Student s Printed Name: _KEY Grading Guidelines 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, smart watch, 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 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 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 1a 3 6b 6 1b 5 6c 7 2 5 7a&b 8 3a 5 8 6 3b 3 9 1 Points Earned 4 4 Free Response 64 5 6 Multiple Choice 36 6a 5 Test Total 100 Page 1 of 13
Multiple Choice. There are 15 multiple choice questions. Each question is worth 2 3 points and has one correct answer. The multiple choice problems will count as 36% of the total grade. Use a number 2 pencil and bubble in the letter of your response on the scantron sheet for problems 1 15. 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. If the tangent line to f(x) at the point ( 2,1) also passes through the point (0,7), what is f + ( 2)? a) -. b) 3 c) 4 d) 2 e) 1 2. Find the derivative of y = e 34567 (89). a) 8:;<=67 (>?) -AB9 > c) ln F - -A89 >G b) : >? 345(89) d) 2e 345(89) cos(2x) e) 2e 3456> (89) 3. Find f (B8) (x) of sin(x). (3 pts.) a) sin(x) b) cos(x) c) 0 d) sin(x) e) cos(x) Page 2 of 13
4. The graph of a function f(x) is given below. Select the graph of f (x). a) b) c) d) e) There is intentionally no choice e due to space. Page 3 of 13
5. If y = 4 is a horizontal asymptote of f(x), then a) lim 9 O f(x) = 4 b) f(4) = 0 c) lim 9 B f(x) = d) f(4) is undefined e) f + (x) = 4 6. On the interval [0,2π] at what values of x does f(x) = x 2 sin x + π have a (3 pts.) horizontal tangent line? a) π 3, WX Y, ZX Y, --X Y c) X Y, --X Y b) X Y, WX Y d) X [, BX [ e) X [, WX [ 7. Which of the following horizontal or vertical asymptotes does y = 8 + 3 have? 9A- (3 pts.) I. x = 1 II. x = 1 III. y = 2 IV. y = 3 V. x = 0 a) IV and V b) I and IV c) I, II, and V d) I and III e) II and III Page 4 of 13
8. If g(x) = f(cos 2x), find g (x). a) 2f(cos 2x) sin 2x b) 2f (cos 2x) sin 2x c) 2f (sin 2x) d) f (cos 2x) sin 2x e) 2f (cos x) sin x 9. Find ^_ ^9`([,[) for x[ + y [ = 6xy (3 pts.) a) 1 b) - [ c) 1 d) 0 e) None of the above. 10. Find the slope of the tangent line to f(x) = cot(3x) at x = X Y. (3 pts.) a) - [ b) - [ c) d) 3 e) 0 11. Suppose that the functions f(x) and g(x) and their derivatives with respect to x have the following values at the given values of x. x f(x) g(x) f (x) g (x) 3 1 4 6 5 4 3 3 5 4 If h(x) = f(g(x)), find h (4). a) 0 b) 18 c) 24 d) 2 e) 6 Page 5 of 13
Use the following graph of f(x) to help answer the next two questions. 12. Evaluate f + ( 1) + f (2). (3 pts.) a) W 8 b) 3 2 c) W [ d) 0 e) undefined 13. At how many points in the interval ( 1,3) is f NOT differentiable? a) 1 b) 4 c) 3 d) 0 e) 2 Page 6 of 13
14. Evaluate lim [345 [e e f e. a) 1 b) 0 c) 3 d) 9 e) DNE 15. Find the third derivative of f(x) = 3x B + 2x 8 + x a) 36x + 4 b) 36x 8 + 4 c) 72 d) 24x 8 + 2x e) 72x 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 13
Free Response. The Free Response questions will count as 64% 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 following limits. Work must be shown on the limits to earn full credit no shortcuts h9 a. (3 pts.) lim lim 9 AO = 9 AO B9 > AW 8x 4x 8 5 b. (5 pts.) lim 7 W9 lim? 9 AO 89l 9 > 7 l-? 