Math2413-TestReview2-Fall2016
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1 Class: Date: Math413-TestReview-Fall016 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the value of the derivative (if it exists) of the function fx ( ) 15 x at the extremum point Ê Á0, 15ˆ. a. 13 b. 0 c. 15 d. 15 e. does not exist. Find all critical numbers of the function g( x) x 4 4x. a. critical numbers: x 0, x, x b. critical numbers: x 0, x, x c. critical numbers: x, x d. critical numbers: x, x e. no critical numbers 3. Locate the absolute extrema of the function f(x) 3x 1x 3 on the closed interval [ 4, 4]. a. absolute max: Ê Á4, 99ˆ ; no absolute min b. no absolute max or min c. no absolute max; absolute min: (4, 99) d. absolute max: Ê 4, 99 Ê Á, 9 ˆ e. absolute max: Ê, 9 Ê Á 4, 99 ˆ 1
2 4. Determine whether Rolle's Theorem can be applied to the function fx ( ) ( x 3) ( x ) on the closed È interval ÎÍ 3,. If Rolle's Theorem can be applied, find all numbers c in the open interval Ê Á 3, ˆ such that f () c 0. a. Rolle's Theorem applies; 8 3 b. Rolle's Theorem applies; 8 5 c. Rolle's Theorem applies; 4 5 d. Rolle's Theorem applies; 4 3 e. Rolle's Theorem does not apply 5. The height of an object t seconds after it is dropped from a height of 50 meters is st () 4.9t 50. Find the time during the first 8 seconds of fall at which the instantaneous velocity equals the average velocity. a. 3 seconds b seconds c seconds d. 4 seconds e..45 seconds
3 6. Use the graph of the function y x3 4 is increasing or decreasing. 3x given below to estimate the open intervals on which the function a. increasing on Ê Á, ˆ and Ê Á, ˆ ; decreasing on Ê Á, ˆ b. increasing on Ê Á, ˆ and Ê Á, ˆ ; decreasing on Ê Á, ˆ c. increasing on Ê Á,ˆ ; decreasing on Ê Á, ˆ and Ê Á, ˆ d. increasing onê Á, ˆ and Ê Á, ˆ ; decreasing on Ê Á, ˆ e. increasing on Ê Á, ˆ and Ê Á, ˆ ; decreasing on Ê Á, ˆ 7. Find the open interval(s) on which fx ( ) x 1x 8 is increasing or decreasing. a. increasing on Ê Á,6 ˆ ; decreasing on Ê Á 6, ˆ b. increasing on Ê Á,16 ˆ ; decreasing on Ê Á 16, ˆ c. increasing on Ê Á,3 ˆ ; decreasing on Ê Á 3, ˆ d. increasing on Ê Á,4 ˆ ; decreasing on Ê Á 4, ˆ e. increasing on Ê Á,3 ˆ ; decreasing on Ê Á 3, ˆ 3
4 8. For the function fx ( ) 4x 3 48x 6: (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. Then use a graphing utility to confirm your results. a. (a) x = 0, (b) increasing: Ê Á,0 ˆ Ê Á, ˆ ; decreasing: Ê Á 0, ˆ (c) relative max: f() 0 6 ; relative min: f() 154 b. (a) x = 0, (b) decreasing: Ê Á,0 ˆ Ê Á, ˆ ; increasing: Ê Á 0, ˆ (c) relative min: f() 0 6 ; relative max: f() 154 c. (a) x = 0, (b) increasing: Ê Á,0 ˆ Ê Á, ˆ ; decreasing: Ê Á 0, ˆ (c) relative max: f() 0 6 ; no relative min. d. (a) x = 0, 8 (b) increasing: Ê Á,0 ˆ Ê Á8, ˆ ; decreasing: Ê Á 0,8 ˆ (c) relative max: f() 0 6 ; relative min: f() e. (a) x = 0,8 (b) decreasing: Ê Á,0 ˆ Ê Á8, ˆ ; increasing: Ê Á 0,8 ˆ (c) relative min: f() 0 6 ; relative max: f() Determine the open intervals on which the graph of fx ( ) 3x 7x 3 is concave downward or concave upward. a. concave downward on Ê Á, ˆ b. concave upward on Ê Á,0 ˆ ; concave downward on Ê Á 0, ˆ c. concave upward on Ê Á,1 ˆ ; concave downward on Ê Á 1, ˆ d. concave upward on Ê Á, ˆ e. concave downward on Ê Á,0 ˆ ; concave upward on Ê Á 0, ˆ 4
5 10. Find all points of inflection on the graph of the function fx ( ) 1 x4 x 3. a. Ê Á, 8ˆ b. Ê Á0,0ˆ c. Ê Á 0,0 ˆ, Ê Á 4,0 ˆ d. Ê Á0,0ˆ Ê Á, 8ˆ e. Ê Á, 8ˆ 11. Find the points of inflection and discuss the concavity of the function fx ( ) x x 16. a. no inflection points; concave up on Ê Á 16, ˆ b. no inflection points; concave down on Ê Á 16, ˆ c. inflection point at x = 16; concave up on Ê Á 16, ˆ d. inflection point at x = 0; concave up on Ê Á 16,0 ˆ ; concave down on Ê Á 0, ˆ e. inflection point at x = 16; concave down on Ê Á 16, ˆ 1. Find the points of inflection and discuss the concavity of the function fx ( ) sinx cosx on the interval Ê Á 0, ˆ. a. no inflection points. concave up on Ê Á0, ˆ Ê b. concave upward on 0, 1 Á ˆ ; concave downward on Ê 5 Á, ˆ inflection point at ; Ê 0, 1 Á ˆ c. no inflection points. concave down on Ê Á0, ˆ Ê d. concave downward on 0, 1 Á ˆ ; concave upward on Ê 5 Á, ˆ inflection point at ; Ê 0, 1 Á ˆ e. none of the above 5
