Test 3 Review. fx ( ) ( x 2) 4/5 at the indicated extremum. y x 2 3x 2. Name: Class: Date: Short Answer
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1 Name: Class: Date: ID: A Test 3 Review Short Answer 1. Find the value of the derivative (if it exists) of fx ( ) ( x 2) 4/5 at the indicated extremum. 7. A rectangle is bounded by the x- and y-axes and the graph of y ( 5 x) (see figure). What length and 2 width should the rectangle have so that its area is a maximum? 2. Find the differential dy of the function y x 2 3x Find the differential dy of the function y x cos( 7x). 4. Find the differential dy of the function y 2x 3/7. 8. A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 38 feet. 5. Find the equation of the tangent line T to the graph of fx ( ) 19 Ê 19 ˆ at the given point 2, 2 x Á Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 36%. Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.) 1
2 Name: ID: A 9. The graph of f is shown below. For which values of x is f ( x) zero? 13. Analyze and sketch a graph of the function fx ( ) x 2 2x 9. x 14. Analyze and sketch a graph of the function fx ( ) x x Determine the slant asymptote of the graph of fx ( ) 5x 2 9x 5. x The graph of a function f is is shown below. Sketch the graph of the derivative f. 10. The graph of f is shown below. For which value of x is f ( x) zero? 11. A rectangular page is to contain 144 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used. 12. Determine the slant asymptote of the graph of fx ( ) x2 8x 14. x 6 2
3 Name: ID: A 17. The graph of a function f is is shown below. Sketch the graph of the derivative f. 21. Find the limit. lim x 8x 2 5x Find the limit. lim x Ê Á 5 3 x 2 ˆ 23. Find all critical numbers of the function gx ( ) x 4 4x Find the limit. 18. The graph of a function f is is shown below. Sketch the graph of the derivative f. lim x 6x 64x A business has a cost of C 1.5x 500 for producing x units. The average cost per unit is C C. Find the limit of C as x approaches x infinity. 26. Match the function fx ( ) 2sinx x 2 1 following graphs. with one of the 19. Sketch the graph of the relation xy 2 2 using any extrema, intercepts, symmetry, and asymptotes. 20. Sketch the graph of the function fx ( ) x2 x 2 1 using any extrema, intercepts, symmetry, and asymptotes. 27. Find the cubic function of the form fx ( ) ax 3 bx 2 cx d, where a 0 and the coefficients a,b,c,d are real numbers, which satisfies the conditions given below. Relative maximum: Ê Á3,0ˆ Relative minimum: Ê Á5, 2ˆ Inflection point: Ê Á4, 1ˆ 28. Find all relative extrema of the function fx ( ) x 2/3 6. Use the Second Derivative Test where applicable. 3
4 Name: ID: A 29. The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes. 36. The graph of f is shown in the figure. Sketch a graph of the derivative of f. 30. Determine the open intervals on which the graph of y 6x 3 8x 2 6x 5 is concave downward or concave upward. 31. Find the points of inflection and discuss the concavity of the function fx ( ) x x Find all points of inflection on the graph of the function fx ( ) 1 2 x4 2x Find all relative extrema of the function fx ( ) 4x 2 32x 62. Use the Second Derivative Test where applicable. 34. Find the points of inflection and discuss the concavity of the function fx ( ) sinx cosx on the interval Ê Á0,2 ˆ. 37. Find the relative extremum of fx ( ) 9x 2 54x 2 by applying the First Derivative Test. 38. Find the open interval(s) on which fx ( ) 2x 2 12x 8 is increasing or decreasing For the function fx ( ) ( x 1) 3 : (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Use a graphing utility to confirm your results. 35. Determine the open intervals on which the graph of fx ( ) 3x 2 7x 3 is concave downward or concave upward. 4
