Math E-15 Homework 3
1. Consider the function
f
(
x
)=5
x
3
. Do not round o
ff
your answers to the following.
(a) What is the average rate of change of
f
on the interval [2, 2.1]?
(b) What is the average rate of change of
f
on the interval [2, 2.01]?
(c) What is the average rate of change of
f
on the interval [2, 2.001]?
(d) Use the limit definition of the derivative to compute
f
‘
(
x
).
(e) Use your answer to (d) to compute the instantaneous rate of change of
f
at
x
= 2.
(f) Find the equation of the tangent line to
f
at
x
= 2.
2. Let
f
(
x
)=
2
4
–
x
.
(a) Use the limit definition of the derivative to find
f
‘
(
x
).
(b) Find the equation of the tangent line to
f
at
x
= 1.
3. Let
f
(
x
)=1
–
2
x
–
3
x
2
.
(a) Use the limit definition of the derivative to find
f
‘
(
x
).
(b) At what
x
-value(s) does
f
have a horizontal tangent line?
4. Shown below is a graph of
f
‘
on its entire domain. The graph is NOT
f
.
At which labelled
x
-value(s) (that is, the letters
a
through
k
)
(a) is
f
greatest?
(b) is
f
least?
(c) is
f
‘
greatest?
(d) is
f
‘
least?
(e) is the slope of
f
‘
greatest?
(f) is the slope of
f
‘
least?
On what interval(s) is
(a)
f
increasing?
(b)
f
decreasing?
(c)
f
‘
increasing?
(d)
f
‘
decreasing?
f ’ (x)
d
c
b
a
f
g
h
e
i
j
k
