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1 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. Student s Printed Name: Instructor: XID: C Section: No questions will be answered during this eam. If ou consider a question to be ambiguous, state our assumptions in the margin and do the best ou can to provide the correct answer. Instructions: You are not permitted to use a calculator on an portion of this test. You are not allowed to use a tetbook, notes, cell phone, computer, or an other technolog on an portion of this test. All devices must be turned off and stored awa while ou are in the testing room. During this test, an kind of communication with an person other than the instructor or a designated proctor is understood to be a violation of academic integrit. No part of this test ma be removed from the eamination room. Read each question carefull. To receive full credit for the free response portion of the test, ou must:. Show legible, logical, and relevant justification which supports our final answer.. Use complete and correct mathematical notation.. Include proper units wherever appropriate. 4. Give answers as eact 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.a. 4 4.a. 4.b. 4 4.b..a. 5 4.c. 4.b a. 6 6.a. 4.b. 4 6.b. 6.c (Scantron) Free Response 58 Multiple Choice 4 Test Total 00 Version A KEY Page of

2 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. Multiple Choice: There are 4 multiple choice questions. Each question is worth points and has one correct answer. The multiple choice problems will be 4% of the total grade. Circle our choice on our test paper and bubble the corresponding answer on our Scantron. An questions involving inverse trigonometric functions should be answered based on the domain restrictions for trigonometric functions used in Section.4.. Evaluate 5sin 6. lim C) 0 0. Use Implicit Differentiation to find d given sec( ). sec( ) tan( ) cos( )cot( ) d sec( ) tan( ) cos( )cot( ) d sec( ) tan( ) C) ( )sec( ) tan( ) d + sec( ) tan( ) cos( )cot( ) d sec( ) tan( ) Version A KEY Page of

3 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. d. d. Use the table to evaluate [ f ( g( )) ] 6 0 C) f () f () 6 8 g() 4 g () Find all values of on the graph of g( ) 8 4 at which the tangent line is horizontal., 0, C) 0,, 0,, Find the derivative of ( ) d ( ) d ( ) d 5 C) ( ) d ( ) 9 Version A KEY Page of

4 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. 6. Find the derivative of tan ( ). C) ( + ) Find the derivative of + where n is a positive integer. n n n e n ln + e n n n n n C) ln n n + n n n n e 8. Differentiate. ( ) ln ln C) ( )ln ( ) 9. Find the equation of the tangent line to the graph of ( ) + C) + sin + at 0. Version A KEY Page 4 of

5 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. 0. Find d d given + 4. C) d d d d 4 d d d d Use Logarithmic Differentiation to find the derivative of e ( + 5) 7. C) d d d e 7 e ( 5) ( + ) 6 e ( 5) d + 5. Consider the cost function C( ) , (C() is $ when items produced). Determine the average cost and marginal cost when 000 items are produced. Average cost is $.0 per item and marginal cost is $.00 per item. Average cost is $0.60 per item and marginal cost is $0.0 per item. C) Average cost is $0.0 per item and marginal cost is $0.00 per item. Average cost is $00.00 per item and marginal cost is $00.00 per item. Version A KEY Page 5 of

6 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections... Let h( ) be an everwhere differentiable function where f ( ) h( ). g( ) Use the table to find the slope of the line tangent to h( ) at. 4 f () f () 6 8 g() 4 g () m tan m tan 6 m C) tan 7 6 m tan 6 4. An observer stands 00 meters from the launch site of a hot-air balloon at an elevation equal to the elevation of the launch site. The balloon rises verticall at a constant rate of 4 m/s. The figure below shows the geometr of the launch. As the balloon rises, its distance from the ground and its angle of elevation Ɵ change simultaneousl. Write an equation epressing the relationship between these variables. 00 cosθ tanθ 00 C) sinθ tanθ Version A KEY Page 6 of