5 Name: ID: A 40. Use the graph of the function y x3 3x given 4 below to estimate the open intervals on which the function is increasing or decreasing. 45. Determine whether Rolle's Theorem can be applied to fx ( ) x 2 È 10x on the closed interval ÎÍ 0,10. If Rolle's Theorem can be applied, find all values of c in the open interval Ê Á0, 10ˆ such that f () c Find the critical number of the function fx ( ) 6x 2 108x Find a function f that has derivative f ( x) 12x 6 and with graph passing through the point (5,6). 48. The graph of f is shown in the figure. Sketch a graph of the derivative of f. 41. Determine whether the Mean Value Theorem can be applied to the function fx ( ) x 2 on the closed interval [3,9]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (3,9) such that f () c f() 9 f() 3. 9 () Which of the following functions passes through the point Ê Á0,10ˆ and satisfies f ( x) 12? 43. Sketch the graph of the function Ï 10x 25 0 x 5 f(x) Ô Ì ÓÔ x 2 5 x 8 and locate the absolute extrema of the function on È the interval ÎÍ 0, Identify the open intervals where the function fx ( ) 6x 2 6x 4 is increasing or decreasing. 49. The height of an object t seconds after it is dropped from a height of 250 meters is st () 4.9t Find the time during the first 8 seconds of fall at which the instantaneous velocity equals the average velocity. 50. Locate the absolute extrema of the function f(x) 3x 2 12x 3 on the closed interval [ 4, 4]. 51. Determine whether Rolle's Theorem can be applied to the function fx ( ) ( x 5) ( x 6) ( x 7)on the È closed interval ÎÍ 5,7. If Rolle's Theorem can be applied, find all numbers c in the open interval Ê Á 5,7 ˆ such that f () c 0. 5
6 Test 3 Review Answer Section SHORT ANSWER 1. ANS: f (2) is undefined. PTS: 1 DIF: Easy REF: OBJ: Understand the relationship between the value of the derivative and the extremum of a function NOT: Section ANS: ( 2x 3)dx PTS: 1 DIF: Medium REF: OBJ: Calculate the differential of y for a given function NOT: Section ANS: cos( 7x) 7x sin( 7x)dx PTS: 1 DIF: Medium REF: OBJ: Calculate the differential of y for a given function NOT: Section ANS: 6 7 x 4/7 dx PTS: 1 DIF: Medium REF: OBJ: Calculate the differential of y for a given function NOT: Section ANS: y 19x PTS: 1 DIF: Easy REF: OBJ: Write an equation of a line tangent to the graph of a function at a specified point NOT: Section ANS: 24 % PTS: 1 DIF: Difficult REF: OBJ: Apply calculus techniques to solve a minimum/maximum problem involving interest rates MSC: Application NOT: Section 3.7 1
7 7. ANS: x 2.5; y 1.25 PTS: 1 DIF: Difficult REF: OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the area of a rectangle bounded beneath a line MSC: Application NOT: Section ANS: x feet; y 4 4 feet PTS: 1 DIF: Medium REF: OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the area of a Norman window MSC: Application NOT: Section ANS: x 0; x 4 PTS: 1 DIF: Easy REF: a OBJ: Identify properties of the derivative of a function given the graph of the function NOT: Section ANS: x 2 PTS: 1 DIF: Easy REF: b OBJ: Identify properties of the second derivative of a function given the graph of the function NOT: Section ANS: 14,14 PTS: 1 DIF: Medium REF: OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the print area on a page MSC: Application NOT: Section ANS: y x 2 PTS: 1 DIF: Medium REF: OBJ: Identify the slant asymptote of the graph of a function NOT: Section ANS: none of the above PTS: 1 DIF: Medium REF: OBJ: Graph a function using extrema, intercepts, symmetry, and asymptotes NOT: Section 3.6 2
8 14. ANS: PTS: 1 DIF: Medium REF: OBJ: Graph a function using extrema, intercepts, symmetry, and asymptotes NOT: Section ANS: y 5x 4 PTS: 1 DIF: Medium REF: OBJ: Identify the slant asymptote of the graph of a function NOT: Section ANS: PTS: 1 DIF: Medium REF: OBJ: Graph a function's derivative given the graph of the function NOT: Section 3.6 3
9 17. ANS: PTS: 1 DIF: Medium REF: OBJ: Graph a function's derivative given the graph of the function NOT: Section ANS: PTS: 1 DIF: Medium REF: OBJ: Graph a function's derivative given the graph of the function NOT: Section 3.6 4
10 19. ANS: PTS: 1 DIF: Medium REF: OBJ: Graph a function using extrema, intercepts, symmetry, and asymptotes NOT: Section ANS: PTS: 1 DIF: Medium REF: OBJ: Graph a function using extrema, intercepts, symmetry, and asymptotes NOT: Section ANS: 0 PTS: 1 DIF: Medium REF: OBJ: Evaluate the limit of a function at infinity NOT: Section 3.5 5
