Name _______________
Chapter 2
t 2  2 m. compute its average velocity over [2, 5] and
estimate its instantaneous velocity at t  2 (showing both left- and right-hand values).
1.
A particle’s position at time t is s (t ) =
2.
Evaluate or state that it does not exist.
a.
3x 2  4 x  1
x 1
x  1
lim( 3  x1 / 2 )
b.
x 4
c.
x3  2x
x 1
lim
x 1
3.
lim
Evaluate.
lim sec 


4
sin 2 t
4. Evaluate. lim
t3
t 0
5.
Sketch the graph of a function f(x) such that
lim f ( x)  1 ,
x2
6.

lim f ( x)  3 , lim f ( x)  2
x  2
x 4
Sketch the graph of a function g(x) such that lim g ( x )  
x  3 
7.
Discuss the discontinuity of the function F(x) defined by
 x, for x  1

F(x) = 3, for 1  x  3
 x, for x  3

8. Evaluate lim
x 2
x4
x 4
9.
Calculate lim
x
10.

tan x
sec x
2
Evaluate lim
sin 2h sin 3h
h2
h 0
11.
Use the Bisection Method to locate a zero (root) of x  7 =0 to two decimal places.
2
x2  2
12. Use the IVT to prove that f ( x)  x 
has a root (zero) in the interval [0, 2].
cos x  2
3
Chapter 3
1. Supposing that f ( x)  3x 2  2 x , use the definition of the derivative to find f’(-1).
2. Supposing the f ( x ) 
1
, use the definition of the derivative to find f’(2) and then find
x
the equation for the tangent line at x = 2.
3. Find the equation of the tangent line to f ( x) 
x 1
at x = 16.
x
Use the Product or Quotient Rules to solve problems 4 – 6.
4. Find the Derivative of f ( x) 
5. Find the Derivative of f ( x) 
x (1  x 4 ) .
x4
.
x  x 1
2
6. Find the equation of the tangent line to f ( x) 
7. Find the derivative of the following functions.
x2
xx
at x = 9.
a. g ( z )  7 z 3  z 2  5
b. f ( x)  4 x +
3
x
c. P( z )  (3z  1)( 2 z  1)
8.
The population P(t) of Freedonia in 1933 was P(1933) = 5 million.
a. What is the meaning of P’(1933)?
b. Estimate P(1934) if P’(1933) = 0.2. What if P’(1933) = 0?
9. Find the Rate of Change of the volume of a cube with respect to the length of its side s
when s = 3 and s = 5.
10. The height in feet of a helicopter at time t in minutes is s(t )  3t 3  400t for
0  t  10 .
a. Plot the graphs of height s(t) and velocity v(t).
y
x
2
4
6
8
10
12
b. Find the velocity at t = 6 and t = 7.
c. Find the maximum height of the helicopter.
11. It takes a stone 3 seconds to hit the ground when dropped from the top of a building.
How high is the building and what is the stone’s velocity on impact?
Calculate the second and third derivative.
1. y =7 – 2x
2. 𝑦 = 4𝑥 3 − 9𝑥 2 + 7
Calculate the indicated derivative.
3.
𝑑4 𝑓
𝑑𝑡 4
t=1 𝑓(𝑡) = 3𝑡 5 − 2𝑡 4 + 𝑡 3
Find the equation of the tangent line on the point indicated.
4.
y=cos 𝑥, 𝑥 =
𝜋
3
Find the derivative.
sin 𝑥
𝜋
5. 𝑓(𝑥) = 𝑥 , 𝑥 = 4
Calculate the second derivative.
6. 𝑓(𝑥) = tan 𝑥
Find the derivative of the following:
7. 𝑦 = 𝑠𝑖𝑛5 𝑥
9.
8. 𝑦 = (7𝑥 − 9)5
10. 𝑦 = 𝑥 2 tan(2𝑥)
𝑦 = √1 − 𝑡 2
Calculate the derivative of y with respect to x.
11. 𝑦 4 − 𝑦 = 𝑥 3 + 𝑥
𝑦
𝑥
12.
+ = 2𝑦
𝑥
13.
𝑦
𝑒 𝑥𝑦 = 𝑥 2 + 𝑦 2
Find the equation of the tangent line at the given point.
14. 𝑥 2 𝑦 3 + 2𝑦 = 3𝑥, (2, 1)
Find the derivative.
16. 𝑦 = 𝑥 tan−1 𝑥
Find the derivative.
1. y = x lnx _______________
15. 𝐺(𝑠) = tan−1 (√𝑠 )
17. 𝑦 = arcsin(𝑒 𝑥 )
2. y = ln (tan x)
Find an equation of the tangent line at the point indicated.
3. 𝑓(𝑥) = 4𝑥 , 𝑥 = 3
______________
_______________
Related Rates
4. A 16-ft ladder leans against a wall. The bottom of the ladder is 5 ft. from the wall at
time
t = 0 and slides away from the wall at a rate of 4 ft/s. Find the velocity of the
top of the ladder at time t = 1. (Draw a picture and define variables.)
5. Water pours into a fish tank at a rate of 4 ft3/min. How fast is the water level rising if
the base of the tank is a rectangle of dimensions 3 X 4 ft?
6. The radius of a circular oil slick expands at a rate of 3 m/min.
a. How fast is the area of the oil slick increasing when the radius is 30 m?
b. If the radius is 3 m at time t = 0, how fast is the area increasing after 3 minutes?
7.
4
The Volume of a sphere is 𝑉 = 3 𝜋𝑟 3 . Assume that its radius r is expanding at a rate of
15 in./min. Determine the rate at which the Volume is changing with respect to time when
r = 8 in.
I. Multiple Choice
_____ 1. The largest interval on which f  x   x 2  4 x  2 is increasing is
(A) 0, 
(B) 0,   
(C)  2,  
(D)  , 2
(E)  ,   
_____ 2. f  x   2 x 2  2 has an absolute minimum on   3,3 of
(A) 2
(B) –2
(C) 52
(D) –52
(E) 0
______3. Determine the value of c that satisfies the Mean Value Theorem for f  x   x 3 on
0,1 .
(A) 0
(B)
3
3
(C)
3
2
(D)
2
2
(E)
2
3
_____ 4. f  x   x 2  8x  7 has
(A) a relative maximum at x = 4
(B) a relative minimum at x = 4
(C) a relative maximum at x = -4
(D) a relative minimum at x = -4
(E) a relative minimum at x = 0
5-6. Use the graph of f '  x  shown in the figure to answer the
questions below.
_____ 5. Determine the x-coordinates where f  x  has a relative
minima.
(A) x  0
(B) x  1
(C) x  0 and x  2
(D) x  2
(E) x  3
_____ 6. Determine the x-coordinates where f  x  has a point of inflection.
(A) x  0
(B) x  1
(C) x  0 and x  2
(D) x  2
(E)
x  1 and x  3
_____ 7. f  x    x 4  6 x 2 is concave up on

3, 3

(A)  ,   
(B)  , 1
(C)  1, 1
(D) nowhere
(E)
_____ 8. Use differentials to approximate √4.9.
(A)2.225
(B)2.250
(C)2.214
II. Free Response
(D)2.450
9 . Give the following information for the function:
y  x3  3x 2  4
Derivative: __________________________________
Increasing on ( ___ , ___ ) and ( ___ , ___ )
Decreasing on ( ___ , ___ )
Relative Maximum at ( ___ , ___ )
Relative Minimum at ( ___ , ___ )
Second Derivative: ___________________________
Concave Up on ( ___ , ___ )
Concave Down on ( ___ , ___ )
Points of Inflection at( ___ , ___ )
(E) None of these