Work the following on notebook paper. You may use your calculator to find

Transcription

1 CALCULUS WORKSHEET ON 3.1 Work the following on notebook paper. You may use your calculator to find f values. 1. For each of the labeled points, state whether the function whose graph is shown has an absolute maimum, absolute minimum, local maimum, local minimum, or neither. A C E B D F G. Sketch the graph of a function f that is continuous on [0, 6] and has the given properties: absolute maimum at = 0, absolute minimum at = 6, local minimum at =, and local maimum at = 4. Find the absolute maimums and absolute minimums of f on the given closed interval by using the Candidates Test, and state where these values occur. f , f sin cos 0, f 1, 4 7. f 6 1, , 0 5. f 3 y on, 3? 8. What is the smallest possible slope to 4 9. A particle moves along a straight line according to the position function st t 4t 3 6t 0. Find the maimum and minimum velocity on 0, 3. For what values of t do the maimum and minimum occur? Remember that velocity is the derivative of position.

2 Answers to Worksheet on A abs. min. B local ma. C neither D local min. E local ma. F local min. G neither. Graph 3. Ma.: (, 9). Min.: 1,0 4. Ma.:, Min.: 1 1, 3 5. Ma.: (8, 48). Min.: (0, 0) and (0, 0) 6. Ma.: 3, , 1 Min.: 7. Ma.: 1, 8. Min.: (3, 0) 8. The smallest slope on, 3 is. 9. Ma. velocity = 36 when t = 3. Min. velocity = 0 when t = 0.

3 CALCULUS WORKSHEET ON ROLLE S TH. & MEAN VALUE THEOREM Work the following on notebook paper. Do not use your calculator on problems 1 3. On problems 1, determine whether or not the conditions of Rolle s Theorem are met. If the conditions are met, find the value of c that the theorem guarantees. f , f 4 3 4, 3 On problems 3 4, determine whether or not the conditions of the Mean Value Theorem are met. If the conditions are met, find the value of c that the theorem guarantees. 3 f 1, f cos cos 0, (Use your calculator.) You may use your calculator on problems 5 6, but be sure you show supporting work. 5. A trucker handed in a ticket at a toll booth showing that in hours, the truck had covered 159 miles on a toll road on which the speed limit was 65 mph. The trucker was cited for speeding. Why? 6. A marathoner ran the 6. mile New York marathon in hours, 1 minutes. Show that at least twice, the marathoner was running at eactly 11 mph Given f 3. f 3 f in the following window: 8. Use your calculator to graph f and : 7.9, 7.9, y: 10, 10. Sketch below. 9. The relative maimum and minimum values of f occur at = 10. f = 0 or f is undefined at = 11. f is increasing on what interval(s)? 1. f is positive on what interval(s)? 13. f is decreasing on what interval(s)? 14. f is negative on what interval(s)? TURN->>>

4 15. Given f Use your calculator to graph f and f : 1.975, 1.975, y:,. Sketch below. f in the following window: 17. The relative maimum and minimum values of f occur at = 18. f = 0 or f is undefined at = f f f f is increasing on what interval(s)? is positive on what interval(s)? is decreasing on what interval(s)? is negative on what interval(s)? Use your answers to problems 7 to complete the following statements: 3. The relative maimum and minimum values of f occurred when 4. The function f is increasing when 5. The function f is decreasing when

5 Answers to Worksheet on Rolle s Theorem and the Mean Value Theorem 1. Rolle s Th. applies.. Rolle;s Th. applies. 3. MVT applies. st 1 c 3 11 c c 3 4. MVT applies. c = 0.17 and s s0 5. Let distance traveled in t hours. s0 0, s 159, 79.5mph. 0 Between t = 0 and t = hr, there must be a time t where the speed was 79.5 mph by the MVT. Since 79.5 > 65, the trucker was speeding. s. s0 6. Let st distance he ran in t hours. s 0 0, s. 6., mph.. 0 Between t = 0 and t =. hr, there must be a time where the speed was mph by the MVT. By the IVT, there must be a time another time 7. t f 3 8. Sketch 9. 1 and and 3, 1 3, , 1 3, f Sketch , , ,1.,1 3. f 0 or f f 4. 0 f 5. 0 between t m is undefined t m t 1 between 0 and and. hr when the speed was 11 mph. t m when the speed was 11 mph and

