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, 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.. 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 Earned Free Response Problem Possible 1a 5 5c 3 1b 4 5d 3 a 6 5e 3 b 5 5f 3 3a 6 5g 3 3b 4 6 1 4 8 Free Response 60 5a 3 Multiple Choice 40 Earned 5b 3 Test Total 100 Page 1 of 15
Multiple Choice. There are 17 multiple choice questions. Each question is worth 3 points and has one correct answer. The multiple choice problems will count as 40% of the total grade. Use a number pencil and bubble in the letter of your response on the scantron sheet for problems 1 17. 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. Find the derivative of tan xy = 5x!. (3 pts.) a) dy dx = b) dy dx c) dy dx d) dy dx 10x xsec xy y x 10x sec xy = xsec xy 10x ysec xy = xsec xy 10x sec xy = x y. x + 5x+ 6 Consider the function f( x) =. Which of the following statements x 4 describes the asymptotes of f( x )? a) f( x ) has vertical asymptotes at x =± and a horizontal asymptote at y = 1. b) f( x ) has a vertical asymptote at x = and a horizontal asymptote at y = 1. c) f( x ) has a vertical asymptote at x = and a slant asymptote at y = x. d) f( x ) has vertical asymptotes at x =± and a slant asymptote at y = x. 3. (3 pts.) Given f ( x) = x + 3, identify the interval on which f is concave down. x 5 b) ( 3, 5) a),5 c) ( 3, ) d) (, 3) Page of 15
4. The graph of a function is given. Choose the answer that represents the graph of its derivative. a. b. c. d. Page 3 of 15
5. Find lim sec x. a) 1 b) 0 c) d) x π + 6. Consider the function y = f(x) defined on the interval [a, b] shown in the figure below. Which of the following statements about the point c is/are true? I. The point c is a critical point of f(x). II. The point c is an inflection point of f(x). III. The point c is a location of a local minimum of f(x). IV. The point c is the location of an absolute minimum of f(x) on [a, b]. a) III and IV are true. b) I, II, III, and IV are all true. c) I, III, and IV are true. d) None of the statements are true. Page 4 of 15
A ball is thrown vertically upward on a planet. The height (in meters) of the ball is given by H t = t! + 7t + 30, where t is the number of seconds the ball is in the air. Use this information to answer questions #7-8. 7. What is the acceleration of the ball after two seconds? a) m/s! b) 9 m/s! c) 1 m/s! d) 4 m/s! 8. When does the ball reach its maximum height? a)!! sec b) 6 sec c) sec d)!! sec 9. Suppose that the functions f (x) and g(x) and their derivatives with respect to x (3 pts.) have the following values at the given values of x. x f (x) g(x) f (x) g (x) 6-3 9 3 1 4 6 5 4 3 3 5-4 If k(x) = f ( x ), find k (). 3x a) 1 4 b) c) 1 4 1 d) 3 4 Page 5 of 15
10. Using the graph of the derivative f! (x) shown below, identify the intervals on which the function f(x) is increasing and decreasing. a) Increasing on, 3 and 3, ; Decreasing on 3, 3 b) Increasing on, 0 and, ; Decreasing on, and 0, c) Increasing on 3, 3 ; Decreasing on, 3 and 3, d) Increasing on, and 0, ; Decreasing on, 0 and, 11. Suppose the population of bacteria growing in a petri dish is given by Pt () = t + t+ 1, where t is hours and P is billions of bacteria. Find the average rate of change of the population over the interval [0,]. (3 pts.) a) 0 billion/hour b) 6 billion/hour c) 4 billion/hour d) 8 billion/hour 1. (3 pts.) Find d y dx for y = 3x + π 4. a) c) d y dx = 3x b) d y dx = 4 3x 3 d y dx = 1 x 3 d) d y dx = 4 3x 3 +1π Page 6 of 15
