Student s Printed Name: _Key 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:. Show legible and logical (relevant) justification which supports your final answer. 2. Use complete and correct mathematical notation.. 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 Earned Free Response Problem Possible 6 5 6 2 8 6a 5 a 2 6b 4 b,c,d 7 6c 7 e,f 6 7 g,h 6 Free Response 67 i 2 Multiple Choice Earned 4 7 Test Total 00 Page of 5
Multiple Choice. There are 2 multiple choice questions. Each question is worth 2 points and has one correct answer. The multiple choice problems will count as % of the total grade. Use a number 2 pencil and bubble in the letter of your response on the scantron sheet for problems 2. 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.. Approximate the change in the volume of a sphere when its radius increases from 4 r = 0 inches to r = 0.2 inches. Note: The volume of a sphere is V() r = π r, where r is the radius in inches. ( pts.) a) 400π in b) 2000π in c) 800π in d) 80 π in e) 80π in 2. If you want to use a linear approximation to estimate 27 which function and center ( pts.) should you use? a) f ( x) = x a = 25 b) f ( x) = x + 27 a = 0 c) f ( x) = x a = 25 d) f ( x) = x + 25 a = 25 e) f ( x) = x a = 27 Page 2 of 5
. Identify the location of any absolute extrema. (2 pts.) a) x =, 5 only b) x = 2, only c) x =,, 2,, 5 only d) x =, 2, 5 only e) x = 2 only 4. ( pts.) Let f (x) = x2 + 2 x. Choose the equation of the slant asymptote. a) y = x b) y = x c) y = x d) y = x 2 e) y = x + 5. Evaluate ( pts.) π x 2 x!! sec x. a) b) c) 2 d) 0 e) DNE Page of 5
6. ( pts.) Find dy given y = 4cot 4 x4. a) dy = 4csc ( 2 x )dx b) dy = 4sec ( 2 x )dx c) dy = 4x sec 2 e) dy = 4x csc 2 4 x4 4 x4 dx d) dy = 4x csc 2 dx 4 x4 dx 7. Let f x (2 pts.) f ( ) be an everywhere differentiable function such that f ( x) > 0 for x <, ( x) < 0 for x >, and f ( ) = 0. What is true about the point, f ( ) a) It is a relative minimum. b) It is a hole. c) It is a relative maximum. d) It is a cusp. e) It is a point of inflection. ( )? 8. (2 pts.) Does the Mean Value Theorem apply to the function f (x) = x2 x 6 on the interval e x [ 2,]? a) The Mean Value Theorem does not apply because f ( 2) f ( ). b) The Mean Value Theorem does not apply because the function is not continuous over the interval 2, [ ]. c) Yes, the Mean Value Theorem applies. d) There is not enough information given to determine if the Mean Value Theorem applies. e) The Mean Value Theorem does not apply because f ( ) 0. Page 4 of 5
The derivative graph f ( x) of a function f x questions #9-0. ( ) is given below. Use it to answer the following 9. Identify the interval(s) over which f ( x) is increasing. ( pts.) a), b) 2, 4 4 c) (, 2),, d) (, ) 4 e), 0. Identify the interval(s) over which f ( x) is concave down. ( pts.) a) 2, 4 b), c), 4 d) (, 2),, e), ( ) Page 5 of 5
. ( pts.) x 4 Evaluate x 0 2. x a) 4 b) 0 c) ln2 d) 4 ln2 e) DNE 2. Find the x -value(s) that satisfy the conclusion of the Mean Value Theorem for the ( pts.) function f ( x) = 5 x2 4 x on the interval [ 0,5 ] 5. a) x = 2 b) x = c) x = d) x = 5 2 e) x = 5 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 6 of 5
Free Response. The Free Response questions will count as 67% 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.. Consider the function f (x) = 2x a. (4 pts.) Find the linearization L(x) of f (x) = 2x You must show step by step work. f (0) = 2(0) = f ( x) = ( 2x 2 ) 2 f ( 0) = 2 = 2 y = 2 ( x 0) L(x) = 2 x + 2 ( ) = ( 2x) 2 b. (2 pts.) Use the linearization to estimate at a = 0. 0.8 = 2 0. Your final answer should be presented as a fraction containing integers only. L 0 = 2 0 + = 5 + = 4 5 function value 0.5 derivative.5 derivative evaluated presented as linearization Evaluating linearization in part b 2 -.5 derivative incorrect - derivative instead of slope into L(x) -0.5 poor or missing notation, ex: der=eval -0.5 minor math error -0.5 to - lack of simplification or error in -2 evaluating the linearization at inappropriate values simplification depending on severity Page 7 of 5