6 CALCULUS WORKSHEET ON DERIVATIVES Work the following on notebook paper. Do not use your calculator. 1. Given 3 f Use the Second Derivative Test to find whether f has a local maimum or a local minimum at = 4. Justify your answer.. Given f 3 sin. Use the Second Derivative Test to find whether f has a local maimum or a local minimum at =. 6 Justify your answer. On problems 3 5, find the critical points of each function, and determine whether they are relative maimums or relative minimums by using the Second Derivative Test whenever possible f f f 5. sin cos, 0 6. Consider the curve given by 4y 7 3 y. dy 3y (a) Show that. d 8y 3 (b) Show that there is a point P with -coordinate 3 at which the line tangent to the curve at P is horizontal. Find the y-coordinate of P. (c) Find the value of d y d at the point P found in part (b). Does the curve have a local maimum, a local minimum, or neither at point P? Justify your answer. On problems 7 8, the graph of the derivative,, of a function f is shown. (a) On what interval(s) is f increasing or decreasing? Justify your answer. (b) At what value(s) of does f have a local maimum or local minimum? Justify your answer. f 7. y 8. f y f TURN->>>

7 9. The y graph of the second derivative, of the inflection points of f. Justify your answer. f, of a function f is shown. State the -coordinates f For what values of a and b does the function f a b have a local maimum when 11. The graph of a function f is shown on the right. Fill in the chart with +,, or 0. Point A B C f 3 and a local minimum when f f A 1? B y C

8 Answers to Worksheet on Derivatives 1. f has a local maimum at 4. f has a local minimum at 3. Crit. pts: 0, 3 and, 1 0, 3 since f f and since f 0 and f is a relative maimum because f0 0 and f0 6. is a relative minimum because f 0 and f Crit. pts:, 4 and, 4 is a relative maimum because f f (, 4) is a relative minimum because f 0 and f 1., 1, 4 0 and Crit. pts:, and, 4 4 3, 4 is a relative maimum because 3 3 f 0 and f , 4 is a relative minimum because 7 7 f 0 and f AB 4/BC 4 See AP Central for full solution. (a) Show with implicit diff. (b) y = d y dy (c). At (3, ), 0 d 7 d and d y so the curve has a local maimum at (3, ) by the d 7 Second Derivative Test. 7. (a) f is increasing on,0 and 3, because f 0. f is decreasing on 0, 3 because f 0. (b) f has a relative maimum at = 0 because f has a relative minimum at = 3 because 8. (a) f is decreasing on, 1 and 3, 5 f is increasing on and because f 0. 1, 3 f changes from positive to negative there. f changes from negative to positive there. f 5, because 0. (b) f has a relative minimum at 1 and = 5 because positive there. f has a relative maimum at = 3 because f changes from negative to 9. f has an inflection point at = 0 and at = 7 because f 0 and f f changes from positive to negative there. changes from positive to negative or vice versa there. f does not have an inflection point at = 4 because f does not change signs at = a = 6, b = Point f f f A + + B + 0 C +

9 CALCULUS WORKSHEET A SUMMARY OF CURVE SKETCHING Work the following on notebook paper. On each problem, find: 1) the - and y-intercepts ) symmetry 3) asymptotes 4) intervals where f is increasing and decreasing 5) relative etrema 6) intervals where f is concave up and concave down 7) inflection points Justify your answers and use the information found above to sketch the graph. 1. y 9. y 1 3. y 4

10 Answers to Worksheet A Summary of Curve Sketching See bottom of page for graphs 1. Intercepts: (0, 0) Vert. : 3, 3 Horiz.: y = 0 Slant: none y-ais symmetry y is increasing for 3 and 3 0 because is positive there. y is decreasing for 0 3 and 3 because y is negative there. y has a relative maimum at (0, 0) because changes from positive to negative there. y is concave up on 3 and 3 because is positive there. y is concave down on because is negative there. y has no inflection points. 3 < 3. Intercepts: (0, 0) Vert. : 1, 1 Horiz.: y = 1 Slant: none origin symmetry y is increasing for 1, 1 1, and 1 because is positive there. y is never decreasing. y has no relative etrema. y is concave up on 1 and 0 1 because is positive there. y is concave down on 1 < 0 and 1 because is negative there. y has an inflection point at (0, 0) because y changes from negative to positive there. 3. Intercepts: none Domain: Vert. :, Horiz.: y = 1, y 1 Slant: none y-ais symmetry y is decreasing for and because is negative there. y has no relative etrema. y is concave down on because y is negative there. y is concave up on because y is positive there. y has no inflection points. y y y y y y y y y y y

11 CALCULUS WORKSHEET ON f, f, f 1. Graph of Let f be a function that has domain the closed interval [ 1, 4] and range the closed f 1 1, f 0 0, and f 4 1. Also let f has the derivative interval [ 1, ]. Let f function that is continuous and that has the graph shown in the figure above. (a) Find all values of for which f assumes a relative maimum. Justify your answer. f (b) Find all values of for which f assumes its absolute minimum. Justify your answer. (c) Find the intervals on which f is concave downward. (d) Give all values of for which f has a point of inflection. (e) Sketch the graph of f. f 3 4 and f A function f is continuous on the closed interval [ 3, 3] such that The functions f f f and f have the properties given in the table below = Positive Fails to Negative 0 Negative eist Positive Fails to Positive 0 Negative eist (a) What are the -coordinates of all absolute maimum and absolute minimum points of f on the interval [ 3, 3]? Justify your answer. (b) What are the -coordinates of all points of inflection of f on the interval [ 3, 3]? Justify your answer. (c) Sketch a graph that satisfies the given properties of f.

