy5 x 5 dy 2y dx (y 5 ) 5 = = 2xy 5 + 2x 5 y x5 y 5 ž y O 2xy 5 2xO y y O = 2x(yK + x K ) y J = 2xyK 2x O y J = 2x y J Quotient rule 2.5 Substitute dy/dx 1 Simplify (substitution of original not 2.5 required) OK to use y -.5 notational error -5 no derivative or implicit not used -1. for no bottom squared -3. for quotient rule but no implicit -3. for product rule but no implicit -2. for incorrect product rule applied -1.5 for backwards quotient rule -1. for bad algebra in simplifying 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 MATH 17 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 B both written at the top and bubbled in below my CUID; and shows my correct CUID both written and bubbled in (bubble in a in place of the C). Page 13 of 13