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1 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 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: 1. Show legible, logical, and relevant justification which supports our final answer.. Use complete and correct mathematical notation.. Include proper units, if necessar. 4. Give answers as eact values whenever possible. You have 90 minutes to complete the entire test. On m honor, I have neither given nor received inappropriate or unauthorized information at an time before or during this test. Student s Signature: Do not write below this line. Free Response Problem Possible Earned Free Response Problem Possible Earned a b c. 4. a (Scantron) 1 4. b. 6 Free Response c. 6 Multiple Choice. 9 Test Total 100 Version A Page 1 of 1

2 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections..9 Multiple Choice. There are 11 multiple choice questions. Each question is worth points and has one correct answer. The multiple choice problems will count % of the total grade. Use a number pencil and bubble in the letter of our response on the scantron sheet for problems For our own record, also circle our choice on our test since the scantron will not be returned to ou. Onl the responses recorded on our scantron sheet will be graded. 1. ( pts.) Find the second derivative d if 7 e. a) d b) d c) d d) d. Let h( ) g( f ( )). Use the table to evaluate h (4). ( pts.) =1 = = =4 f () 1 f () g() g () a) 10 b) 8 c) d) 10 Version A Page of 1

3 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections..9. The figure shows the velocit v (t) of an object moving along a coordinate line as a function ( pts.) of time t, where 0 t 10. Which statement about the velocit and acceleration of the object at t 1is true? a) v(1) 0 and a(1) 0 b) v(1) 0 and a(1) 0 c) v(1) 0 and a(1) 0 d) v(1) 0 and a(1) 0 4. ( pts.) Find t t 1. if a) 1 t t 1 t 1 b) 1 t 1 t 1 c) 1 t t 1 d) t 1 1 t Version A Page of 1

4 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections..9. ( pts.) Find the derivative of sin cos. sec a) 1 sin b) 1 cos c) sin cos ' d) sec tan sec tan 6. A triangle has a base with a length of b cm and a height of h cm. The area of the triangle is ( pts.) 1 given b A bh cm. Find the rate at which the area of the triangle is changing when the length of the base is 10 cm the length of the base is increasing at a rate of cm/min the height of the triangle is cm the height is decreasing at a rate of 1 cm/min a) The area of the triangle is increasing at a rate of cm /min. b) The area of the triangle is increasing at a rate of cm /min. c) The area of the triangle is decreasing at a rate of cm /min. d) The area of the triangle is decreasing at a rate of cm /min. Version A Page 4 of 1

5 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections ( pts.) Find if cos. a) c) cos b) sin 1 sin cos d) cos sin 1 sin 8. ( pts.) 6sin( ) cos( ) Evaluate lim. 0 0 a) 6 b) c) d) 0 Version A Page of 1

6 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections Find the equation of the line normal to the graph of ( pts.) sin at. a) b) c) d) 10. The area of a circle is increasing at 4 square feet per second. At what rate is the radius of the ( pts.) circle increasing when the radius is eactl 10 feet? dr dr a) 4 ft / s b) 0 ft / s dr dr 1 c) 100 ft / s d) ft / s Version A Page 6 of 1

7 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections Find the derivative of ( pts.) 1 sin. a) (1 ) b) 1 c) 1 1 d) 1 1 Version A Page 7 of 1

8 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections..9 Free Response. The Free Response questions will count 67% 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. 1. (9 pts.) Consider the curve defined b the equation cos( ). a. ( pts.) Find. cos( ) sin( ) 1 sin( ) sin( ) sin( ) sin( ) 1 sin( ) sin( ) sin( ) 1 sin( ) Derivative of left side Derivative of right side Solves for points points b. (4 pts.) Find the equation of the tangent line at the point,0. sin 0 1 1,0 m 11 1sin 0 tan Calculates slope Equation of tangent line points points Version A Page 8 of 1

9 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections..9. (10 pts.) A baseball tossed straight up reaches a height of s( t) 16t 80t 6, where s is feet above ground and t is seconds after leaving the plaer s hand. Include units on all answers. a) ( pts.) Find the velocit and the acceleration of the baseball the moment it leaves the plaers hand. s t t t ( ) v( t) t 80 v(0) (0) ft/sec at ( ) a(0) ft/sec Finds velocit function Finds v(0) Finds acceleration function Finds a(0) 1/ point 1/ point 1/ point 1/ point b) (4 pts.) At what time(s) will the baseball be 70 feet above the ground? 70 16t 80t 6 16t 80t t 80t ( t t 4) 0 t t 4 0 (t1)(t 4) 0 t 1,4 seconds Sets the position function equal to 70 Solves for t ( pts. for factoring, ½ pt. each solution points c) (4 pts.) At what time will the baseball reach its maimum height? s t t t ( ) v( t) t 80 Solve vt ( ) 0 t t sec Finds velocit function (no work needed) Sets velocit equal to zero Solves for t points s Version A Page 9 of 1