11 22. ANS: 5 PTS: 1 DIF: Medium REF: OBJ: Evaluate the limit of a function at infinity NOT: Section ANS: critical numbers: x 0, x 2, x 2 PTS: 1 DIF: Easy REF: OBJ: Identify the critical numbers of a function NOT: Section ANS: 3 4 PTS: 1 DIF: Medium REF: OBJ: Evaluate the limit of a function at infinity NOT: Section ANS: 1.5 PTS: 1 DIF: Easy REF: OBJ: Evaluate limits at infinity in applications MSC: Application NOT: Section ANS: PTS: 1 DIF: Medium REF: OBJ: Identify the graph that matches the given function NOT: Section 3.5 6
12 27. ANS: fx ( ) 1 2 x3 6x x 27 PTS: 1 DIF: Medium REF: OBJ: Construct a function that has the specified relative extrema and inflection points NOT: Section ANS: relative minimum: Ê Á0, 6ˆ PTS: 1 DIF: Medium REF: OBJ: Identify all relative extrema for a function using the Second Derivative Test NOT: Section ANS: PTS: 1 DIF: Medium REF: OBJ: Graph a function's derivative and second derivative given the graph of the function NOT: Section ANS: Ê concave upward on, 4 ˆ Á 9 ; concave downward on Ê 4 9, ˆ Á PTS: 1 DIF: Medium REF: OBJ: Identify the intervals on which a function is concave up or concave down NOT: Section ANS: no inflection points; concave up on Ê Á 16, ˆ PTS: 1 DIF: Medium REF: OBJ: Identify all points of inflection for a function and discuss the concavity NOT: Section 3.4 7
13 32. ANS: Ê Á 0,0 ˆ Ê Á 2, 8ˆ PTS: 1 DIF: Easy REF: OBJ: Identify all points of inflection for a function NOT: Section ANS: relative min: f( 4) 2; no relative max PTS: 1 DIF: Medium REF: OBJ: Identify all relative extrema for a function using the Second Derivative Test NOT: Section ANS: none of the above PTS: 1 DIF: Medium REF: OBJ: Identify all points of inflection for a function and discuss the concavity NOT: Section ANS: concave upward on Ê Á, ˆ PTS: 1 DIF: Medium REF: OBJ: Identify the intervals on which a function is concave up or concave down NOT: Section ANS: PTS: 1 DIF: Medium REF: OBJ: Graph a function's derivative given the graph of the function NOT: Section 3.3 8
14 37. ANS: relative maximum: Ê Á3,83ˆ PTS: 1 DIF: Easy REF: c OBJ: Identify the relative extrema of a function by applying the First Derivative Test NOT: Section ANS: increasing on Ê Á,3 ˆ ; decreasing on Ê Á 3, ˆ PTS: 1 DIF: Easy REF: b OBJ: Identify the intervals on which the function is increasing or decreasing NOT: Section ANS: (a) x 1 (b) decreasing: Ê Á,1 ˆ ; increasing: Ê Á 1, ˆ (c) relative min: f() 1 0 PTS: 1 DIF: Medium REF: c OBJ: Identify the intervals on which the function is increasing or decreasing; Identify the relative extrema of a function by applying the First Derivative Test NOT: Section ANS: increasing on Ê Á, 2ˆ and Ê Á 2, ˆ ; decreasing on Ê Á 2,2 ˆ PTS: 1 DIF: Easy REF: OBJ: Estimate the intervals where a function is increasing and decreasing from a graph NOT: Section ANS: MVT applies; c = 6 PTS: 1 DIF: Medium REF: OBJ: Identify all values of c guaranteed by the Mean Value Theorem NOT: Section ANS: fx ( ) 12x 10 PTS: 1 DIF: Easy REF: OBJ: Construct a function that has a given derivative and passes through a given point NOT: Section ANS: left endpoint: Ê Á0, 25ˆ absolute minimum right endpoint: Ê Á8, 64ˆ absolute maximum PTS: 1 DIF: Medium REF: OBJ: Graph a function and locate the absolute extrema on a given closed interval NOT: Section 3.1 9
15 44. ANS: Ê decreasing:, 1 ˆ Á 2 ; increasing: Ê 1 2, ˆ Á PTS: 1 DIF: Medium REF: OBJ: Identify the intervals on which a function is increasing or decreasing NOT: Section ANS: Rolle's Theorem applies; c = 5 PTS: 1 DIF: Easy REF: OBJ: Identify all values of c guaranteed by Rolle's Theorem NOT: Section ANS: x 9 PTS: 1 DIF: Easy REF: a OBJ: Identify the critical numbers of a function NOT: Section ANS: fx ( ) 6x 2 6x 114 PTS: 1 DIF: Medium REF: OBJ: Construct a function that has a given derivative and passes through a given point NOT: Section ANS: PTS: 1 DIF: Medium REF: OBJ: Graph a function's derivative given the graph of the function NOT: Section
16 49. ANS: 4 seconds PTS: 1 DIF: Easy REF: b OBJ: Identify all values of c guaranteed by the Mean Value Theorem in applications MSC: Application NOT: Section ANS: absolute max: Ê Á 4, 99 ˆ ; absolute min: Ê Á 2, 9 ˆ PTS: 1 DIF: Medium REF: OBJ: Locate the absolute extrema of a function on a given closed interval NOT: Section ANS: Rolle's Theorem applies; c 6 3 3,6 3 3 PTS: 1 DIF: Medium REF: OBJ: Identify all values of c guaranteed by Rolle's Theorem NOT: Section
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