Chapter (AB/BC, non-calculator) (a) Write an equation of the line tangent to the graph of f at x 2.

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1 Chapter 1. (AB/BC, non-calculator) Let f( x) x 3 4. (a) Write an equation of the line tangent to the graph of f at x. (b) Find the values of x for which the graph of f has a horizontal tangent. (c) Find f ''( x ).

2 . (AB/BC, non-calculator) Let f( x) 4x 3 and f ( x) gx ( ). x (a) What is the slope of the graph of f at x 3? Show the work that leads to your answer. (b) Write an equation of the line tangent to the graph of g at x 3. (c) What is the slope of the line normal to the graph of g at x 3?

3 3. (AB/BC, non-calculator) Evaluate each limit analytically. (Note: Finding the answer should not involve a lengthy algebraic process.) (a) sin( x h) sin x lim h0 h (b) lim h0 x h x h 3 3 (c) lim h 0 16 h 4 h (d) 1 1 lim 5 h 5 h0 h

4 4. (AB/BC, calculator neutral) Given: x f ( x ) f ( x) gx ( ) g ( x) (a) If f ( x) hx ( ), find h '(). gx ( ) (b) If j( x) f( g( x)), find j '(). (c) If kx ( ) f( x), find k '(5).

5 5. (AB/BC, non-calculator) Given: f ( x) x (a) Find the slope of the normal line to the graph of f at x 3. (b) Two lines passing through the point (3,8) will be tangent to the graph of f. Find an equation for each of these lines.

6 6. (AB/BC, calculator neutral) The accompanying diagram shows the graph of the velocity in the line x 4. ft for a particle moving along sec v(t) t (a) During which time interval is the particle: (i) moving upward. (ii) moving downward. (iii) at rest. (b) State the acceleration of the particle at the specified times. Include units. (i) t 0.75 (ii) t 4.

7 7. (AB/BC, non-calculator) Given: gx ( ) f( x) tan x kx, where k is a real number. f is differentiable for all x; f 4 ; f. 4 4 (a) For what values of x, if any, in the interval 0 x will the derivative of g fail to exist? Justify your answer. (b) If g 6, find the value of k. 4

8 8. (AB/BC, calculator neutral) The table provided below shows the position of a particle, S, at several times, t, as the particle moves along a straight line, where t is measured in seconds and S is measured in meters. t St () Which of the following best estimates the velocity of the particle at t 3? (a) m 9. s (b) m 7.8 s (c) m 5.6 s

9 9. (AB/BC, non-calculator) If y xyx, then dy dx x (a) 6y 3 x 3y (b) 3x 6y x 3 (c) 6y x (d) 6y 3x 3y x (e) 6y

10 10. (AB/BC, non-calculator) The volume of a cylinder with radius r and height h is given by V r h. The radius and height of the cylinder are increasing at constant rates. The radius is expanding at 1cm and the height is 3sec increasing at 1cm. At what rate, in cubic cm per second, is the volume of the cylinder sec increasing when its height is 9 cm and the radius is 4 cm? (a) 3 (b) 6 (c) 8 3 (d) 4 3 (e) 18

11 Chapter (Solutions) Question 1 f x x. Let ( ) 3 4 (a) Write an equation of the line tangent to the graph of f at x. (b) Find the values of x for which the graph of f has a horizontal tangent. (c) Find f ''( x ). f x x x (a) '( ) Point:,1 1: derivative 4: 1: point 1: slope 1: equation m 16 y1 16( x ) (b) 8 xx ( 3) 0 3: 1: derivative equal to 0 : answers x0; x 3 3 (c) f ''( x) 48x x 3 8x 3 : answer

12 Question Let f( x) 4x3and f ( x) gx ( ). x (a) What is the slope of the graph of f at x 3? Show the work that leads to your answer. (b) Write an equation of the line tangent to the graph of g at x 3. (c) What is the slope of the line normal to the graph of g at x 3? (a) f '( x) 4x 3 3: : derivative 1: answer f '(3) 3 (b) x g xf x f x x 1 g '(3) 9 :derivative 5: 1: evaluates g(3) 1: (3,1) 1: equation 3, g 3 3,1 y1 1 x3 y 1 x (c) 9 1: answer

13 Question 3 Evaluate each limit analytically. (Note: Finding the answer should not involve a lengthy algebraic process.) (a) sin( x h) sin x lim h 0 h (b) lim h0 x h x h 3 3 (c) lim h0 16 h 4 h (d) 1 1 lim 5 h 5 h0 h (a) f ( x) cos x 1:answer (b) f( x) 1 : answer 3 3x (c) 1 f( x) 3: :derivative x 1: answer 1 f (16) 8

14 Question 3 (cont.) 1 (d) f( x) 3: x : derivative 1: answer 1 f (5) 5

15 Question 4 Given: x f ( x ) f ( x) gx ( ) g ( x) (a) If f ( x) hx ( ), find h '(). gx ( ) (b) If jx ( ) f( gx ( )), find j '(). (c) If kx ( ) f( x), find k '(5). (a) gxf ( ) ( x) f( xg ) ( x) h( x) 3: : derivative 1: answer gx ( ) 1 h() 5 (b) j( x) f( g( x)) g( x) 3: :derivative 1: answer j() 14

16 Question 4 (cont.) (c) 1 k( x) f( x) 3: :derivative f( x) 1: answer k '(5) 1

17 Question 5 Given: f ( x) x (a) Find the slope of the normal line to the graph of f at x 3. (b) Two lines passing through the point (3,8) will be tangent to the graph of f. Find an equation for each of these lines. (a) f '( x) x f '( 3) 6 : 1: derivative 1: slope 1 m 6 (b) 8 x 3 x x x4; x 1: equation 7: : points : slopes : equations 4,16 ;, 4 m8; m 4 y16 8( x4); y4 4( x )

18 Question 6 The accompanying diagram shows the graph of the velocity in the line x 4. ft for a particle moving along sec v(t) t (a) During which time interval is the particle: (i) moving upward. (ii) moving downward. (iii) at rest. (b) State the acceleration of the particle at the specified times. Include units. (i) t 0.75 (ii) t 4.

19 Question 6 (cont.) (a) (i) 0t 1; t 5 5 4: : intervals : find t-intercept on 4,5 (ii) t 1: answer 5 (iii) 1t 1: answer ft (b) (i) : sec 1:answer 1: units ft (ii) 5 1: answer sec

20 Question 7 Given: gx ( ) f( x) tan x kx, where k is a real number. f is differentiable for all x; f 4 ; f. 4 4 (a) For what values of x, if any, in the interval 0 x will the derivative of g fail to exist? Justify your answer. (b) If g 6, find the value of k. 4 (a) The derivative of g will fail to exist at x and 3 x 4: :values : justification because g is not continuous at these values. (b) g'( x) f( x)sec xtan x f '( x) k sec tan ' f f k 5: 3: derivative : solution 4 1 k 6 k 0

21 Questions c S(3.) S(.7) m s 9. b dy dy y x y x dx dx dy x 3y dx 3x 6y (3 3 ) a V r h dv dh dr r rh dt dt dt dv dt 3 dv 3 dt

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