2. In preparation for laying new tile, Michelle measures the floor of a large conference room and finds it to be square, measuring 00 ft. by 00 ft. Suppose her measurements have a possible error of at most 6 inches (0.5 ft.). a. (4 pts.) Use differentials to estimate the maximum error in the area da due to the possible error in measurement. Be sure to include units on your answer. A x ( ) = da = 2xdx x = 00 dx = 2 da = 2( 00) 2 = 00 ft 2 b. (4 pts.) Calculate the relative error in the area. State the percentage error. da A = 2xdx = 2dx x 2 da A = 2 00 = 00 = 0.0 % general area formula 0.5 Differential Instant substituted Answer (only if correct) unit 0.5-0.5 to - for notation errors - wrong formula, but rest correct based on general relative error 0.5 Simplified instant substituted 0.5 Relative error (only if correct) Percentage error -0.5 to - for notation errors Relative error: 0.0 Percentage error: % Page 8 of 5
. Consider the function f (x) = x2 4 Use this information to answer the following.. Given that f ( x) = 8x 2 and 4 f x ( ) a. (2 pts.) State the domain of f (x). Also state the x -intercept(s). 4 0 x 2,2 ( ). ( 4) ( ) = 8 x2 + 4 4 = 0 x = 0 Domain: (, 2) ( 2,2) 2, ( ) each right or wrong x -intercept(s) stated as point(s): 0,0 ( ) b. ( pts.) Where does f (x) have vertical asymptote(s)? State the equation for the asymptote(s). Support your answer with it(s). x 2 4 = 4 = + small pos x 2 + 4 = 4 = small neg x 2 4 = 4 = small neg x 2 + 4 = 4 = + small pos State vertical asymptote as 0.5 each equation Setting up a correct it 0.5 each Correct evaluation of it 0.5 each - did not choose a side for the it(s) -0.5 to - notational errors student does NOT have to show both its for each point if 4 are shown: forgive wrong, -0.5 for two wrong sign y = 2 and y = 2 are both vertical asymptotes c. (2 pts.) Does f (x) have a horizontal asymptote? If so, state the equation for the asymptote. Support your answer with it(s). x 4 = x 4 = 0 = y = is horizontal asymptote State horizontal asymptote as equation Limit (do not have to show work) -/2 for x= instead of y= d. (2 pts.) Does f (x) have a slant asymptote? If yes, find it and give the equation for the slant asymptote. Show your work you need not use its. If the function f (x) has no slant asymptote, just say so. NO slant asymptote right or wrong Page 9 of 5
continued: x2 ( ) = 8x Still considering the function f (x) = 4. Given f x 2 and ( 4) f x e. (4 pts.) Identify the intervals on which f (x) is increasing and decreasing. f ( x) und? x = 2,2 f ( x) = 0? 8x = 0 (, 2) f x x = 0 ( ) > 0 ( 2, 0) f ( x) > 0 ( 0,2) f ( x) < 0 ( 2, ) f ( x) < 0 ( ). ( 4) ( ) = 8 x2 + 4 Increasing: _(, 2), ( 2,0) Decreasing: ( 0,2), ( 2, ) f. (2 pts.) State the coordinates ( x - and y -values) of any local extrema, identifying each as a local maximum or local minimum. If the function f (x) has no local extreme values, just say so. local max of 0 at x = 0 g. (4 pts.) Identify the intervals on which f (x) is concave up and concave down. f ( x) und? x = 2,2 f ( x) = 0? + 4 = 0 (, 2) f x never ( ) > 0 ( 2,2) f ( x) < 0 ( 2, ) f ( x) > 0 critical numbers 0.5 correct sign chart (or similar 0.5 work) Summary -0.5 notation error -0.5 did not show breaks in domain Point MAX - extra extrema critical numbers 0.5 correct sign chart 0.5 Summary -0.5 more than one notation error -4 any attempt that used a non-real solution to x^2+4=0 Concave Up: (, 2), ( 2, ) Concave Down: ( 2,2) h. (2 pts.) State the coordinates ( x - and y -values) of any inflection points. If the function f (x) has no inflection points, just say so. None right or wrong Page 0 of 5
