Student s Printed Name: Instructor: XID: C Section: No questions will be answered during this exam. If you consider a question to be ambiguous, state your assumptions in the margin and do the best you can to provide the correct answer. Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use a textbook, notes, cell phone, computer, or any other technology on any portion of this test. All devices must be turned off and stored away while you are in the testing room. During this test, any kind of communication with any person other than the instructor or a designated proctor is understood to be a violation of academic integrity. No part of this test may be removed from the examination room. Read each question carefully. To receive full credit for the free response portion of the test, you must: 1. Show legible, logical, and relevant justification which supports your final answer.. Use complete and correct mathematical notation. 3. Include proper units wherever appropriate. 4. Give answers as exact values whenever possible. You have 90 minutes to complete the entire test. Do not write below this line. Free Response Problem Possible Earned Free Response Problem Possible Earned 1.a. 4 4. 8 1.b. 4 5.a. 1.c. 4 5.b. 4 1.d. 4 5.c. 4.a. 6. (Scantron) 1.b. 3.c. 3 3.a. 4 3.b. Free Response 5 3.c. 1 Multiple Choice 48 3.d. Test Total 100 Version A KEY Page 1 of 14
Multiple Choice: There are 16 multiple choice questions. Each question is worth 3 points and has one correct answer. The multiple choice problems will be 48% of the total grade. Circle your choice on your test paper and bubble the corresponding answer on your Scantron. Any questions involving inverse trigonometric functions should be answered based on the domain restrictions for trigonometric functions used in Section 1.6. 1. Find the derivative of y sin ( 1 3x ). y 6x ( ) 1+ 3x y 6x 1 x C) y y 6x ( ) 1 3x 6x ( ) 1 3x 1. The radius of a sphere is measured at 5 inches, with a possible error of inch. Use differentials 0 to approximate the maximum possible error in calculating the surface area of the sphere. Hint: The formula for the surface area of a sphere is S 4π r. C) 0 π in 4 π in 100 π in π in Version A KEY Page of 14
3. Find all values of x on the graph of 3 x f ( x) at which the tangent line is horizontal. x 3 9 x 0, 9 x, 3 C) x 0, 3 9 x 0,, 3 4. Find the locations of the absolute extrema of 4 3 F( x) 3x 4x 8 on the interval [, ]. Maximum at x, minimum at x 1 Maximum at x, minimum at x 1 C) Maximum at x, minimum at x 0 Maximum at x, minimum at x 5. Use the table to evaluate [ f ( x) g( x) ] 14 14 C) 1 d x 3 x 1 x x 3 x 4 f (x) 3 5 f (x) 6 1 4 1 g(x) 6 4 3 g (x) 4 5 3 7. Version A KEY Page 3 of 14
y t t + 4. 9 5 6. Find the derivative of ( ) 3 dt + + + t 8 ( t 5 4) ( 9t 5 3t 36) dt + + 3t 8 ( t 5 4) ( 8t 5 5) 7t t 4 dt + 8 5 C) ( ) dt + + 1t 8 ( t 5 4) ( t 5 3) NOTE: All students were given credit for this problem. 7. Evaluate d y for x + y 5 at the point (, 1). C) d y (,1) d y (,1) d y (,1) 5 0 5 d y (,1) 5 8 8. Find the derivative of 5 1 y sin x + cot x. C) y 5csc x cot x sec x + y 5csc x cot x sec x y 5csc x cot x csc x y 5cos x csc x Version A KEY Page 4 of 14
9. Find the derivative of (ln3 t) y 8. C) dt dt ln8 8 (ln 3 t) t 3ln8 8 (ln3 t) t dt 8(ln 3t ) dt 3ln8 t 10. Use implicit differentiation to find for xy+ x+ y x y. C) xy + y x y x xy y x y + x xy y x y x + + 1 1 xy y x y + x + 1 1 d u 11. Suppose u and v are differentiable functions of x. Find the value of at x 1. v u(1) 5, u (1) 5, v(1) 6, v (1) 10 3 5 9 C) 13 36 Version A KEY Page 5 of 14
1. Determine from the graph whether the function has any absolute extrema on the interval [0, 3.5]. Absolute minimum only Absolute maximum only C) Absolute minimum and absolute maximum No absolute extrema 13. Evaluate lim x 0 5 4sin x. x 0 C) 5 4 4 5 Does Not Exist (DNE) 14. Find the linearization L(x) of 3x f ( x) centered at 1. x L( x) 3x + 1 L( x) 6x + 3 C) L( x) 6x + 6 L( x) 3x + 17 Version A KEY Page 6 of 14
15. Use the table to evaluate [ f ( g( x)) ] 3 8 C) 1 40 d x 4 x 3 x 4 f (x) 1 3 f (x) 8 g(x) 16 3 g (x) 5 4. 16. Find the derivative of y ln( cos( lnθ) ). C) dθ dθ dθ dθ tan( lnθ) tan( lnθ) tan( lnθ) θ tan lnθ ( ) θ Version A KEY Page 7 of 14
