Chapter (AB/BC, non-calculator) (a) Find the critical numbers of g. (b) For what values of x is g increasing? Justify your answer.

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1 Chapter 3 1. (AB/BC, non-calculator) Given g ( ) : (a) Find the critical numbers of g. (b) For what values of is g increasing? Justify your answer. (c) Identify the -coordinate of the critical points at which g has a relative minimum. Justify your answer.

2 2. (AB/BC, non-calculator) Let f ( ) 2 cos2. (a) Find the maimum value of f for 0. Justify your answer. (b) Eplain how the conditions of the Mean Value Theorem are satisfied by f for 0. Find the value of, 0, whose eistence is guaranteed by the Mean Value Theorem.

3 3. (AB/BC, non-calculator) Let f( ). (a) State f ( ) and identify the value(s) of for which f does not eist. (b) For what values of is f decreasing? Justify your answer. (c) For what values of is the graph of f concave down? Show the work that leads to your answer. (d) Does the graph of f contain an inflection point? Justify your answer.

4 4. (AB/BC, calculator neutral) y In the figure above, f, the derivative of function f, is shown. f is a twice differentiable function on,. f ''( 0.8) 0 and f ''(1.3) 0. (a) Name the value(s) of for which f has a relative minimum. Justify your answer. (b) For what values of is f increasing? Justify your answer. (c) For what values of is the graph of f concave down? Justify your answer. (d) Is f ( 0.5) f (0) positive or negative? Justify your answer

5 5. (AB/BC, calculator neutral) The depth of the water at the end of a pier is shown in the table below and is modeled by differentiable function D for t 0. Selected values of D are shown in the table below. D is epressed in meters and t is the number of hours since midnight t 0. t (hours) Dt ()(meters) (a) Use the data in the table to estimate the rate at which the depth of the water is changing at 3:30 am and 7:40 am. Include units. (b) What is the least number of times in the interval 0 t 12for which D() t 0? Justify your answer. (c) Use the method of linear approimation to estimate the depth of the water at 2:30 am ( t 2.5). Show the work that leads to your answer.

6 6. (AB/BC, non-calculator) y The graph of f, the derivative of f, is shown above. The function f is differentiable on the interval 5 4. f ( 4) 0. (a) Find f ( 1). (b) Find f ( 1). (c) Find the coordinate of each inflection point for the graph of f on the interval 5 4. (d) If 2 g ( ) f( ) sin, is g increasing or decreasing at? Justify your answer. 4

7 7. (AB/BC, calculator neutral) Given: f is continuous for, ; f (2) 4 ; lim f( ) f ( ) positive does not eist negative f ( ) negative does not eist positive (a) For what values of is f increasing? (b) Does f have a relative maimum at 4? Eplain. (c) If possible, name the -coordinate of an inflection point on the graph of f. Justify your answer. (d) Does the Mean Value Theorem apply over the interval [3,5]? Justify your answer. (e) Sketch a possible graph of f using the information from the table.

8 8. (AB/BC, calculator neutral) y Consider the graph of y f( ) shown above. If f is a function such that f and f are defined in a region around = 2, then which of the following must be true? (a) f (2) f (2) (b) f(2) f(2) (c) f(2) f(2) (d) f (2) f (2) (e) f(2) f(2)

9 9. (AB/BC, non-calculator) 3 2 The position of an object along a vertical line is given by st () t 3t 9t 5, where s is measured in feet and t in seconds. The maimum velocity of the object in the time interval 0t 4is (a) ft 32 sec (b) ft 16 sec (c) ft 12 sec (d) ft 9 sec (e) ft 15 sec

10 10. (AB/BC, non-calculator) Which of the following is true for the graph of 4 f( )? 2 I. 2 is a vertical asymptote of the graph of f. II. f is decreasing for,. III. f is concave down for,2 (a) None (b) I and II only (c) I and III only (d) III only (e) I, II and III

11 Chapter 3 (Solutions) Question 1 Given g ( ) : (a) Find the critical numbers of g. (b) For what values of is g increasing? Justify your answer. (c) Identify the -coordinate of the critical points at which g has a relative minimum. Justify your answer. 3 2 (a) g( ) 24 6( 6) : derivative 5 : 1: equation 2: solutions ( 6) ; 4, because g( ) 0. 2: 1: answer 1: justification (b) 4, (c) Relative minimum at 4 because g changes from 2: 1: critical number 1: justification negative to positive at 4.

12 Question 2 Let f ( ) 2 cos2. (a) Find the maimum value of f for 0. Justify your answer. (b) Eplain how the conditions of the Mean Value Theorem are satisfied by f for 0. Find the value of, 0, whose eistence is guaranteed by the Mean Value Theorem. (a) f ( ) 2 2sin(2 ) 22sin(2 ) 0 4 1: derivative 1: equation 5: 1: solution 1: eliminates = as a candidate 4 1: answer f() Maimum value of f is 2 1.

