Calculus Problem Solving: Derivatives and Limits, Exams of Mathematics

Download Calculus Problem Solving: Derivatives and Limits and more Exams Mathematics in PDF only on Docsity! MthSc 106L Test 1 Spring 2009 Version B UC 2.1 – 3.8 Page 1 of 15 Student’s Printed Name: _______________________ CUID:___________________ Instructor: ______________________ Section # :_________ Read each question very carefully. You are NOT permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cellphone, or laptop on either portion of the test. No part of this test may be removed from the examination room. In order to receive full credit for the free response portion of the test, you must: 1. Show legible and logical (relevant) justification which supports your final answer. 2. Use complete and correct mathematical notation. 3. Include proper units, if necessary. 4. Give exact numerical 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 during this test. Student’s Signature: ________________________________________________ Do not write below this line. Free Response Problem Possible Points Points Earned Free Response Problem Possible Points Points Earned 1a-f 6 7b 3 1g 2.5 7c 3 1h 2.5 8 5 2 4 9a 2 3a 2 9b 2 3b 2 9c 3 4 4 10 1 5 6 Free Response 55 6 5 Multiple Choice 45 7a 2 Test Total 100 MthSc 106L Test 1 Spring 2009 Version B UC 2.1 – 3.8 Page 2 of 15 Multiple Choice. There are 19 multiple choice questions. Each question is worth 2 – 3 points and has one correct answer. The multiple choice problems will count 45% of the total grade. Use a number 2 pencil and bubble in the letter of your response on the scantron sheet for problems 1 – 19. For your own record, also circle your choice on your test since the scantron will not be returned to you. Only the responses recorded on your scantron sheet will be graded. You are NOT permitted to use a calculator on any portion of this test. 1. (3 pts.) Consider x 3 + 3x 2 y + y 3 = 8 . Find dy dx . a) dy dx = !x2 ! 2xy + 8 x2 + y2 b) dy dx = ! x + 2xy x2 + y c) dy dx = ! x2 + 3xy x2 + y2 d) dy dx = ! x2 + 2xy x2 + y2 2. (3 pts.) Find the derivative of h(x) = 10 + sec8x . a) !h (x) = sec2 8x 2 10 + sec8x b) !h (x) = 1 10 + sec8x c) !h (x) = 4sec8x tan8x 10 + sec8x d) !h (x) = sec8x tan8x 2 10 + sec8x 3. (2 pts.) Find the derivative of y = 4 (3x +1)5 . a) dy dx = !60 (3x +1)6 b) dy dx = !60 (3x +1)4 c) dy dx = 60 (3x +1)6 d) dy dx = !20 (3x +1)6 MthSc 106L Test 1 Spring 2009 Version B UC 2.1 – 3.8 Page 5 of 15 12. (2 pts.) Evaluate cos !1 2 2 . a) ! 6 b) ! 4 c) 7! 4 d) 11! 6 13. (2 pts.) Find lim x!5 x 2 " 25 x 2 " 7x +10 . a) 0 b) 1 c) 10 3 d) DNE 14. (2 pts.) Find the derivative of y = ln!( ) " . a) !y = 1 " # $% & '( ) ln) b) !y = " # ln#( ) " $1 c) !y = 1 " ln"( ) # d) !y = " ln#( ) " $1 15. (2 pts.) Find the derivative of y = tan !1 ln2x( ) . a) dy dx = 1 x 1+ ln2x( ) 2 b) dy dx = 1 1+ ln2x( ) 2 c) dy dx = 2 x 1+ ln2x( ) 2( ) d) dy dx = 1 x 1+ ln2x( ) 2( ) MthSc 106L Test 1 Spring 2009 Version B UC 2.1 – 3.8 Page 6 of 15 16. (2 pts.) The graph of f (x) is given below. Choose the answer that represents the graph of the derivative of f (x) . a) b) c) d) MthSc 106L Test 1 Spring 2009 Version B UC 2.1 – 3.8 Page 7 of 15 17. (2 pts.) Given f (x) = 3x3 + 3x2 + 7x + 3 3x2 + 2 , use long division to find the equation of the slant asymptote. a) y = 3x 2 + 2 b) y = x +1 c) y = x d) y = 5x +1 18. (2 pts.) Suppose that the functions f (x) and g(x) and their derivatives with respect to x have the following values at the given values of x . x f (x) g(x) !f (x) !g (x) 3 1 4 6 5 4 3 3 5 -4 If h(x) = f ! g( )(x) = f g(x)( ) , find !h (4) . a) 6 b) -20 c) 18 d) -24 19. (2 pts.) Find lim x! " 2 cos x # sin x cos2x . a) 1 b) !1 c) 0 d) DNE MthSc 106L Test 1 Spring 2009 Version B UC 2.1 – 3.8 Page 10 of 15 4. (4 pts) Find the derivative of y = csc 3 7x 2( ) . Simplify. y = csc 3 7x 2( ) = csc 7x2( )!" #$ 3 so %y = 3 csc 7x2( )!" #$ 2 & 'csc 7x2( )cot 7x2( )( ) &14x = '42xcsc3 7x2( )cot 7x2( ) 5. (6 pts) Use logarithmic differentiation to find the derivative of y = (x ! 