Free Response: The Free Response questions will be 5% of the total grade. Read each question carefully. To receive full credit, you must show legible, logical, and relevant justification which supports your final answer. Give answers as exact values. Questions involving inverse trigonometric functions should be answered based on the domain restrictions in Section 1.6. 1. (16 pts.) Find the derivative of each of the following functions. Do not simplify. SHOW ALL WORK to receive full credit. {Simplification provided for information only.} Guidelines: (ALL PARTS) Subtract ½ point for incorrect simplification (even though simplification is not required) a. (4 pts.) ln x y x + 6 1 ( x 6) (ln x ) 1 x + y ( x + 6) x + 6 x ln x x( x + 6) b. (4 pts.) y sec( 5t + 6) [ ] Applies the quotient rule points Finds the derivative of the numerator Finds the derivative of the denominator Subtract points for not using the quotient rule, but finding the correct derivative or each factor Award full credit for converting quotient into a product and correctly applying the product rule 1 ( ( ) ( )) ( ) ( t + ) ( t + ) 1/ y sec 5t + 6 tan 5t + 6 5t + 6 5 5sec 5 6 tan 5 6 5t + 6 Applies the chain rule and finds the derivative of the outside function, i.e. takes the points derivative of sec(function of t) leaving the inside function unchanged Finds the derivative of the inside function applying the chain rule again, i.e. takes the 1.5 points derivative of (function of t)^(1/) leaving the inside function unchanged Finds the derivative of the most inside function 0.5 point Subtract points for applying the chain rule incorrectly by substituting the derivative inside instead of multiplying by the derivative Version A KEY Page 8 of 14
1. (16 pts.) Find the derivative of each of the following functions. Do not simplify. (Continued) Guidelines: (ALL PARTS) Subtract ½ point for incorrect simplification (even though simplification is not required) 1 c. (4 pts.) g( x) tan ( 5x) 1 g ( x) 5 1 + (5 x) 5 1+ 5x Applies the chain rule and finds the derivative of the outside function, i.e. takes the points derivative of tan 1 (function of x) leaving the inside unchanged Finds the derivative of the inside function points Subtract points for applying the chain rule incorrectly by substituting the derivative inside instead of multiplying by the derivative d. (4 pts.) f (x) e (x3 5x) 3 ( x 5x) f ( x) e 6x 5 ( ) ( ) ( 3 x 5x { 6x 5 e ) } Applies the chain rule and finds the derivative of the outside function, i.e. takes the points derivative of e^(function of x) leaving the inside unchanged Finds the derivative of the inside function points Subtract points for applying the chain rule incorrectly by substituting the derivative inside instead of multiplying by the derivative Version A KEY Page 9 of 14
. (8 pts.) The driver of a car traveling at 60 ft/sec suddenly applies the brakes. The position of the car is s( t) 60t 3t, t seconds after the driver applies the brakes until the car stops. a. ( pts.) Find the velocity, v (t), and the acceleration, a (t), of the car at any time t. v( t) s ( t) 60 6 t ft/sec a( t) v ( t) 6 ft/sec Differentiates s(t) to find v(t) Differentiates v(t) to find a(t) Units are not necessary Subtract ½ point for missing or incorrect derivative notation b. (3 pts.) How many seconds after the driver applies the brakes does the car come to a stop? v( t) 0 when 60 6t 0 t 10 sec Sets velocity equal to zero points Solves for time t Subtract ½ point for missing or incorrect units If velocity function found in part a is used in part b, follow work and award up to full credit c. (3 pts.) How far does the car travel before it stops? s (10) 60(10) 3(10) 600 300 300 ft Substitutes the value of time t (found in part b) into the function s(t) Calculates the distance (½ pt for number, ½ pt for units) If time found in part b is used in part c, follow work and award up to full credit points Version A KEY Page 10 of 14
3. (9 pts.) Consider the equation 3x y π cos y 4π. a. (4 pts.) Find. Use implicit differentiation. d d ( 3x y π cos y) ( 4π ) [ x] y x y ( 3x + π sin y) 6xy 6xy + 6 + 3 π ( sin ) 0 3x π sin y Explicitly or implicitly knows to take the derivative of both sides with respect to x Applies the product rule to differentiate the first term on the left-hand side Applies the chain rule to differentiate the second term on the left-hand side Differentiates the right-hand side Solves for / (Prime notation okay) Subtract ½ point for notation errors with a maximum deduction of b. ( pts.) Find the slope of the curve at (1, π). 