1 x 8 = lim 1 9 AO x 8 W9 = 9 AO 89l 9 > l- = lim 8 x 4 5 x 8 = 0 4 0 = 0 W9 7? W 9 AO 89 7? l 9> 7 = lim = W = W l- 6 m? > 9 AO 8An-l 7 8A -lf? > Correct WORK for a limit going to 1.5 infinity Limit as go to infinity applied 0.5 Correct answer (award only if correct) 1-1 including inf/inf in work -0.5 missing equals -0.5 missing limits -0.5 limit notation carried too far -1.5 if correct limits, but student showed some work and used "degree comparison" with no algebraic support -3 if correct limits with no clear support for limits -3 if plugged in infinity 8A- = 5 Correct WORK for a limit going to 1.5 infinity Correct use of negative with root 1.5 function Limit as go to infinity applied 1 Correct answer (award only if correct) 1-1 including inf/inf in work -0.5 missing equals -0.5 missing limits -0.5 limit notation carried too far -1.5 if correct limits, but student showed some work and used "degree comparison" with no algebraic support -3 if correct limits with no clear support for limits -3 if plugged in infinity Page 8 of 13
2. (5 pts.) Consider the function f(x) = - 9 >. o(9)ao(a-) Use lim to help to find the slope of the tangent line to the graph of f(x) at x = 1. 9 A- 9l- Detailed limit work must be shown. Use of any shortcut derivatives rules will receive no credit. o(9)ao(a-) lim for f(x) = - 7? 9 A- 9l- 9 > is lim >A- 9 A- 9llim 9 A- 1 x 8 1 x + 1 = lim 9 A- So m rst = 2 1 x 8 x 8 (x + 1) = lim 9 A- (1 x)(1 + x) x 8 1 x + 1 1 x = lim 9 A- x 8 = 2 ( 1) 8 = 2 Set up for this function 0.5 Common denominator 1 Flip 1 Factor 1 Cancel 0.5 Limit evaluation (only award if correct) 1-1 including 0/0 in work -1 missing equals (-0.5 for only one missing equals) -1 missing limits (-0.5 for only one missing) Excused 1 missing = sign ONLY if it was at a line break -0.5 limit notation carried too far up to -1 for notational errors -0.5 lack of limit rewrite after cancellation -0.5-5 appeared due to lack of parentheses -2 set up limit but poor evalataion Page 9 of 13
3. Consider the function f(x) = x + 7 for x 7. a. (5 pts.) Use the limit definition of the derivative to find the derivative of f(x) = x + 7. Detailed limit work must be shown. Use of any shortcut derivatives rules will receive no credit. Hint: the derivative is f + (x) = - 8 9lZ. x + h + 7 x + 7 x + h + 7 + x + 7 lim e f h x + h + 7 + x + 7 x + h + 7 (x + 7) = lim e f hv x + h + 7 + x + 7w h = lim e f hv x + h + 7 + x + 7w = lim 1 e f x + h + 7 + x + 7 = 1 2 x + 7 Points Awarded Conjugate 1 Correctly simplifies numerator 2 Correctly eliminates 0/0 issue 1 Correct limit (award only if correct) 1-5 only set up, no work -3 definition reversed -2 pt for no limit notation -1 pt for poor limit notation -1/2 missing equals -1/2 pt for limit notation carried too far -2 dropping the denominator -1 each skipping a step -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 b. (3 pts.) Find the equation of the tangent line to f(x) = x + 7 at the point x = 3. f( 3) = 3 + 7 = 4 = 2 f + 1 ( 3) = 2 3 + 7 = 1 2 2 = 1 4 So equation of tangent line at x = 3 is y 2 = - (x + 3) B y value 1 Slope of tangent 1 Putting line together properly 1-3 answer is not linear ie used derivative function instead of number for slope of tangent -1 simplified tangent line incorrectly -3 took derivative again and it is incorrect Page 10 of 13
