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1 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., 4. Student s Printed Name: Instructor: CUID: Section: Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any tetbook, 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 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 eamination room. Read each question very carefully. In order to receive full credit for the free response portion of the test, you must:. Show legible, logical, and relevant justification which supports your final answer.. Use complete and correct mathematical notation.. Include proper units, if necessary. 4. Give eact 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 w rite below this line. Free Response Problem Possible Earned Free Response Problem Possible Earned.a. 8 5.a. 4.b. 8 5.b. 4.c. 8 5.c. 4.d a b. 5 Test Total a. 7 4.b. 8 Version A Page of

2 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., 4. Read each question carefully. In order to receive full credit you must show legible, logical, and relevant justification which supports your final answer. Give answers as eact values. You are NOT permitted to use a calculator on any portion of this test.. (8 pts. each) Find the indicated derivatives. Assume g() is a differentiable function. 5 a) Find f ( ) if f ( ) g(sec(7 )). f 5 ( ) g(sec(7 )) 5 f g(sec(7 )) sec(7 ) tan(7 ) (7 )(ln 7)() Derivative of g Derivative of secant function Derivative of eponential function points points points b) Find dy d if y tan( y) e. tan( y) e y dy y dy sec ( y) e ( y ) d d dy y dy sec ( y) sec ( y) e ( y ) d d dy y dy sec ( y) e ( y ) sec ( y) d d dy y sec ( y) e ( y ) sec ( y) d dy sec ( y) y d sec ( y) y e or dy sec ( y) y d y e sec ( y) Derivative of left side Derivative of right side Solves for dy d points points points Other equivalent forms of the answer obtained with logarithmic differentiation can get full credit. Version A Page of

3 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., 4. c) Find f ( ) if f ( ) ln sin(5 ) f( ) f( ) ln sin(5 ). Simplify your answer. sin(5 )( ln ) (cos(5 )(5)) sin (5 ) sin(5 ) sin(5 ) sin(5 )( ln ) (cos(5 )(5)) sin (5 ) sin(5 )(ln ) 5cos(5 ) sin(5 ) Alternative Solution: ln ln sin 5 f f ln cos55 sin 5 f ln 5cot 5 Work on Problem Derivative of natural log function Quotient Rule applied to argument for ln pt for derivative of sin 5 pts for derivative of pts for quotient rule Simplify No credit is awarded if a mistake leads to a derivative that cannot be simplified 5 Point Subtract / point for missing derivative notation Subtract / point for etraneous terms in the derivative Subtract if quotient rule is applied to sin 5 Work on Problem (Alternative Solution) Apply properties of logarithm pts for derivative of ln ln sin 5 pts for derivative of Simplify Second term must be simplified to 5cot 5 to receive credit 5 Point Version A Page of

4 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., 4. d) Find dy d if sin y (also written y sin (logaritmic differentiation) sin ln( y) ln( ) ln( ) (sin )ln y dy y d sin (ln ) dy sin (ln ) y d dy sin (ln ) d sin y arcsin ) Work on Problem Natural log of both sides No credit is awarded for the entire problem if the natural log is not taken for the right side Simplifies right side with log properties Derivative of left side Derivative of right side pt for product rule pt for derivative of ln pt for derivative of arcsin Solves for dy d Replaces y Subtract / point for missing derivative notation\ Subtract / point if derivative is taken for one side but not the other e.g. dy arcsin ln y d Subtract points for y arcsin ln, but then follow subsequent work Point Point points Version A Page 4 of

5 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., 4.. ( pts.) A modern art ehibit at a museum contains a steel sphere with a radius of 8 meters. a) (7 pts.) Heating causes the sphere to epand. Use a differential to estimate the change in the surface area of the sphere if heat causes the radius to increase from 8 meters to 8. meters. Note: The surface area S of a sphere of radius r is S 4 r. S 4 r ds 8 r dr ds (8 r) dr when r 8 and dr 0. ds (8 (8))(0.) 6.4 m Differentiates both sides Solves for differential ds Substitutes values for r and dr Calculates ds Units points points b) (5 pts.) Assume the paint on the surface of the sphere will crack if the volume of the sphere increases by 64 cubic meters. Use the differential to estimate the increase in radius that would cause the paint to begin to crack. 4 Note: The volume V of a sphere of radius r is V r. V 4 r dv (4 r ) dr dv 4 r dr when r 8 and dv (4 )(8 ) dr 64 (4 )(8 ) dr m 4 dr Differentiates both sides Solves for differential dv Substitutes values for r and dv Calculates dr Units s s Version A Page 5 of

MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., 4.. ( pts.) A tank in the shape of an inverted cone has a radius of 4 meters and a height of 5 meters.

