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1 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. 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 a tetbook, 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 eamination room. Read each question carefully. 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 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 b a b (Scantron) 4. c. 6 Free Response 70. a. 4 Multiple Choice 0. b. Test Total 00. c. Version A Page of 4

2 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. Multiple Choice: There are 0 multiple choice questions. Each question is worth points and has one correct answer. The multiple choice problems will be 0% of the total grade. Circle your choice on your test paper. Any questions involving inverse trigonometric functions should be answered based on the domain restrictions for trigonometric functions used in Section.6.. Solve the equation. Hint: Don t forget the domain for the natural log function. ( pts.) ln( ) ln( ) 0 a) No real solutions b) c) d). Suppose the population P (in billions) of bacteria in a culture is given by the function ( pts.) P( t) t t, where t is measured in hours. Find the average rate of change of Pt () over the interval t. a) billion/hour b) billion/hour c) billion/hour d) 6 billion/hour. ( pts.) Evaluate ln. a) b) c) Does Not Eist d) 0 Version A Page of 4

3 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. 4. ( pts.) Evaluate. 7 a) b) c) d) Does Not Eist. Solve the equation for. ( pts.) 4 a) c) ln 4 b) ln ln 4 y d) No real solutions. 4ln 6. ( pts.) Evaluate. a) b) c) d) Version A Page of 4

4 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. 7. ( pts.) Let g( ) f ( ) h ( ). Use the table to evaluate h (4). = = = = 4 f () f () 6 8 g() 4 4 g () a) 8 b) c) 6 d) 6 8. ( pts.) Evaluate tan 6. a) b) c) d) Version A Page 4 of 4

5 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. 9. ( pts.) Find f if ln( ) e if f 7 if 8 if a) b) c) 8 d) Does Not Eist 0. ( pts.) Find the intervals on which the function f( ) e ( 6 8) is continuous. a) (, ), (, 4), (4, ) b) (, 4), ( 4, ), (, ) c) (, ), (, ) d) (, ), (, 4), (4, ) Version A Page of 4

6 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. Free Response: The Free Response questions will be 70% 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 eact values. Any questions involving inverse trigonometric functions should be answered based on the domain restrictions for trigonometric functions used in Section.6.. (8 pts.) Use the graph of f( ) to find each of the following its, if it eists. ( pt. each) Infinite its should be answered with = or =, whichever is appropriate. If the it does not eist (and cannot be answered as or ), state DNE. a. b. f( ) e. f f( ) d d f. f ( ) 0 ( ) Note: Grade all or nothing. No penalty for missing or inappropriate use of equal signs, but note errors. c. ( ) tan f ( e ) 0 g. 4 f( ) d. f( ) DNE h. f (0 h) f (0) h h0 Version A Page 6 of 4

7 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. 8. ( pts.) Find cos 0, if it eists. Show your work, citing any theorems that you use in finding the value of the it. we know - cos cos 8 checking its in the two outside funcntions: 8 8 ( ) and Set up the necessary inequality points Limits of two outside functions (/ each) Mentions Squeeze (Sandwich) Theorem Final answer -/ point for it notation errors Maimum of deduction for all notation errors By the Squeeze (Sandwich) Theorem, we know 8 cos 0 0. ( pts.) For what value of a is appropriate left and right its. f continuous at? Show all supporting work, including the f a, a, Left it. points Right it or f() Sets its equal and solves for a. points -/ point for it notation errors Maimum of deduction for all notation errors OK if use f() rather than right-side it in work Subtract one point for no left-hand it notation, but correct epression used in solving the problem. left: f ( ) ( a ) a() () 4a 4 right: f( ) ( ) () () 8 setting left it equal to right it: 4a 4 8 a 6a 4 4 a 6 a a a Version A Page 7 of 4

8 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. 4. (8 pts.) Find the following its. If a it does not eist, state does not eist and provide a brief eplanation. Show all work. Do not use L Hopital s Rule. a. (6 pts.) ( ) ( ) 4 6 ( ) ( ) 8 4 ( ) ( ) ( ) ( ) Recognizes implicitly or eplicitly the indeterminate form 0/0. Algebra to get a reduced form for which substitution works 4 points Correctly substitutes and evaluates it. f( ) 0 Subtract ½ point for the untrue statement:. a g ( ) 0 Subtract for not presenting the correct reduced form before substitution -/ for notation errors such as missing equal sign, it without argument, failure to drop it on substitution, etc Maimum of deduction for all notation errors Version A Page 8 of 4