10 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections..9. (10 pts.) At a sand and gravel plant, sand falls from a conveor belt and accumulates in a conical pile where the radius is alwas three times the height. Suppose the height of the pile is increasing at a rate of meters per minute when the pile is meters high. At what rate is the volume of sand in the pile increasing at that moment? Include units on our answer. The volume V of a cone is V r h, where r is the radius of the base and h is the height. V V h r h V ( h) h dv dh 9 h dh when h and dv 9 () () dv 16 m / min Finds volume in terms of h onl Finds dv Substitutes for h and dv Final answer points 4 points points OR V r h dv dh dr r h( r) dr dh know r h dh dr h, and r 9 and 6 Finds dv Substitutes for h, r, dh, dr Final answer and dv points 4 points dv dv dv dv () ()()(9)(6) 16 m / min Version A Page 10 of 1

11 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections (18 pts.) Find the indicated derivatives. Assume g() is a differentiable function wherever it appears. DO NOT TRY TO SIMPLIFY YOUR ANSWERS. a. (6 pts.) Find if tan 1 g( ) g 1 tan ( ) 1 g( ) 1 g( ) Derivative of arctangent Derivative of function g Derivative of points points ( ) 4 b. (6 pts.) Find h ( ) if ( ) 4 ( e 1) h ( ) ( e 1) h ( ) ( ) ( ) ( ) 4 4( e 1) e ( ) ( e 1) ( )(ln ) h( ) ( ) Correct numerator Correct denominator points c. (6 pts.) Find f () t if f ( t) 7t sec(4 t) f t t t ( ) 7 sec(4 ) 1 f ( t) 7t sec(4 t) 7 sec(4 t) tan(4 t)(4)) 4 Derivative of 1/ power function Derivative of 7 Derivative of secant function Derivative of 4t points Version A Page 11 of 1

12 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections..9. (9 pts.) In man cases, throid cancer can be treated with radioactive iodine, I-11. This form of iodine has a half-life of 8 das, and is given in small doses measured in millicuries. a) ( pts.) If a patient is given an initial dose of 100 millicuries, how much I-11 will be present after 1 das? Give our final answer without using the number e and natural logarithms; for eample, 1. e 0 100e solve for k appl (8) e 1 e k (8) kt kt (deca model) k (8) k (8) ln 1 ln( e ) ln 1 8k ln 1 k 8 ( t) 100e (1) 100e 100e 100e ln(1 ) 8 t ln(1 ) (1) 8 (/)ln(1 ) / ln((1 ) ) millicuries Knows the basic deca model Finds k Substitutes t =1 and gets answer in terms of e and natural logarithm Simplifies answer points Version A Page 1 of 1

13 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections..9 b) (4 pts.) Suppose a patient is given an initial dose of 100 millicuries, and is considered cured when the amount has dropped to 17 millicuries. How man das will it take for the patient to be cured? Give our final answer in terms of natural logarithms. ( t) 100e e e ln(1 ) 8 ln(1 ) 8 ln(1 ) 8 ln(1 ) 17 t 8 ln ln e ln(1 ) ln t ln(17 100) t das ln(1 ) t t t Knows the basic deca model Finds k Substitutes t =1 and gets answer in terms of e and natural logarithm Simplifies answer points Version A Page 1 of 1

14 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections (10 pts.) Let sin( g( )), where ( ) g is a differentiable function with i) g( ) ii) g( h) g( ) lim 1 h h0 a) (4 pts.) Find. sin( g( )) logarithmic differentiation sin( g( )) ln ln( ) ln sin( g( ))ln( ) 1 1 sin( g( )) ln( )cos( g( ))g ( ) 1 sin( g( )) ln( )cos( g( ))g ( ) sin( g( )) ln( )cos( g( ))g ( ) sin( g( )) Takes natural log of both sides Log properties to simplif right side Derivative of left side Derivative of right side Multiplies b to get final answer in terms of onl ½ point ½ point ½ point points ½ point b) (4 pts.) Evaluate at. sin( g( )) ln( )cos( g( ))g ( ) sin( g( )) ln( )cos( g( ))g ( ) sin( ) ln( )cos( )(1) 1 (1) ln( )(0)(1) sin( ) sin( g ( )) sin( g ( )) Substitutes into the derivative from (b) Finds correct value for the derivative points Version A Page 14 of 1

15 MATH 1060 Test Answer Ke Spring 016 Calculus of One Variable I Version A Sections..9 c) ( pts.) Find the equation of the line tangent to the curve sin( g( )) at. sin( g( )) find the point on the curve ( ) 0 0 sin( g ( )) sin( ) 1 (, ) (, ) tangent line: - - Substitutes into the function Calculates the value Finds the equation of the line ½ point ½ point Version A Page 1 of 1