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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