Chapter 2

1. (AB/BC, non-calculator)

 

Let 42( ) 3f x x .

(a) Write an equation of the line tangent to the graph of f at 2x .

(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 ( ) 4 3f x x and ( )

( )f x

g xx

.

(a) What is the slope of the graph of f at 3x ? Show the work that leads to your answer.

(b) Write an equation of the line tangent to the graph of g at 3x .

(c) What is the slope of the line normal to the graph of g at 3x ?

3. (AB/BC, non-calculator)

Evaluate each limit analytically.

(Note: Finding the answer should not involve a lengthy algebraic process.)

(a) 0

sin( ) sinlimh

x h x

h

(b) 3 3

0limh

x h x

h

(c) 0

16 4limh

h

h

(d) 0

1 15 5lim

h

hh

4. (AB/BC, calculator neutral)

Given:

x ( )f x ( )f x ( )g x ( )g x

2 3 1 5 2

5 4 7 1 2

(a) If ( )

( )( )

f xh x

g x , find '(2)h .

(b) If ( ) ( ( ))j x f g x , find '(2)j .

(c) If ( ) ( )k x f x , find '(5)k .

5. (AB/BC, non-calculator)

Given: 2( )f x x

(a) Find the slope of the normal line to the graph of f at 3x .

(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. (AB/BC, calculator neutral)

The accompanying diagram shows the graph of the velocity in ft

secfor a particle moving along

the line 4x .

t

v(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) 0.75t

(ii) 4.2t

7. (AB/BC, non-calculator)

Given: ( ) ( ) tang x f x x kx , where k is a real number.

f is differentiable for all x; 44

f

; 24

f

.

(a) For what values of x, if any, in the interval 0 2x will the derivative of g fail to exist? Justify your answer.

(b) If 64

g

, find the value of k.

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 2.0 2.7 3.2 3.8

( )S t 5.2 7.8 10.6 12.2

Which of the following best estimates the velocity of the particle at 3t ?

 

(a) m

9.2s

(b) m

7.8s

(c) m

5.6s

9. (AB/BC, non-calculator)

If 3 22 3 4y xy x , then dy

dx

(a) 2

2

6 3

x

y

(b) 2

2 3

3 6

x y

x y

(c) 2

2 3

6

x

y

(d) 2

2

6 3

x

y x

(e) 2

3 2

6

y x

y

10. (AB/BC, non-calculator)

The volume of a cylinder with radius r and height h is given by 2V r h . The radius and height

of the cylinder are increasing at constant rates. The radius is expanding at 1 cm

3 secand the height is

increasing at 1 cm

2 sec. At what rate, in cubic cm per second, is the volume of the cylinder

increasing when its height is 9 cm and the radius is 4 cm?

 

(a) 32

(b) 6

(c) 8

3

(d) 4

3

(e) 18

Chapter 2 (Solutions)

Question 1

Let 42( ) 3f x x .

(a) Write an equation of the line tangent to the graph of f at 2x .

(b) Find the values of x for which the graph of f has a horizontal tangent.

(c) Find ''( )f x .

 

(a) 32'( ) 8 3f x x x

Point: 2,1

1: derivative1: point4 : 1: slope1: equation

16m

1 16( 2)y x

(b) 28 ( 3) 0x x 1: derivative equal to 03: 2: answers

0; 3x x

(c) 2 32 2 2''( ) 48 3 8 3f x x x x 2: answer 

Question 2

Let ( ) 4 3f x x and ( )

( )f x

g xx

.

(a) What is the slope of the graph of f at 3x ? Show the work that leads to your answer.

(b) Write an equation of the line tangent to the graph of g at 3x .

(c) What is the slope of the line normal to the graph of g at 3x ?

 

(a) 2

'( )4 3

f xx

2 : derivative3: 1: answer

2

'(3)3

f

(b) 2

xf x f xg x

x

1

'(3)9

g

2 : derivative1: evaluates (3)5 : 1: (3,1)1: equation

g

3, 3 3,1g

1 1 41 3

9 9 3y x y x

(c) 9 1: answer

Question 3

Evaluate each limit analytically.

(Note: Finding the answer should not involve a lengthy algebraic process.)

(a) 0

sin( ) sinlimh

x h x

h

(b) 3 3

0limh

x h x

h

(c) 0

16 4limh

h

h

(d) 0

1 15 5lim

h

hh

(a) ( ) cosf x x 1: answer

(b) 2

3

1( )

3

f x

x

2: answer

(c) 1

( )2

f xx

2 : derivative3: 1: answer

1

(16)8

f

Question 3 (cont.)

(d) 2

1( )f x

x 2 : derivative3: 1: answer

1(5)

25f

Question 4

Given:

(a) If ( )

( )( )

f xh x

g x , find '(2)h .

(b) If ( ) ( ( ))j x f g x , find '(2)j .

(c) If ( ) ( )k x f x , find '(5)k .

x ( )f x ( )f x ( )g x ( )g x

2 3 1 5 2

5 4 7 1 2

(a) 2

( ) ( ) ( ) ( )( )

( )

g x f x f x g xh x

g x

2 : derivative3: 1: answer

1

(2)25

h

(b) ( ) ( ( )) ( )j x f g x g x                                                                                               2 : derivative3: 1: answer  

       (2) 14j  

 

 

Question 4 (cont.)

(c) 1

( ) ( )2 ( )

k x f xf x

2 : derivative3: 1: answer

'(5) 1k

 

Question 5

Given: 2( )f x x

(a) Find the slope of the normal line to the graph of f at 3x .

(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) '( ) 2f x x

'( 3) 6f 1: derivative2 : 1: slope

1

6m

(b) 28

23

xx

x

4; 2x x

1: equation2 : points7 : 2 : slopes2: equations

4,16 ; 2, 4

8; 4m m

16 8( 4); 4 4( 2)y x y x

Question 6

The accompanying diagram shows the graph of the velocity in ft

secfor a particle moving along

the line 4x .

t

v(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) 0.75t

(ii) 4.2t

Question 6 (cont.)

(a) (i) 0 1t ; 22

55

t 2 : find -intercept on 4,54 :2 : intervals

t

(ii) 22

25

t 1: answer

(iii) 1 2t 1: answer

(b) (i) 2

ft2

sec 1: answer2 : 1: units

(ii) 2

ft5

sec 1: answer

Question 7

Given: ( ) ( ) tang x f x x kx , where k is a real number.

f is differentiable for all x; 44

f

; 24

f

.

(a) For what values of x, if any, in the interval 0 2x will the derivative of g fail to exist? Justify your answer.

(b) If 64

g

, find the value of k.

(a) The derivative of g will fail to exist at 2

x

and 3

2x

2 : values4 : 2: justification

because g is not continuous at these values .

(b) 2'( ) ( )sec tan '( )g x f x x x f x k

        2sec tan ' 64 4 4 4

f f k

                                                                3: derivative5 : 2: solution  

         4 2 1 2 6

0

k

k

 

Questions 8-10

8. c (3.2) (2.7) 10.6 7.8 m

5.63.2 2.7 0.5 s

S S

9. b

2

2

2 (3 3 ) 2 0

2 3 3 6

dy dyy x y x

dx dxdy x y

dx x y

10. a   

2

2

2

2

1 14 2 9 4

2 3

32

V r h

dV dh drr rh

dt dt dt

dV

dt

dV

dt