13 Question 2 (cont.) (b) f is continuous on 0, and differentiable on 0, f( ) f(0) 22sin(2 ) 0 1: justification 4: 1: equation 1: rejecting 0 and 1: solution 2

14 Question 3 Let f( ). (a) State f ( ) and identify the value(s) of for which f does not eist. (b) For what values of is f decreasing? Justify your answer. (c) For what values of is the graph of f concave down? Show the work that leads to your answer. (d) Does the graph of f contain an inflection point? Justify your answer. 1 (a) f( ) 4 3: 2 2 : derivative 1: value f does not eist at 0. because f( ) 0. 2: 1: answer 1: justification (b),0 0, 2 (c) f( ) 2: 1: answer 3 1: justification,0because f( ) 0. (d) No, because although f changes sign at 0, 2: 1: answer 1: justification f is discontinuous at = 0.

15 Question 4 y f( ) y In the figure above, f, the derivative of function f, is shown. f is a twice differentiable function on,. f ( 0.8) 0 and f (1.3) 0. (a) Name the value(s) of for which f has a relative minimum. Justify your answer. (b) For what values of is f increasing? Justify your answer. (c) For what values of is the graph of f concave down? Justify your answer. (d) Is f(0) f( 0.5) positive or negative? Justify your answer (a) f has a relative minimum at 0 since f changes 2: 1: answer 1: justification from negative to positive at 0. (b) f is increasing on the interval 0, 2 because f( ) 0. 2: 1: answer 1: justification

16 Question 4 (cont.) (c) f is concave down on the interval 2, , 3:1: 2:answer justification because f( ) 0 (d) Negative because the Mean Value Theorem can be applied 2: 1: answer 1: justification and f( ) 0for

17 Question 5 The depth of the water at the end of a pier is shown in the table below and is modeled by differentiable function D for t 0. Selected values of D are shown in the table below. D is epressed in meters and t is the number of hours since midnight t 0. t (hours) Dt () (meters) (a) Use the data in the table to estimate the rate at which the depth of the water is changing at 3:30 am and 7:40 am. Include units. (b) What is the least number of times in the interval 0 t 12for which D() t 0? Justify your answer. (c) Use the method of linear approimation to estimate the depth of the water at 2:30 am ( t 2.5). Show the work that leads to your answer. (a) The depth of the water is changing at approimately m 0.6 at 3:30 am. 3: hr 2:answers 1: units and m 0.8 hr at 7:40 am.

18 Question 5 (cont.) D(2) D(0) (b) 0 and 2 0 D(5) D(2) 0 so the Mean Value and 5 2 3: 1: answer 2: justification the Intermediate Value Theorems indicate D( t) 0 for some 0,5 t. D(7) D(5) 0 and 7 5 D(8) D(7) 0 so for the same reason 8 7 as shown above D( t) 0 for some 5,8 t. (c) D(2.5) D(2)(2.52) D(2) D D(2.5) : linear approimation 3: equation 1: answer

19 Question 6 y The graph of f, the derivative of f, is shown above. The function f is differentiable on the interval 5 4. f ( 4) 0. (a) Find f ( 1). (b) Find f ( 1). (c) Find the coordinate of each inflection point for the graph of f on the interval 5 4. (d) If 2 g ( ) f( ) sin, is g increasing or decreasing at? Justify your answer. 4 (a) 1 f ( ) 1for 3 0 2: 1: equation 3 1: answer 2 f ( 1) 3

20 Question 6 (cont.) (b) 1 f ( 1) 1: answer 3 (c) Two conditions must be met for = c to correspond to an inflection point of f : i). there must be a change in concavity at = c (f must change sign), and ii). the tangent 3: answers line at f(c) must eist (f must be continuous at = c). The values 4, 0, and = 1 satisfy these two conditions. (d) g( ) f( ) 2sin cos At 0.785, both f and 2sin cos 4 are negative. Therefore, 1: derivative 3: 1: answer 1: justification g ' 0 and g is decreasing. 4

21 Question 7 Given: f is continuous for, ; f (2) 4 ; lim f( ) f ( ) positive does not eist negative f ( ) negative does not eist positive (a) For what values of is f increasing? (b) Does f have a relative maimum at 4? Eplain. (c) If possible, name the -coordinate of an inflection point on the graph of f. Justify your answer. (d) Does the Mean Value Theorem apply over the interval [3,5]? Justify your answer. (e) Sketch a possible graph of f using the information from the table. (a) (,4) 1: answer (b) Yes, f (4) eists and f ( ) changes sign from positive 2: 1:answer 1: justification to negative at 4.

22 Question 7 (cont.) (c) Two conditions must be met for = c to correspond to an inflection point of f : i). there must be a change 2: 1: answer 1: justification in concavity at = c (f must change sign), and ii). the tangent line at f(c) must eist (f must be continuous at = c). Only = 4 satisfies the first condition, but it fails to satisfy the second. There is no inflection point. (d) No, the Mean Value Theorem does not apply because 2: 1: answer 1: justification f is not differentiable at a point on 3,5. (e) y 1: increasing and decreasing 2 : in correct intervals 1: concavity correct

23 Questions d f ( t) must be positive, since the graph is concave up. 9. c Apply the first derivative test to s ( t) 10. c f is negative for,2 and lim f( ) 2

. Show the work that leads to your answer. (c) State the equation(s) for the vertical asymptote(s) for the graph of y f( x).

Chapter 1 1. (AB/BC, non-calculator) The function f is defined as follows: f( ) 5 6. 7 3 (a) State the value(s) of for which f is not continuous. (b) Evaluate f ( ). Show the work that leads to your answer.