6(1)( π ) 6π π π 3(1) + π sin( π ) 3 + π (0) (1, ) If derivative in part a is incorrect due to multiple egregious differentiation and algebra errors, work should NOT be followed through parts b d. 0.5 point 0.5 point Substitutes given point, following work from part a Evaluates the derivative Subtract ½ point for notation if the point is substituted into the expression for the derivative without appropriate notation for evaluating the derivative at a point Subtract for substituting values into opposite variables Subtract for evaluating sin(π) incorrectly c. (1 pt.) Find the equation of the tangent line to the curve at (1, π). y π π ( x 1) y π x + 3π States any form of the equation of the tangent line, following work from parts a and b Subtract ½ point for incorrectly simplifying a correct equation of the tangent line d. ( pts.) Find the equation of the normal line to the curve at (1, π). 1 1 π 1 y π ( x 1) y x + π π π Explicitly or implicitly finds the slope of the normal line, following work States any form of the equation of the normal line, following work Subtract ½ point for incorrect sign of slope of the normal line Subtract ½ point for incorrectly simplifying a correct equation of the normal line Version A KEY Page 11 of 14
4. (8 pts.) Use logarithmic differentiation to find for 4 y (sin x) x. ln y ln(sin x) 4x ln y 4x ln(sin x) d d ( ln y) ( 4x ln(sin x) ) 1 1 [4]ln(sin x) + 4x cos x y sin x y[ 4 ln(sin x) + 4x cot x] 4 x (sin x) 4 ln(sin x) + 4x cot x [ ] 4 x[ ] { 4(sin x) ln(sin x) + x cot x } Takes the natural log (ln) of both sides Simplifies the right-hand side using properties of logs Finds the derivative of the left-hand side using the chain rule points Finds the derivative of the right-hand side using the product rule points Solves for / Substitutes the expression for y back into the solution No credit awarded if the natural log is not used Maximum of half-credit (4 points) awarded if the natural log is applied to only one side of the equation Subtract points for taking the derivative of one side of the equation and not the other in the same step, then taking the derivative of the other side at a later step, regardless of whether the derivatives are correct or not Subtract 4 points for incorrect use of logarithm properties (egregious algebra errors) Version A KEY Page 1 of 14
5. (10 pts.) Boris was very excited about a big snowstorm and decided to make a large snowball, which was, of course, a perfect sphere. Unfortunately, it warmed up the next day and the snowball 3 started to melt at a rate of 5 in /min. 4 3 The volume of a sphere is V π r and the surface area of a sphere is 3 a. ( pts.) Identify known rate(s). Use t for time. Use derivative notation. dv 3 5 in / min dt A r 4π. Recognizes the given rate as the rate of change of the volume with respect to time States the rate of change for dv/dt (Units are not necessary.) Subtract ½ point for incorrect sign of rate of change Incorrect sign should be followed through work so that no further deductions are made for this error. Subtract ½ point for the use of V instead of dv/dt due to ambiguity b. (4 pts.) How fast was the radius changing when the radius of the snowball was 10 inches? d d 4 3 ( V ) π r dt dt 3 dv dr 4π r dt dt dr dr 5 1 When r 10 in, 5 4 π (10) in/min dt dt 400π 80π Takes the derivative of both sides of the volume formula w.r.t. time t (1 pt. per side) Substitutes the given information into the result and solves for dr/dt Subtract ½ point for notation errors with a maximum deduction of Subtract ½ point for the use of r instead of dr/dt due to ambiguity Subtract ½ point for missing or incorrect units points points c. (4 pts.) How fast was the surface area of the snowball changing then? d d ( ( 4π r ) dt dt da dr 8π r dt dt da 1 When r 10 in, 8 π (10) 1 in /min dt 80π Takes the derivative of both sides of the volume formula w.r.t. time t (1 pt. per side) Substitutes the given information into the result and solves for da/dt Subtract ½ point for notation errors with a maximum deduction of Subtract ½ point for the use of A instead of da/dt due to ambiguity Subtract ½ point for missing or incorrect units points points Version A KEY Page 13 of 14
Scantron (1 pt.) My Scantron: 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. 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 16 bubbled in answers; has MATH 1060 and my section number written at the top; has my instructor s last name written at the top; has Test No. written at the top; has the correct test version written at the top and bubbled in below my XID; shows my correct XID both written and bubbled in; Bubble a zero for the leading C in your XID. Please read and sign the honor pledge below. On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test. Student s Signature: Version A KEY Page 14 of 14