7 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. Free Response: The Free Response questions will be 58% of the total grade. Read each question carefull. To receive full credit, ou must show legible, logical, and relevant justification which supports our final answer. Give answers as eact values. Questions involving inverse trigonometric functions should be answered based on the domain restrictions in Section.4.. (8 pts.) Find the derivative of each of the following functions. Assume g () is a differentiable function wherever it appears. DO NOT TRY TO SIMPLIFY YOUR ANSWERS. Guidelines: (ALL PARTS) Subtract ½ point for incorrect simplification (even though simplification is not required) a. (4 pts.) g 5 g /5 g g 5 5 g /5 Applies the chain rule and find the derivative of the outside function, i.e. takes the points derivative using the power rule leaving the inside function unchanged Finds the derivative of the inside function appling the chain rule again, i.e. take the derivative of g leaving the inside unchanged Finds the derivative of the most inside function Subtract points for appling the chain rule incorrectl b substituting the derivative inside instead of multipling b the derivative b. (4 pts.) tan e g ( ) sec ( ) ( ) tan tan e g + e g tan tan { e g( )sec + e g ( ) } Applies the product rule points Finds the derivative of the st factor (appling the chain rule) Finds the derivative of the nd factor Subtract points for not using the product rule, but finding the correct derivative or each factor Version A KEY Page 7 of

8 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections... (0 pts.) Find the derivative of each of the following functions. a. (5 pts.) f ( ) ln + 4 Simplif our answer to f ( ) 4. ( )( + ) ( )( ) ( ) d ln d ( )( ) ( ) ( ) ( ) Applies the chain rule and finds the derivative of the outside function, i.e. takes the points derivative of ln (function of ) leaving the inside unchanged Finds the derivative of the inside function using the quotient rule points Simplifies the result to verif the given equation Subtract points for appling the chain rule incorrectl b substituting the derivative inside instead of multipling b the derivative b. (5 pts.) f π for 0 < < Simplif our answer to f ( ). ( ) sin (cos ) d d ( sin (cos ) ) cos ( ) ( sin ) sin sin sin cos sin sin Applies the chain rule and finds the derivative of the outside function, i.e. takes the points derivative of sin - (function of ) leaving the inside unchanged Finds the derivative of the inside function points Uses the Pthagorean Identit to simplif the result and verif the given equation Subtract points for appling the chain rule incorrectl b substituting the derivative inside instead of multipling b the derivative Version A KEY Page 8 of

9 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections... (0 pts.) Consider the equation + 5. a. (6 pts.) Use Implicit Differentiation to find d. d d ( + ) ( 5) d d + () + 0 d d If derivative in part a is incorrect due to multiple egregious differentiation ( + ) and algebra errors, work should NOT d be followed through part b. d + Eplicitl or implicitl knows to take the derivative of both sides with respect to Takes the derivative of the first term on the left-hand side Applies the product rule to differentiate the second term on the left-hand side Differentiates the right-hand side Solves for /d (Prime notation oka) Subtract ½ point for notation errors with a maimum deduction of points b. (4 pts.) Find the equation of the tangent line to the curve at the point (4, ). m tan d (4,) 9 ( 4) Substitutes given point, following work from part a Evaluates the derivative to find the slope of the tangent line States an form of the equation of the tangent line, following work points Subtract ½ point for notation if the point is substituted into the epression for the derivative without appropriate notation for evaluating the derivative at a point Subtract for substituting values into opposite variables Subtract ½ point for incorrectl simplifing a correct equation of the tangent line Version A KEY Page 9 of