3) tan x with respect to x . y = (x ! 3)tan x ln y = ln x ! 3( ) tan x ln y = tan x " ln x ! 3( ) implicit derivative is 1 y " dy dx = tan x " 1 x ! 3 "1+ sec2 x " ln x ! 3( ) so dy dx = y " tan x x ! 3 + sec2 x " ln x ! 3( ) # $ % & ' ( dy dx = x ! 3( ) tan x " tan x x ! 3 + sec2 x " ln x ! 3( ) # $ % & ' ( Work on Problem Points Awarded Derivative of the outside 1 Keep the inside 1/2 Times the derivative of the outside 1/2 2nd derivative of the outside etc. 1 2nd inside derivative 1/2 Simplifying 1/2 Notes: -4 Derivative of the outside evaluated at the derivative of the inside -4 mistakenly trying to use product rule -1/2 notational misuses -1/2 dropping the negative Work on Problem Points Awarded Takes natural log of both sides 0.5 Correctly applies properties of natural logs 0.5 Uses implicit diff. for derivative of y 1 Correctly uses product rule on right hand side (1 point for setting up and 1 point for each summand) 3 Solves for dy/dx in terms of x 1 Notes: -1 fails to change y to function in terms of x at end -3 product rule as f’g’ MthSc 106L Test 1 Spring 2009 Version B UC 2.1 – 3.8 Page 11 of 15 6. (5 pts) Use the limit definition of the derivative to find f ' x( ) when f (x) = 4x2 ! 3x + 7 . No credit will be given for only using differentiation rules. !f (x) = lim h"0 4 x + h( ) 2 # 3 x + h( ) + 7 # 4x2 # 3x + 7( ) h = lim h"0 4x2 + 8xh + 4h2 # 3x # 3h + 7 # 4x2 + 3x # 7 h = lim h"0 8xh + 4h2 # 3h h = lim h"0 h 8x + 4h # 3( ) h = lim h"0 8x + 4h # 3( ) = 8x # 3 Work on Problem Points Awarded Correctly states limit definition of derivative with correct substitution 1.5 Correctly simplifies numerator 1.5 Correctly eliminates h/h 1 Correctly evaluates limit 1 Notes: -1 pt for no limit notation -1/2 to -1 pt for poor limit notation -1/2 pt for limit notation carried too far -1/2 1 or 2 missing equals -1 several missing equals -1 “dropping” the denominator -5 pt for taking the derivative correctly without using the limit definition • award ½ pt for writing the difference quotient without substituting in for the specific function and not going any further • work that jumped from a correct point to the correct answer without showing work in between lost points for whatever steps were not shown MthSc 106L Test 1 Spring 2009 Version B UC 2.1 – 3.8 Page 12 of 15 7. (2, 3, 3 pts respectively) Suppose that a ball is thrown into the air and its position can be described by s(t) = !t 2 + 4t +12 where s is in meters and t is in seconds. Be sure to include units on each answer. a. What is the average velocity of the ball between t = 1 and t = 3 seconds? v avg = s(3) ! s(1) 3!1 = (!9 +12 +12) ! (!1+ 4 +12) 3!1 = 15!15 2 = 0 m/s b. When will the ball reach its maximum height? object reaches max height when v t( ) = 0 v t( ) = !s t( ) = "2t + 4 "2t + 4 = 0 "2t = "4 t = 2 sec c. At what second will the ball strike the ground? What is the velocity of the ball at the instant it hits the ground? object strikes ground when s t( ) = 0 !t 2 + 4t +12 = 0 ! t 2 ! 4t !12( ) = 0 ! t ! 6( ) t + 2( ) = 0 t = 6, t = !2 ! 2 is not a valid answer so ball strikes ground at t = 6 sec v(6) = !2(6) + 4 = !12 + 4 = !8 m/s Work on Problem Points Awarded Finds s(a) and s(b) ½ each Finds a-b, or b-a ½ Finds average velocity with units ½ Notes: -2 Uses velocity function Work on Problem Points Awarded Correct velocity function (can be shown in any part) 1 Sets velocity =0 0.5 Correct time 1 Correct units 0.5 Notes: Work on Problem Points Awarded Sets position function equal to 0 ½ Correctly factors and finds two times 1 Eliminate the negative time ½ Finds velocity at their time 1 Notes: -1/2 no units