Test Review 4 Math 1241 Name:_______________________________________ Chapters/Sections 3.7 - 4.7 Part I - Multiple Choice - You must show ALL your work to receive credit For #1-5, Use L'Hospital's Rule if applicable. 1. Answer Choice 6x 2 2x 7 x 15 2 x 3 x 2 Find the limit if it exists lim (a) 2. (c) 2 (b) (d) 2 (e) 0 sin(3x) x 0 5x Find the limit if it exists lim (a) 3. (b) (c) 1 3 Find the limit if it exists lim 1 x x (a) 12 (b) e12 (d) 5 3 (e) 3 5 4x (c) 16 (d) e16 (e) 4. Find the limit if it exists lim x 2 ln( x) x 0 (a) 0 5. (b) 1 (c) 2 (d) 3 (c) 7 (d) (e) 1 Find lim 7 xe x 7 x x (a) 0 (b) 1 7 (e) 6. The absolute maximum and absolute minimum values of f ( x) x 2 x 3 on the interval [2,3] are: 4 (a) 66, 2 7. (b) 3, -2 (c) 66, 11 (d) 11,3 2 (e) 66, 0 The function f x satisfies the following conditions: f (3) 0 f (3) 2 f ( x) 0 f ( x) 0 f ( x) 0 f ( x) 0 if if if if 0 x3 x3 0 x 1 or x 4 1 x 4 Then the graph of f x is: (a) Concave down if 0 x 3 and Concave up if x 3 (b) Increasing if 1 x 4 and Decreasing if 0 x 1 or x 4 (c) Concave up if 1 x 4 and has a local maximum at x 3 (d) Concave up if 1 x 4 and has a local minimum at x 3 (e) Increasing if 1 x 4 and has a point of inflection at x 3 8. On which interval is the function f ( x) 2 x 9 x 24 x decreasing? 3 (a) ( ,- 4) 9. (b) ( 3,1) (c) ( 4,1) 2 (d) ( 3, 2) (e) (1, ) Suppose that f '( x) ( x 1) ( x 1)( x 2) . (Note that this is f ' NOT f ) Which critical point(s) give a relative maximum for the function f . 2 (a) -2 (b) -1 (c) 1 (d) -2 and -1 (e) -2 and 1 10. For what values of x, if any, does the function f x 3x 32 x 72 x 10 have a local minimum? 4 (a) There is no local minimum 11. (b) Only at x = 0 3 (c) Only at x = 2 2 (d) Only at x = 6 (e) At x = 0 and x = 6. The graph of f '( x ) is shown. On which of the following intervals is f decreasing? (You are being asked about f , but you are shown the graph of the derivative of f .) (a) ( 1,1) (b) (0, 2) (c) ( 2, 2) (d) ( ,0) (e) (0, ) 12. How many inflection points does the function f ( x) (a) 0 13. (c) 2 (d) 3 1 x5 12 x 4 x 1 have? (e) 4 Find the value of a so that x 1 is a critical number for f ( x) ax 2 x 2 x 1 . 3 (a) -2 14. (b) 1 1 20 (b) -1 (c) 0 (d) 1 2 (e) 2 If f ( x ) has a local minimum at x a , then which of the following must be true: (a) f '(a) 0, f ''(a ) 0 (d) f '(a) 0, f ''(a ) 0 (b) f '(a) 0, f ''(a ) 0 (e) f '(a) 0, f ''(a ) 0 (c) f '(a) 0, f ''(a ) 0 15. Find the value of c that is guaranteed by the Mean Value Theorem for the function f ( x) 2 x2 3x 1 on the interval [2,1] . (a) 5 16. (b) (c) - 5 (d) - 1 (e) 1 2 Given that 2 f '( x) 5 for all x in the interval [1,3] , use the Mean Value Theorem to find A and B so that A f (3) f (1) B . (a) A 8, B 20 17. 1 2 (b) A 4, B 10 (c) A 1, B 3 (d) A 2, B 15 Given the graph of the second derivative f ''( x ) , find all interval(s) on which f '( x ) is increasing. (a) ( , 4) (b) ( ,1),(1, 4) (c) (1,3),(3, 4) (d) (1,3) (e) Cannot be determined. 18. (e) A 2, B 5 Find two positive numbers whose product is a maximum and whose sum is 8. (a) 2 and 6 (b) 1 and 7 (c) 4 and 4 (d) 5/2 and 11/2 (e) 14/17 and 122/17 19. A rectangle has a perimeter of 30 inches. Express the area, A, of the rectangle as a function of the length, x, of one of its sides. a) A 30 x x 2 , (b) A 15x x 2 (c) A x 2 30 x , (d) A x 2 15x , (e) A 2 x 20. Find the general anti-derivative of. (a) 4x 1 x 2 2 cos( x ) C 21. 