4. (4 pts.) At what values of the curve y = 5x + e 9 is the tangent line parallel to the line 16x + 2y = 4? 16x + 2y = 4 2y = 16x 4 y = 8x 2 y + = 5 + e 9 Want 5 + e 9 = 8 e 9 = 3 x = ln3 Derivative 1 5. (6 pts.) Use implicit differentiation to find the second derivative ^>_ 2e 8_ dy dx = 1 dy dx = 1 2e 8_ = 1 2 ea8_ ^9 > of e8_ = x. Small error 1, all others -2 Find slope of line 1 Derivative = slope of line 0.5 Solve 0.5 Award last point only if correct 1 Then ^>_ = - ^9 > 8 ea8_ ( 2) ^_ ^_ = ea8_ = ^9 ^9 ea8_ F - 8 ea8_ G = - 8 eab_ Derivative of left side 0.5 Derivative right side 0.5 isolate dy/dx 1 (follow work as long as implicit differentiation used) Second derivative with chain rule 2 Simplify 0.5 Sub dy/dx 1 Simplify again 0.5 OK to use y -1 misuse of ( ) or [ ] -0.5 each notational error -0.5 each minor algebra error Page 11 of 13
6. Find the first derivative of the following functions. Use proper notation to denote the derivative. DO NOT SIMPLIFY. z a. (5 pts.) y = secv 8xw + πe 9 + π 9 e z y = secv 8x z wtanv 8xw 1 8 3 (8x)A [ 8 + πe 9 + lnπ π 9 0 First derivative of first term 2 First derivative of second term 1 First derivative of third term 1 First derivative of fourth term 1-2 incorrectly labeling derivatives or not labeling derivatives b. (6 pts.) h(x) = 3 9l:6? > 9} x 8 W ~ 1 2 (csc - x)a 8 csc x cot x e A9 v csc x + e A9 w 2 [ 5 xa W h (x) = ~x 8 8 W c. (7 pts.) y = 6 W 5(BA9) + 6tan A- x + π : W 9 + sin8 x Keep bottom 0.5 Correct derivative of top first term 2 Correct derivative of top second term 1 Hold top 0.5 Correct derivative of bottom 1 Denominator squared 1-2 incorrectly labeling derivative or not labeling derivative -5 f /g if f and g correct -3 top of quotient rule backwards -3 denominator squared completely missing -0.5 each for missing parentheses y + = ln6 6 W 5(BA9) 5sec 8 (4 x) ( 1) + 6 + 0 + W -l9 > 9 > + 2 sin x cos x - First derivative of first term w chain rule 2 First derivative of second term 1 First derivative of third term 1 First derivative of fourth term 1 First derivative of fifth term w chain rule 2-2 incorrectly labeling derivatives or not labeling derivatives Page 12 of 13
7. Consider the function f(x) = 5 9 9. a. (3 pts.) Find ^ 5 9 of f(x) =. ^9 9 f + (x) = xsec8 x tan x x 8 b. (5 pts.) Write the equation of the normal line of f(x) = 5(9) at x = π. f + (π) = πsec8 π tan π π 8 = π( 1)8 0 π 8 = 1 π So m t ƒ = π Also f(π) = rstx X = 0 Then y 0 = π(x π) Keep bottom 0.5 Correct derivative of top 1 Hold top 0.5 Correct derivative of bottom 0.5 Denominator squared 0.5-1 incorrectly labeling derivative or not labeling derivative -3 f /g if f and g correct -2 top of quotient rule backwards -2 denominator squared completely missing -1 erroneous simplification 9 y value 1 Slope of tangent 1 Slope of normal 1 Putting line together properly 2-4 answer is not linear ie used derivative function instead of number for slope of tangent -1 simplified tangent line incorrectly -3 took derivative again and it is incorrect 8. (6 pts.) Use implicit differentiation to find the derivative of (x + y) 8 = 9> 2(x + y) - ~1 + dy dx = 1 4 2x 2(x + y) + 2(x + y) dy dx = 1 2 x 2(x + y) dy dx = 1 x 2x 2y 2 2(x + y) dy x 2y ^_ = A-.W9A8_ ^9 8(9l_) dx = 3 2 OR ^_ = 9 1 ^9 B(9l_) B Derivative of left side 2 Derivative right side 1 isolate dy/dx 3 (follow work as long as implicit differentiation used) OK to use y -1 misuse of ( ) or [ ] -0.5 each notational error -0.5 each minor algebra error 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 15 bubbled in answers; has MATH 1040 and my Section number written at the top; has my Instructor s name written at the top; has Test No. 3 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 13 of 13