9 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. b. (6 pts.) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) 0 4 ( ) Recognizes implicitly or eplicitly the indeterminate form 0/0. Algebra to get a reduced form for which substitution works 4 points Correctly substitutes and evaluates it. f( ) 0 Subtract ½ point for the untrue statement:. a g ( ) 0 Subtract for not presenting the correct reduced form before substitution -/ for notation errors such as missing equal sign, it without argument, failure to drop it on substitution, etc Maimum of deduction for all notation errors Version A Page 9 of 4

10 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. c. (6 pts.) There are two approaches to this problem. The guidelines apply to both methods OR Recognizes implicitly or eplicitly the indeterminate form Algebra to get a reduced form for which substitution works 4 points Correctly evaluates it. f( ) Subtract ½ point for the untrue statement:. g ( ) Subtract for not presenting the correct reduced form before substitution -/ for notation errors such as missing equal sign, it without argument, failure to drop it on substitution, etc Maimum of deduction for all notation errors Subtract two points for failing to recognize the need for a negative sign in the work and giving an answer of + Subtract two points for a b a b with correct answer Version A Page 0 of 4

11 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,.. (4 pts.) Find the derivatives of the following functions. Assume g is a differentiable function wherever it appears. Do NOT simplify your answers. a) (4 pts.) h( t) t e 48 h t t e h t t e h t ( ) 48 ( ) 48 ( ) t 0 0 h() t t Derivative of t points Derivative of two constants (/ point each) (can be implied) Constant multiple -/ if they incorrectly simplify a correct answer b) ( pts.) f e g ( ) f f e g ( ) g( ) e e e g( ) g ( ) Derivative of numerator (4 pt each) 4 points Derivative of denominator -/ if they incorrectly simplify a correct answer c) ( pts.) f g( ) e f f f e g( ) g( ) e g( ) e e g( ) g( ) e points ( ) points e g -/ if they incorrectly simplify a correct answer Version A Page of 4

12 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. f. 6. (7 pts.) Consider the function a) (4 pts.) Find f f ( h) f ( ) f( ) h0 h using the definition of the derivative. Work on Problem States the formula for the it definition of the derivative. Okay if implied / Point h0 h0 ( h) ( h) ( ) h ( h h ) h0 h h h 4h h h Substitutes h correctly into the definition / Point Correct algebra to reduce to a form for which substitution works Correctly evaluates it (no credit if this doesn t follow from work) Subtract 4 points for not using the it definition of the derivative Ma of total for all notation errors Subtract for ( h) h Point h 4h h h0 h h0 4h 4 b) ( pts.) Calculate f using your function in part (a). Briefly describe what this value represents. f 4() f ) the slope of the tangent line at ) the slope of the curve y f ( ) at ) the instaneous rate of change (IROC) of f ( ) at Calculates f Eplanation full credit for any one points of the three possibilities OK if they have an incorrect derivative from part (a), but calculate f correctly using the correct derivate obtained with derivative rules. Subtract for just saying slope Version A Page of 4

13 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. 7. (6 pts.) Let f ( ) 4. Use the delta-epsilon definition of a it to prove f( ) 6. we want f( ) 6 ( 4) 6 0 ( ) choose = Proof: Let 0 be given. Choose =. Assume 0 =. Then f ( ) 6 ( 4) 6 0 ( ). Determines a value for (OK if stated without work) points Proof 4 points Subtract pt. if epsilon and delta were switched but the student was consistent throughout and the work is otherwise correct. Subtract ½ if "let epsilon > 0 be given" (or something similar) was missing from the proof Alternative proof OK where they work in reverse, starting with the assumption = and manipulate to get ( 4) 6 Subtract for having = where < should be in proof. Subtract if start proof with f ( ) L Version A Page of 4

14 MATH 060 Test Answer Key Fall 06 Calculus of One Variable I Version A Sections.,.6,. -.7,.,. 8. (6 pts.) Find the equation of the line tangent to the graph of f( ) at the given value. e f ( ), 0 e f( ) f f '0 f ' 0 m tan ( ) e e ( ) ( ) 0 0 (0 ) e e ((0)) (0 ) Calculates f () Calculates f (0) Calculates f (0) Equation of tangent line points 0 e f (0) 0 (, y ) (0, ) 0 0 line : y ( 0) Version A Page 4 of 4