6 13. Find all relative extrema of the function fx ( ) x 4 3x 3 4. Use the Second Derivative Test where applicable. a. relative max: Ê Á4,1188ˆ ; no relative min b. relative min: Ê Á1, 1380ˆ ; no relative max c. relative min: Ê Á4,1188ˆ ; no relative max d. relative max: Ê Á1,1380ˆ ; no relative min e. no relative max or min 14. Find the limit. lim x a. b. 3 Ê Á c. d. 3 e x ˆ 15. Find the limit. lim x 8x 5x 4 a. 1 b. 1 c. 0 d. e
7 16. Find the limit. lim x 6x 64x 5 a. 3 3 b. 3 4 c. 1 d. 6 e. 17. Find the limit. lim x 7x 6 64x x a. b c. 1 d. 7 e. 18. A model for the average typing speeds S (words per minute) of a typing student after t weeks of lessons is given by S 81t 1 t, t 0. Find lim S. t a. 96 words per minute b. 1 words per minute c. 56 words per minute d. 16 words per minute e. 81 words per minute 7
8 19. Determine the slant asymptote of the graph of fx ( ) 5x 9x 5. x 1 a. y 5x 4 b. y 5x 4 c. y 5x 4 d. y 5x 4 e. no slant asymptotes 0. The graph of f is shown below. For which value of x is f ( x) zero? a. x b. x 0 c. x d. x 6 e. x 4 8
9 1. The graph of f is shown below. On what interval is f an increasing function? a. Ê Á0, ˆ b. Ê Á 1, ˆ c. Ê Á, ˆ d. Ê Á1, ˆ e. Ê Á, ˆ. Find two positive numbers whose product is 181 and whose sum is a minimum. a. 181, 181 b. 181, c. 0, 181 d. 1, 181 e , Find two positive numbers such that the sum of the first and twice the second is 56 and whose product is a maximum. a. 8 and 8 b. 8 and 14 c. and 17 d. 34 and 11 e. 4 and 16 9
10 4. Find the length and width of a rectangle that has perimeter 48 meters and a maximum area. a. 1 m; 1 m. b. 16 m; 9 m. c. 1m; 3 m. d. 13 m; 11 m. e. 6 m; 18 m. 5. A rectangle is bounded by the x- and y-axes and the graph of y should the rectangle have so that its area is a maximum? ( 5 x) (see figure). What length and width a. x.5; y 3 b. x 3; y 5 c. x.5; y 1.5 d. x 5; y 3 e. x 1.5; y.5 10
11 6. A farmer plans to fence a rectangular pasture adjacent to a river (see figure). The pasture must contain 70,000 square meters in order to provide enough grass for the herd. No fencing is needed along the river. What dimensions will require the least amount of fencing? a. x = 600 and y = 100 b. x = 1000 and y = 70 c. x = 100 and y = 600 d. x = 70 and y = 1000 e. none of the above 7. Use Newton s Method to approximate the zero(s) of the function fx ( ) x 3 x 1 accurate to three decimal places. a b c d e Use Newton's Method to approximate the zero(s) of the function fx ( ) x 5 4x 1 accurate to three decimal places. a b c. 0.9 d e
12 9. Use Newton s Method to approximate the x-value of the indicated point of intersection of the two graphs accurate to three decimal places.continue the process until two successive approximations differ by less than [Hint: Let h( x) fx ( ) gx ( ).] fx ( ) 3x 1 gx ( ) x 5 a b c d e Find the point on the graph of fx ( ) x that is closest to the point Ê Á8, 4ˆ. Use Newton's Method to approximate all numerical values of the solution to three decimal places. a. Ê Á0.168, 0.08ˆ b. Ê Á0.78,0.61ˆ c. Ê Á0.410,0.168ˆ d. Ê Á.059,4.39ˆ e. Ê Á4.39, ˆ 31. Compare dy and y for y x 4 1 at x = 0 with x dx 0.0. Give your answers to four decimal places. a. dy ; y b. dy 0.000; y c. dy ; y d. dy ; y e. dy ; y
13 3. Find the differential dy of the function y x 3x. a. ( x 3)dx Ê b. x 3 3x ˆ x Á dx Ê c. x ˆ 3x Á dx Ê d. 1 3 x3 3 ˆ x x Á dx e. ( x 3)dx 13