13. Find the equation of the asymptote that describes the end behavior of the graph of x + x 7 f( x) =. x 7 a) y = x+ 16 b) y = x 16 c) y = x 1 d) y = 14. 4 x Find lim x 3 3 x x a) b) 4 3 c) d) 0 15. Does the graph of the function y = 4x + 8cos x have any horizontal tangents in the (3 pts.) interval 0 x π? If so, where? a) yes, at x = 7π 6, x = 11π 6 b) yes, at x = π 3, x = π 3 c) yes, at x = π 6, x = 5π 6 d) no Page 7 of 15
Consider the graph of f( x ) shown below. Use this graph to answer questions #16-17. 16. State all of the values of x at which f( x ) is continuous but not differentiable. a) x= 0, x=, x= 3 b) x = 3 c) x= 0, x= 3 d) x= 0, x= 17. Which of the following is true about lim f( x) and lim x 3 f (x)? a) lim f( x) = f(0) and lim x 3 f (x) does not exist b) lim f( x) does not exist and lim x 3 f (x) = 6 c) lim f( x) = f(0) and lim f (x) = 6 x 3 d) lim f( x) does not exist and lim x 3 f (x) does not exist. 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 15
Free Response. The Free Response questions will count as 60% 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. Find the limit of each of the following functions, if it exists. Show all work. You will not be given any credit for using L Hopital s Rule to find the limits. 5 5 + x a. (5 pts.) lim x 0 x 5 lim lim x 5+ x x b. (4 pts.) lim x 3x 7x + x 5+ 5+ x 5+ 5+ x = lim 5 (5+ x) x 5+ 5+ x 1 x 1 x = lim x 3x 7x + x 3 7 + 1 x = = lim 3 7 + 0 = 3 7 x x( 5+ 5+ x ) = lim 1 5+ 5+ x = 1 5+ 5 = 1 10 Awarded Multiplication by 1 w conjugate 0.5 Simplify numerator Cancellation 1.5 Correct answer 1-0.5 poor notation -1 incorrect sign on final answer Awarded Multiplication by 1 (can be implied) 0.5 Fractional work Individual limits 0.5 Correct answer 1-0.5 poor notation -4 Multiplication by (1/x^)/(1/sqrt(x^)) Page 9 of 15
. Find the first derivative of the following functions. Simplify completely. a. (6 pts.) y = sin ( 7 3x 5x) y = 7 sin 3x 6 ( 5x) cos ( 3x 5x) 6x 5 = ( 4x 35)sin ( 6 3x 5x)cos( 3x 5x) cot( 9θ ) + 9 b. (5 pts.) f (θ) = 9 cot( 9θ ) + 9 = 1 cot( 9θ ) +1 9 9 ( θ ) = 1 ( 9 csc 9θ ) 9 + 0 = csc ( 9θ ) f (θ) = f Awarded First outside Derivative of sine Derivative of second inside -1 power of trig misplaced - extra derivative -5 der. outside at der. of inside -6 product rule -1.5 faulty algebra changed problem 7(6x-5) is OK -1 der(sine)= -cosine - lack of derivative notation - incorrect simplification -0.5 poor notation Awarded Simplify (can be implied) 1 Trig derivative Chain rule 1 Simplify 0.5 Derivative of 1 0.5 - lack of derivative notation -0.5 poor notation -1 missing constant multiplier -5 cot(9x)=9cotx -0.5 wrong sign = 9 csc 9θ f θ ( 9) ( cot( 9θ ) + 9) 0 9 ( = 81csc 9θ ) = csc ( 9θ ) 81 Awarded Derivative of numerator Keep denominator 1 Derivative of denominator 1.5 Simplify 0.5-3 bad version of quotient rule except -5 f /g - lack of derivative notation -0.5 poor notation -1 wrong sign on trig Page 10 of 15
3. a. (6 pts.) Use the limit definition of the derivative to find the derivative of f( x) = 4x 3x. (No credit will be given for using derivative theorems in part a.) f x = lim 3( x + h) 4x 3x h 0 4 x + h 4 x + xh + h = lim h 0 h 3 x + h h 4x 3x = lim h 0 4x + 8xh + 4h 3x 3h 4x + 3x h = lim h 0 8xh + 4h 3h h = lim 8x + 4h 3 h 0 = 8x 3 b. (4 pts.) Find the equation of the line that is normal to f( x) = 4x 3x at x = 1. = 4 + 3 = 7 ( 1) = 8 3 = 11 f 1 m tan = f m nor = 1 11 = 1 11 equation of normal line: y 7 = 1 11 x +1 Awarded Correctly shows implicitly/explicitly limit 0.5 definition Correctly uses this function in limit definition 0.5 Correctly expands numerator 1.5 Correctly simplifies numerator 1.5 Correctly simplifies h/h 1 Correctly evaluates limit as h goes to 0 1-1 lack of limits more than once, - 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 -0.5 lack of parentheses on last limit Awarded Function value 1 Slope of tangent line 1 Slope of normal based on slope of tangent 1 Stuff plugged in properly to equation of line 1 - using solution to der=0 as slope of tangent Page 11 of 15