12 f 1, then the graph of f has inflection points when = 3. If (A) (D) 1 only (B) only (C) 1 and 0 only 1 and only (E) 1, 0, and only 3 4. An equation of the line tangent to y 3 at its point of inflection is (A) y 6 6 (B) y 3 1 (C) y 10 (D) (E) y31 y If the graph of y a b 4 has a point of inflection at, what is the value of b? 1, 6 (A) 3 (B) 0 (C) 1 (D) 3 (E) It cannot be determined from the information given. 6. For what value of does the function f 3 (A) 3 (B) 7 3 (C) have a relative maimum? 5 (D) 7 3 (E) 5 7. Let f be the function defined by 3 for 0 f. Which of the following statements for 0 about f is true? (A) f is an odd function. (B) f is discontinuous at = 0. f 0 0 (E) f 0 for 0. (C) f has a relative maimum. (D) 8. If f sin, then there eists a number c in the interval 3 that satisfies the conclusion of the Mean Value Theorem. Which of the following could be c? (A) (B) 3 (C) 5 (D) (E)

13 Answers to Worksheet 1 on f, f, f 1. (a) f has a relative maimum at = because changes from positive to negative there. (b) f doesn t have a relative minimum since f doesn t change from negative to positive so the absolute minimum must occur at an endpoint. f f f 1 1 and 4 1 so the absolute minimum occurs at 1. (c) f is concave down on 1 0 and 1 3 because is decreasing there. (d) f has an IP at = 0 and at = 3 because changes from decreasing to increasing there. f has an IP at = 1 because changes from increasing to decreasing there. (e) f f f. (a) f changes from positive to negative at Since 1 so f has a relative maimum at 1. f 0 on 3 1 and f 0 on 1 3, the absolute maimum of f occurs at 1. f does not change from negative to positive anywhere so f does not have a relative minimum. Therefore, the absolute minimum must occur at an endpoint. f 3 4 and f 3 1 so the absolute minimum occurs at = 3. (b) f has an IP at = 1 because f 1 0 and f (c) changes from positive to negative there. 3. C 4. B 5. B 6. D 7. E 8. D

14 CALCULUS WORKSHEET ON f, f, f Work the following on notebook paper. For problems 1 6, give the signs of the first and second derivatives for each of the following functions. 7. (a) Water is flowing at a constant rate into a cylindrical container standing vertically. Sketch a graph showing the depth of water against time. (b) Water is flowing at a constant rate into a cone-shaped container standing in its verte. Sketch a graph showing the depth of the water against time. 8. If water is flowing at a constant rate into the Grecian urn in the figure on the right, sketch a graph of the depth of the water against time. Mark on the graph the time at which the water reaches the widest point of the urn. a b and f a f b 1, which of the 9. Let f be a polynomial function with degree greater than. If following must be true for a least one value of between a and b? I. f 0 II. f 0 f III. 0 (A) None (B) I only (C) II only (D) I and II only (E) I, II, and III 10. The absolute maimum value of 3 f 3 1 on the closed interval [, 4] occurs at = (A) 4 (B) (C) 1 (D) 0 (E) At what value of does the graph of y have a point of inflection? 3 (A) 0 (B) 1 (C) (D) 3 TURN->>>

15 f 1. The graph of, the derivative of the function f, is shown above. Which of the following statements is true about f? (A) f is decreasing for. (B) f is increasing for. (C) f is increasing for 1. (D) f has a local minimum at = 0. (E) f is not differentiable at 1 and The second derivative of the function f is given by f a b. The graph of shown above. For what values of does the graph of f have a point of inflection? (A) 0 and a only (B) 0 and m only (C) b and j only (D) 0, a, and b (E) b, j, and k 14. For all in the closed interval [, 5], the function f has a positive first derivative and a negative second derivative. Which of the following could be a table of values for f? f is 15. The graphs of the derivatives of the functions f, g, and h are shown above. Which of the functions f, g, or h have a relative maimum on the open interval a b? (A) f only (B) g only (C) h only (D) f and g only (E) f, g, and h

16 Answers to Worksheet on 1. f f 0, 0. f f 0, 0 3. f f 0, 0 4. f f 0, 0 5. f f 0, 0 6. f f 0, 0 f, f, f 7. (a) (b) 9. C 10. A 11. C 1. B 13. A 14. B 15. A