continued: Still considering the function f (x) = 4. Given f x x2 ( ) = 8x 2 and ( 4) f x ( ). ( 4) ( ) = 8 x2 + 4 i. (2 pts.) Choose the correct graph of fx. () (a) right or wrong (b) (c) (d) (e) Page of 5
4. (7 pts.) Find the absolute maximum and minimum values of f ( x) = x + x on the interval 2,5. f ( x) = f ( x) und? x = 0 0 is not in the interval f ( x) = 0? x = 0 2 = = x =, is not in the interval x 2 f ( x) + = + = 2 2 + 2 = 2 + 6 = 6.5 5 5 + 5 = 5.6 Absolute Maximum of _ 6.5 at 2. Derivative Where derivative undefined? 0.5 Where derivative =0?.5 Correct x values in table 0.5 Each calculation 0.5 each Correct answer 0.5 each blank -0.5 each checking additional values Absolute Minimum of 2 at. Page 2 of 5
5. Consider the function f (x) = x 9x +4 on the interval,. a. ( pts.) Verify that the function satisfies the hypotheses (necessary conditions) of Rolle s Theorem on the given interval. State a reason that the function satisfies each condition, not just yes. continuous because polynomial differentiable because polynomial ( ) = 27 27 + 27 +4 = f f () = 27 27 27 +4 = ( ) = f ( ) f b. ( pts.) Find all value(s) of c that satisfy the conclusion of Rolle s Theorem for the function on the given interval. f ( x) = 6x 9 6x 9 = 0 ( ) = 0 2x ( )( x +) = 0 x x = x = does not qualify because it is an endpoint Continuity 0.5 Justification of continuity 0.5 Differentiability 0.5 Justification of 0.5 differentiability Endpoints equal Derivative Derivative = 0 0.5 Solved (follow work).5 Do NOT penalize for not einating the endpoint -0.5 poor or missing notation -0.5 minor math error no partial credit on aroc - trying to solve MVT not Rolle s Page of 5
6. Evaluate the following its. If a it does not exist, state DNE. Use of L Hopital s Rule must be indicated each time it is used, either symbolically or in words. Work must be show, NO shortcuts. a. (5 pts.) 5e x + 5x 5 x 0 5e x + 5x 5 L'H 5e x + 5 L'H +5e x = = x 0 x 0 2x x 0 2 b. (4 pts.) x x sin x ( ) = OR x sin x sin x L'H x ( ) 2x cos( x ) = 2 cos0 = 2 ( ) = ( x ) ( x +) x sin( x ) = x x sin x = 5 2 ( ) ( x +) x Correct derivatives on L H 2.5 Correct derivatives on 2 nd L H.5 Correct it (follow work shown) Indeterminate form is NOT required to be shown -5 using quotient rule -0.5 indeterminate form in line with the problem sentence - lack of L H notation where appropriate -0.5 too much L H notation -2 lack of it notation -0.5 too much it notation -2.5 lack of chain rule on first step created wrong answer -.5 exponent morphed to x instead of -x = 2 = 2 Correct derivative of numerator Correct derivative of denominator Correct it 2 Indeterminate form is NOT required to be shown -0.5 indeterminate form in line with the problem sentence - lack of L H notation where appropriate -0.5 too much L H notation - lack of it notation -0.5 too much it notation -4 using quotient rule Page 4 of 5
c. (7 pts.) x2 x x 0 + y = x ln y = ln x ln y = 2x ln x x 0 2ln x ln y = 2x ln x = + + x 0 x 0 + x L'H = x 0 + x2 x = y = eln y = e ln y x 0 + = e 0 = x 0 + x 0 + x 0 + 2 x = ( 2x) = 0 x 0 + y, lny, rearrange right side Start finding it of lny 0.5 Change form to be able to use L H Correct derivative of numerator 0.5 Correct derivative of denominator Simplify 0.5 Correct it lny Limit of original function w proper step notation Final answer (only if correct) 0.5 Indeterminate form is NOT required to be shown -0.5 indeterminate form in line with the problem sentence - lack of L H notation where appropriate -0.5 too much L H notation -2 lack of it notation -0.5 too much it notation 7. ( 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 2 bubbled in answers; has MATH 070 and my Section number written at the top; has my Instructor s name written at the top; has Test No. 2 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 5 of 5