14 Math413-TestReview-Fall016 Answer Section MULTIPLE CHOICE 1. ANS: E PTS: 1 DIF: Medium REF: Section 3.1 OBJ: Understand the relationship between the value of the derivative and the extremum of a function. ANS: B PTS: 1 DIF: Easy REF: Section 3.1 OBJ: Identify the critical numbers of a function 3. ANS: D PTS: 1 DIF: Medium REF: Section 3.1 OBJ: Locate the absolute extrema of a function on a given closed interval 4. ANS: A PTS: 1 DIF: Medium REF: Section 3. OBJ: Identify all values of c guaranteed by Rolle's Theorem 5. ANS: D PTS: 1 DIF: Easy REF: Section 3. OBJ: Identify all values of c guaranteed by the Mean Value Theorem in applications 6. ANS: B PTS: 1 DIF: Easy REF: Section 3.3 OBJ: Estimate the intervals where a function is increasing and decreasing from a graph 7. ANS: E PTS: 1 DIF: Easy REF: Section 3.3 OBJ: Identify the intervals on which the function is increasing or decreasing 8. ANS: D PTS: 1 DIF: Medium REF: Section 3.3 OBJ: Identify the relative extrema of a function by applying the First Derivative Test; Identify the relative extrema of the function 9. ANS: D PTS: 1 DIF: Medium REF: Section 3.4 OBJ: Identify the intervals on which a function is concave up or concave down 10. ANS: D PTS: 1 DIF: Easy REF: Section 3.4 OBJ: Identify all points of inflection for a function 11. ANS: A PTS: 1 DIF: Medium REF: Section 3.4 OBJ: Identify all points of inflection for a function and discuss the concavity 1. ANS: E PTS: 1 DIF: Medium REF: Section 3.4 OBJ: Identify all points of inflection for a function and discuss the concavity 13. ANS: B PTS: 1 DIF: Medium REF: Section 3.4 OBJ: Identify all relative extrema for a function using the Second Derivative Test 14. ANS: E PTS: 1 DIF: Medium REF: Section 3.5 OBJ: Evaluate the limit of a function at infinity 15. ANS: C PTS: 1 DIF: Medium REF: Section 3.5 OBJ: Evaluate the limit of a function at infinity 16. ANS: B PTS: 1 DIF: Medium REF: Section 3.5 OBJ: Evaluate the limit of a function at infinity 1
15 17. ANS: A PTS: 1 DIF: Medium REF: Section 3.5 OBJ: Evaluate the limit of a function at infinity 18. ANS: E PTS: 1 DIF: Medium REF: Section 3.5 OBJ: Evaluate limits at infinity in applications 19. ANS: B PTS: 1 DIF: Medium REF: Section 3.6 OBJ: Identify the slant asymptote of the graph of a function 0. ANS: C PTS: 1 DIF: Easy REF: Section 3.6 OBJ: Identify properties of the second derivative of a function given the graph of the function 1. ANS: E PTS: 1 DIF: Easy REF: Section 3.6 OBJ: Identify properties of the derivative of a function given the graph of the function. ANS: A PTS: 1 DIF: Easy REF: Section 3.7 OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the sum of two numbers 3. ANS: B PTS: 1 DIF: Easy REF: Section 3.7 OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the product of two numbers 4. ANS: A PTS: 1 DIF: Medium REF: Section 3.7 OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the area of a rectangle 5. ANS: C PTS: 1 DIF: Difficult REF: Section 3.7 OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the area of a rectangle bounded beneath a line 6. ANS: B PTS: 1 DIF: Easy REF: Section 3.7 OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the area of a rectangle 7. ANS: E PTS: 1 DIF: Medium REF: Section 3.8 OBJ: Estimate a zero of a function using Newton's Method 8. ANS: B PTS: 1 DIF: Medium REF: Section 3.8 OBJ: Estimate a zero of a function using Newton's Method 9. ANS: A PTS: 1 DIF: Medium REF: Section 3.8 OBJ: Estimate the intersection point of two graphs using Newton's Method 30. ANS: B PTS: 1 DIF: Medium REF: Section 3.8 OBJ: Estimate an extremum involving distance between points using calculus 31. ANS: C PTS: 1 DIF: Medium REF: Section 3.9 OBJ: Compare the change of y to the differential of y at a given point 3. ANS: A PTS: 1 DIF: Medium REF: Section 3.9 OBJ: Calculate the differential of y for a given function
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