4. (8 pts.) A man six feet tall walks at the rate of 5 ft/sec toward a streetlight that is 16 ft above the ground. At what rate is the length of his shadow changing when he is 10 ft from the base of the light? 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 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. x: distance of man from light (ft) s: length of shadow (ft) t: time (sec) 16 Similar triangles 6 s = 16 x + s 6 x + s = 16s 6x + 6s = 16s 6x = 10s 6 dx dt x = 10 ds dt 30 = 10 ds dt ds = 3 ft/sec dt 6 s dx dt = 5 ft/sec Want ds =? when x = 10 dt Awarded General picture (the only numbers labeled should be constants) 0.5 variables described 0.5 Units on variable 0.5 what rates are given (sign must be correct) 0.5 general equation Derivative of general equation Substitutions 1 Solved 0.5 Sentence conclusion (mathematical or English) 0.5 Only earned if correct, must indicate decreasing If the student's general equation was such that it was ultimately not possible to solve for the rate of change of the shadow (e.g., using the Pythagorean Theorem), at most points could be awarded. The length of the man s shadow is decreasing by 3 ft/sec (when the man is any distance from the light) Page 1 of 15
5. Consider the function f x = x! + 4x! +. a. (3 pts.) Find the critical points for the function on,. f x (You only need to state the x -values.) = 4x 3 +1x f und? never f = 0? 4x 3 +1x = 0 = 0 4x x + 3 x = 0, x = 3 critical points are x = 3,0 b. (3 pts.) Identify the intervals on which f is increasing and decreasing. Use interval notation for your answers. c. - + + -3 0 Increasing: ( 3,0), ( 0, ) Decreasing: (, 3) c. (3 pts.) State the location and value of any local extrema, identifying each point as either a local maximum or a local minimum (if there are no local extrema, say none ). Local min of 5 at x = 3 f x Awarded Derivative 1 undefined 0.5 =0 and solved 1.5 Undefined question helps give partial credit but is not required for full credit Awarded work 1 summary Awarded Minimum 1 Location 1 Value 1 Follow work from b unless too simple -1 each extra answer Page 13 of 15
f x d. (3 pts.) Identify the intervals on which f is concave up or concave down. Use interval notation for your answers. = 1x + 4x f und? never f = 0? 1x + 4x = 0 = 0 1x x + x = 0, x = + - + - 0 f x Awarded work 1.5 summary 1.5 Concave Up: (, ), ( 0, ) Concave Down: (,0) e. (3 pts.) State the coordinates for any points of inflection. (if there are no inflection points, say none ). Inflection points at (, 14) and ( 0,) f. (3 pts.) Consider the function f x = x! + 4x! + on the interval 4,1. Determine the absolute minimum and absolute maximum value of the function on that closed interval. Show clear and logical organized work that supports your conclusion. to save you some calculation time f 4 x f x 3 5 0 4 1 7 = Absolute min of 5 at x = 3 Absolute max of 7 at x = 1 g. (3 pts.) Sketch f x = x! + 4x! + on the interval 4,1. Awarded Location (each) 0.5 y-value (each) 1 Follow work from d unless too simple -1 each extra answer Awarded Correct x values in table 0.5 Each calculation 0.5 each Correct answer 0.5 each -0.5 states point instead of max/min values -0.5 states x coordinate instead of max/min value -0.5 checking additional values Awarded Grade based on correct graph (not based on their incorrect info) Minimum 1 Increasing/decreasing 1 Concavity and inflection points 1-0.5 outside of stated interval Page 14 of 15
6. (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 17 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 15 of 15