6 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., 4.. ( pts.) A tank in the shape of an inverted cone has a radius of 4 meters and a height of 5 meters. It drains into a cylindrical tank with a radius of 4 meters and a height of 6 meters (see figure). a) (8 pts.) Consider the conical tank. When the height of water in the conical tank is eactly one meter, the height of the water is decreasing at 0.5 meters per hour. At what rate is water flowing from the conical tank at this moment? Note: The volume V of a cone is V r h V r h by similar triangles: r 4 4h r,so h 5 5 4h 6 V h h 5 75 differentiating with respect to t dv 6 ( dh h ) dt 75 dt dh when h and dt dv 6 (() ) dt 75 dv 8 m dt 5 hr Finds r=4h/5 Substitutes to get V in terms of h Derivative of left side Derivative of right side Substitutes values for h and dh/dt Solves for dv dt points points Version A Page 6 of

7 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., 4. b) (4 pts.) Consider the cylindrical tank. At what rate is the water level in the cylindrical tank changing when the water level in the conical tank is eactly one meter? (the same moment from part (a)) Note: The volume V of a cylinder is V r h V (4 ) note: the radius of the cylinder is constant V 6 h h differentiating with respect to t dv dh 6 dt dt dv 8 when h (in cone) and dt 5 note: water enters the cylinder at the rate it eits the cone 8 dh 6 5 dt dh = m dt 50 hr Finds V in terms of h; r constant Derivative of left side Derivative of right side Substitutes value for dv/dt Solves for dh dt / point / point Version A Page 7 of

8 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., (5 pts.) Let f and g be differentiable functions. Use the values in the table below to answer the questions that follow. =0 = = = f () f () g() g () ( ) a) (7 pts.) Let h( ) f ( ) e g. Is h increasing or decreasing at? Show work that supports your answer. h( ) f ( ) e g( ) h f g f g ( ) g ( ) ( ) ( ) e ( ) e ( ) h f g f g() g() () () e () e () e ()( e )( ) ( e )() h h () e 0 increasing at. Finds h'( ) Substitutes values simplifies conclusion 4 points points b) (8 pts.) Let h( ) tan g( ). Find the equation of the line tangent to h ( ) at 0. Put your final answer in slope-intercept form ( y m b ). h( ) tan g( ) h(0) tan g(0) tan 4 h( ) g( ) g ( ) h(0) g(0) g(0) 4 4 y y m( ) 0 0 y ( 0) 4 y 4 Finds h'( ) points Finds h (0) Finds h (0) points Finds tangent line points Version A Page 8 of

9 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., 4. kt 5. ( pts.) The eponential growth and decay function ( y y0e ) can be used to model the amount of a drug in the bloodstream of a patient t hours after being administered. a) (4 pts.) The half-life of morphine is hours. Assuming an initial dose of 40 mg, find the eponential function that models y, the amount of morphine in the blood t hours after being administered. y 0 40 and y() 0 Solves for k points k () 0 40e Finds equation for y e k k ln ln e ln k ln k ln y 40e ln t b) (4 pts.) How long will it take for an initial dose of 40 mg of morphine to decay to mg? Give your final answer in terms of natural logarithms. y 40e 40e 40 e ln t ln t ln t ln t ln ln e 40 ln ln t 40 ln 40 t ln Sets Solves for t y points Version A Page 9 of

10 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., 4. c) (4 pts.) At what rate is the amount of morphine eiting the blood 6 hours after the administration of a 40 mg dose? Simplify your final answer to the form a ln(/ ), where a is a constant. y 40e ln t dy ln t 40e ln dt dy ln 6 (6) 40e ln dt ln 40e ln ln 4 40e ln 40 ln 4 0 ln Finds dy/dt Evaluates at t=6, interprets points Version A Page 0 of

11 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., (0 pts.) Find the absolute maimum and absolute minimum values of the function on the given interval and the -values where they occur. Put your final answers in the blanks below, with supporting work below the answers. 5 f ( ) ( 5),, Answers: Absolute minimum of -7 at = -. Absolute maimum of 0 at = 0, 5/. f ( ) ( 5) f ( ) () ( 5) () ( 5) ( ) critical numbers: f( ) 0 0( ) 0 ; f ( ) undefined at 0 check function values at endpoints and critical points f ( ) 7 f (0) 0 f () f (5 / ) 0 Finds f Sets f =0 Finds critical numbers Answers points points 5 points Version A Page of

12 MATH 060 Test Answer Key Fall 05 Calculus of One Variable I Version A Sections.., (7 pts.) Prove the following identity. sinh( ) sinh cosh e e e e sinh cosh ( e e )( e e ) 0 0 e e e e e e e e sinh( ) Rewrites one side of equation in terms of e Algebra to get other side of equation in terms of e point 5 points Subtract for improper use of equal signs Version A Page of

Student s Printed Name:

MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections..9 Student s Printed Name: Instructor: CUID: Section: Instructions: You are not permitted to use a calculator on an portion