10 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. 4. (0 pts.) The position of an object moving horizontall after t seconds is given b the function s 48t t for t > 0, where s is measured in feet, with s > 0 corresponding to positions right of the origin. a. (4 pts.) Determine the value(s) of t when the object is stationar and the time intervals when the object is moving to the left and moving to the right. v( t) s ( t) 48 t ft/sec v t ( ) 0 when 48 t 0 v( t) > 0 when 0 < t < 4 v( t) < 0 when t > 4 (6 t ) 0 t 4 or t 4 seconds Object is stationar when t 4 seconds. Object is moving to the left on time interval(s) (4, ) seconds and moving to the right on time interval(s) (0, 4) seconds. Differentiates the position function to find the velocit function Sets velocit equal to zero and finds time t when object is stationar States interval when object is moving to the left States interval when object is moving to the right Subtract ½ point for including an endpoint in an interval b. ( pts.) Determine the velocit and acceleration of the object at t seconds. v() 48 () 48 6 ft/sec a( t) v ( t) 6 t ft/sec a() 6() ft/sec Evaluates the velocit function at t Differentiates the velocit function to find the acceleration function and evaluates at t Subtract ½ point for missing or incorrect units c. (4 pts.) Determine the time intervals when the object is speeding up and slowing down. v( t) > 0, a( t) < 0 when 0 < t < 4 v( t) < 0, a( t) < 0 when t > 4 Object is speeding up on time interval(s) (4, ) seconds and slowing down on time interval(s) (0, 4) seconds. States the interval when object is speeding up States the interval when object is slowing down Subtract ½ point for including an endpoint in an interval points points Version A KEY Page 0 of

11 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. 5. (9 pts.) Find d for ( + 7) There are methods for solving this problem. Either will receive full credit when correct. Method I. Logarithmic Differentiation ( + ) ( + ) d ( ln ) ( ln ( + 7) ) ln ln 7 ln ln 7 d d d () ln ( + 7) + 6 d + 7 ln ( + 7) + d ( + 7) ln ( + 7) 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 points chain rule Finds the derivative of the right-hand side using the points product rule (and chain rule) Solves for /d Substitutes the epression for back into the solution No credit awarded if the natural log is not used Maimum of half-credit (4 points) awarded if the natural log is applied to onl 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) Method II. Epress ( + ) 7 as ( + ) e ln 7 ln( + 7) ln( + 7) e e d ln( + 7) ( e d d ) ln( + 7) e () ln ( + 7) ( + 7) ln ( + 7) Epresses the right-hand side using e and ln points Simplifies the eponent using properties of logs Finds the derivative of the outside function using the points chain rule Finds the derivative of the inside function (eponent) points using the product rule (and chain rule) Substitutes the epression for back into the solution No credit awarded if the natural log is not used Subtract 4 points for incorrect use of logarithm properties (egregious algebra errors) Version A KEY Page of

12 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. 6. (0 pts.) A street light is fastened to the top of a 5-ft-high pole. If a 5-ft-tall woman walks awa from the pole in a straight line over level ground at a rate of ft/s, how fast is the length of her shadow changing when she is ft awa from the pole? Variables: * Use for the distance from the woman to the pole (in ft). * Use for the length of the woman s shadow (in ft). * Use t for time (in s). a. (4 pts.) Identif the rate(s) that are given and the rate that is to be determined, using derivative notation and units. Make a sketch to organize the variables and given information. ft/s Given rate(s): dt d? ft/s when ft Rate to be determined: dt Sketch: (Include labels for variables and an constants.) 5 ft 5 ft points * for right triangle figure * for labels b. ( pts.) Write an equation that epresses the basic relationship among the variables and b similar triangles points for setting up equation c. (4 pts.) At what rate is the length of the woman s shadow changing when she is ft awa from the pole? (Use Calculus to answer the question. Include units with our answer.) + + d dt dt When ft, d dt d dt () ft/sec Takes the derivative of both sides w.r.t. time t points Substitutes the given information into the result and points solves for d/dt Subtract ½ point for notation errors with a maimum deduction of Subtract ½ point for the use of instead of d/dt due to ambiguit Subtract ½ point for missing or incorrect units Version A KEY Page of

13 MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. Scantron ( pt.) M Scantron: Check to make sure our Scantron form meets the following criteria. If an of the items are NOT satisfied when our Scantron is handed in and/or when our Scantron is processed one point will be subtracted from our test total. is bubbled with firm marks so that the form can be machine read; is not damaged and has no stra marks (the form can be machine read); has 4 bubbled in answers; has MATH 060 and m section number written at the top; has m 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 m XID; shows m correct XID both written and bubbled in; Bubble a zero for the leading C in our XID. Please read and sign the honor pledge below. On m honor, I have neither given nor received inappropriate or unauthorized information at an time before or during this test. Student s Signature: Version A KEY Page of