2 sin x 1 x2 (b) (d) 2arctan( x) cos( x) C 2 1 x 2 2 cos( x ) C (c) 2arctan( x) cos( x) C (e) 2arctan( x) cos( x) C Find the general anti-derivative of the function f x e cos x 2 x 5 x (a) f x e x sin x 2 x 4 5x C (c) f x e x sin x (d) f x e x sin x x 4 5 C (e) 3 f x e x sin x 6 x C (b) 22. 60 x f x e x sin x 6 x 2 C x4 5x C 2 Find the general anti-derivative of f ( x ) 4 x e x e x 1 1 2 2 C (a) 4 e 3 C (b) 2 x x 1 x x x 1 x2 (c) 2 x 2 e x 3 1 C (d) 2 x 2 e x C (e) None 3 x x 23. Find the function f ( x ) whose second derivative is f ''( x) 6 x cos( x) 2 and satisfies f (0) 1 and f '(0) 2 . (a) x cos( x) x 2 x 3 2 (b) x cos( x) x 2 x 3 (d) x cos( x) x 2 x 1 3 24. 2 (c) x cos( x) x 2 x 3 2 (e) x cos( x) x 2 x 3 2 1 The derivative of the function f x is given by f x 20 x 6 x 2 . Find a formula for the function f x given that f 1 25 3 (a) f x 10 x 2 4 x 2 11 (d) f x 40 x 9 x 2 25. 2 3 2 24 3 (b) f x 20 x 2 6 x 2 1 (e) f x 3x 1 2 3 (c) f x 40 x 2 6 x 2 21 22 If f '( x) sin( x ) x and f (0) 4 , then f ( x ) equals 3 (a) cos( x) 3x 2 (d) cos( x) 1 x 4 5 4 (b) sin( x) 3x 5 2 5 (e) cos( x ) x 3 2 (c) cos( x) 1 x 4 4 4 Test Review 4 Name:_______________________________________ Chapters/Sections 3.7 - 4.7 Part II - Free Response - You must show ALL your work to receive credit Math 1241 26. Find the limit if it exists using L'Hospital's Rule if applicable. a. cos(18 x) 1 x 0 2x 2 lim b. lim 12 x 2e x 2 x c. 2 lim 1 x x 3x 27. Find the Absolute Maximum and Absolute Minimum values of the function f ( x) x 3 x 2 1 on the interval [1,1] . Absolute Max = Absolute Min = 28. Find the intervals on the x-axis where the function f ( x ) x is increasing and where it is decreasing. Use x 9 2 interval notation (a,b) for your answers. Increasing = Decreasing = 29. Find the intervals on the x-axis where the graph of f ( x) x 3 3x 2 45x 10 is concave upward and where it is concave downward. Use interval notation (a,b) for your answers. Then find any x values of inflection points. Concave Up = Concave Down = Inflection Points at x = 30. A farmer wants to fence in a rectangular pen that is to enclose 20,000 square feet. He only needs fence on three sides of the pen since a river will be the fourth side. Find the dimensions of the pen that minimizes the total length of fence. x= y= 31. A box with a square base and an open top is to have a surface area of 1200 square meters. Find the dimensions of the box that has maximum volume. Bottom Edge = Height = 32. Find the function f ( x ) that has the given derivative and the given value. (a) f '( x) 3x 2 5x 2 and f (1) 1 f x (b) f '( x) sin( x) 2sec2 ( x) and f ( ) 1 f x (c) f '( x ) e x 1 1 and f (1) e x f x 33. Find the coordinates of the point on the graph of y x that is closest to the point (1,0) on the x-axis. 34. Sketch a continuous function on [-5,6] which satisfies all of the following conditions: f '(-3) = f '(1) = f '(4) = 0 f '(x) > 0 if x < -3, or 1 < x < 4 f '(x) < 0 if -3 < x< 1, or x > 4 f ''(x) > 0 if -1 < x < 3 f ''(x) < 0 if x < -1 or x > 3 35. For the graph of f(x) given below, sketch a graph of f '(x). (You may sketch it on the given coordinate axis)