Methods of Calculus (MAC 2233) Spring 2010 Sample Final Exam

Transcription

1 Methods of Calculus (MAC ) Spring 1 Sample Final Eam The following questions are meant to check your readiness for the final eam, as well as to give you an idea of its level of difficulty. There may be questions on the final eam that are different from any of these questions. Answers are included at the bottom of page The slope of the line 4 +y =8is: 4 (B) Find the slope of the line joining the points (5, 1) and (9, 5). (B) 1 1. Find the equation of the line parallel to y = + 5 and passing through (, ). y = + (B)y = + y = 1 + y = Which of the following is the derivative of y =? (B) Let f() = 4 + e.findf (). 1 (B) 1 +e 1 + e Let f(t) =t t +6. Findf (t). t 6t +6 (B)t 6t t 6 6t 6 7. Let f(t) =t t +6. Findf (t), the second derivative of f(t).. t 6t +6 (B)t 6t t 6 6t 6 8. Find the derivative of y =( +1). ( +1) (B) ( +1) (6) ( +1) () ( +1) (6) Page 1 of 8

2 9. The line y = m + b is tangent to the graph of y = at the point (1, 1). Find m and b. m =1,b = (B)m = 1, b = m =,b = m =, b =4 1. Let f() = f( + h) f() +. Find, where h. h + h + (B) h + + h Find lim. +5 (B) 4 4 no limit 1. Find lim 1 1. (B) 1 no limit 1. Find the slope of the line tangent to the curve y = at the point (1, ). 1 (B) 14. Let y = e.find d y, the second derivative of the function. d ( +)e (B) ( +1)e e e 15. The function h() = 4 can be viewed as a composition fucntion h() =f(g()). Find f() andg(). f() =4,g() = (B) f() =4, g() = f() =, g() = f() =, g() =4 16. Find the derivative of the function h() = 4. 4 (B) Page of 8

3 17. Let s =(4p +1).Find ds dp. p=1 1 (B) 4 (4p + 1) 8(4p + 1) 18. What is the rate of change of the function f() = 1 at =? 1 4 (B) The following word problem relates to Questions 19 1: A farmer wants to fence off a rectangular region along a river (which serves as a natural boundary requiring no fence). The enclosed area must be 18 square feet. Let y be the length of the fence used on the side parallel to the river, and the length of each of the other two sides. Consider the problem of finding the dimensions of the region so that the least amount of fencing is required. 19. What is the Objective Equation? A = y (B) P = + y P = +y P = +y (E) y = 18. What is the Constraint Equation? P = y (B) +y = 18 P = +y +y = 18 (E) y = What are the dimensions of the rectangular region in feet that minimize the use of fencing? =5, y =6 (B) =6, y = 5 =6, y = =,y= 6. Differentiate the function y =( + 4)( 1). 6 (B) Differentiate the function y = 5. 4 ( 5) (B) ( 5) 6 ( 5) 1 Page of 8

4 4. Find the equation of the tangent line of the function f() = at the point (, 4). +1 y = (B)y = 4 y = 4 y = 5. Compute dy d if y = u u + 4 and u = 5. u (B) Given + y + y =. Find dy d + y +y (B) + y +y by using implicit differentiation. +y + y +y + y 7. Determine the slope of the line tangent to the curve 4y = 5 at the point (, 1). 1 (B) Solve the equation e 1 = (B) 8 ln ln Solve the equation 4 ln( +1) = 1. e1 1 (B) e 1 1 e 1. Find the derivative of the function y = e. e (B) e e e 1. Find the derivative of the function y = e. (B) e e ln e. Let y =ln(e + ). Find dy d. 1 e + (B) e + e + e e + e + e Page 4 of 8

5 . Differentiate e e +1 e e +1. (B) e (e +1) e (e +1) 1 4. Find the derivative of the function y = ln( +1). ln( +1) (B) ln( +1)+ +1 ln( +1)+ +1 ln( +1) 5. Let P (t) be the population (in millions) of a certain city t years after 199, and suppose that P (t) satisfies the differential equation P (t) =.P (t) andp () = 4. Find the formula for P (t). 4e.t (B) 4e.t 4e.1t 4e.1t 6. Refer to the question above. How fast is the population growing when the population reaches 5 million?.1 million per year (B).4 million per year.5 million per year 5 million per year 7. The decay constant of a radioactive substance is.5 when time is measured in years. What is the half-life of the substance? ln.5 ln.5 (B) ln 8. Three thousand dollars is deposited into a savings account at 6.5% interest compounded continuously. After how many years will the balance reach $6,? ln.5.65 (B) ln 6.5 ln.65 ln An investment earns 4.% interest compounded continuously. How fast is the investment growing (in dollars per year) when its value is $7,?,1 (B) 1,68 1,1 4. Which of the following is an anti-derivative of y =? 6 (B) 6 1 Page 5 of 8

6 41. Find a function f() such that f () = and f() =. f() = (B) f() = f() = + f() = 4. Let ( +1) 5 = k( +1) 6 + C. Find the constant k. 1 (B) Evaluate e d. 1 e + C (B) e + C e + C e 1 + C 44. Evaluate (4 + 5) d / + C (B) 4 + / 5 + C 4 + / + C 4 + / + C 45. Evaluate 1 d. 1 + C (B) 1 + C ln + C ln + C 46. Find ( + +5)d. (B) Find the area of the region between y = and y = +1from =to =. 1 (B) Find the area of the region bounded by the curves y = and y = +6. (B) 4 9 Page 6 of 8

7 49. Use the substitution method, or any other method, to evaluate 4 ( 4 +5) 5 d. (4 +5) C (B) 4 ( 4 +5) 6 + C 4 4 ( 4 +5) 6 + C (4 +5) 6 + C (ln ) 5 5. Use the substitution method, or any other method, to evaluate d. (ln ) 6 + C (B) (ln )6 6 + C (ln )5 + C ln((ln ) 5 )+C 51. By using Integration by Parts, or any other method, find e d. e + C (B) ( 1)e + C ( +1)e + C e + C 5. By using Integration by Parts, or any other method, find ln d. 1 ln + C (B) 1 ( ln ) + C 1 ln C 1 ln C 5. Set up the definite integral that gives the area of the shaded region. 1 ( ) ( d (B) + 1 ) d ( + 1 ) d 1 ( + 1 ) d y (1, ) (4, ) y = + 1 Page 7 of 8

8 54. Set up the definite integral that gives the area of the shaded region. ( )( 4) d (B) ( +1 16) d ( )( 4) d ( ) d y y =( )( 4) y =( )( 4) 55. Which of the following gives the area of region between y = f() (the thinner one) and y = g() (the thicker one) from =to =? (f() g()) d (B) 1 (f() g()) d + 1 (g() f()) d y y = g() (g() f()) d 1 (g() f()) d + 1 (f() g()) d y = f() 1 Answers: 1, (B),, 4(B), 5, 6(B), 7, 8(B), 9(E), 1, 11, 1(B), 1, 14, 15, 16, 17(B), 18, 19(B), (E), 1,,, 4(B), 5, 6, 7, 8, 9(B),, 1,,, 4(B), 5(B), 6, 7, 8, 9, 4, 41(B), 4, 4, 44(B), 45, 46, 47, 48(B), 49, 5(B), 51(B), 5, 5(B), 54, 55. Page 8 of 8