SECTION 1.3
NEW FUNCTIONS FROM OLD FUNCTIONS

45
57.
The
Heaviside function
H
is defined by
It is used in the study of electric circuits to represent the sudden
surge of electric current, or voltage, when a switch is instantane
ously turned on.
(a)
Sketch the graph of the Heaviside function.
(b) Sketch the graph of the voltage
in a circuit if the
switch is turned on at time
and 120 volts are applied
instantaneously to the circuit. Write a formula for
in
terms of
.
(c)
Sketch the graph of the voltage
in a circuit if the switch
is turned on at time
seconds and 240 volts are applied
instantaneously to the circuit. Write a formula for
in
terms of
. (Note that starting at
corresponds to a
translation.)
58.
The Heaviside function defined in Exercise 57 can also be used
to define the
ramp function
, which represents a
gradual increase in voltage or current in a circuit.
(a)
Sketch the graph of the ramp function
.
(b) Sketch the graph of the voltage
in a circuit if the switch
is turned on at time
and the voltage is gradually
increased to 120 volts over a 60second time interval. Write
a formula for
in terms of
for
.
(c)
Sketch the graph of the voltage
in a circuit if the switch
is turned on at time
seconds and the voltage is gradu
ally increased to 100 volts over a period of 25 seconds.
Write a formula for
in terms of
for
.
59.
Let
and
be linear functions with equations
and
. Is
also a linear function? If so, what
is the slope of its graph?
60.
If you invest
dollars at 4% interest compounded annually, then
the amount
of the investment after one year is
.
Find
,
, and
. What do these compo
sitions represent? Find a formula for the composition of
copies of
.
61.
(a) If
and
, find a function
such that
. (Think about what operations you
would have to perform on the formula for
to end up with
the formula for .)
(b) If
and
, find a function
such that
.
62.
If
and
, find a function
such that
.
63.
(a) Suppose
and
are even functions. What can you say about
and
?
(b) What if
and
are both odd?
64.
Suppose
is even and
is odd. What can you say about
?
Suppose
t
is an even function and let
. Is
h
always an
even function?
66.
Suppose
t
is an odd function and let
. Is
h
always an
odd function? What if
is odd? What if
is even?
f
f
h
f
t
h
f
t
65.
f
t
t
f
t
f
f
t
f
t
t
f
t
f
h
t
h x
4
x
1
f x
x
4
f
t
h
t
h x
3
x
2
3
x
2
f x
3
x
5
h
t
f
t
h
f
h x
4
x
2
4
x
7
t
x
2
x
1
A
n
A
A
A
A
A
A
A
A
A
A x
1.04
x
A x
x
f
t
t
x
m
2
x
b
2
f x
m
1
x
b
1
t
f
t
32
H t
V t
t
7
V t
t
60
H t
V t
t
0
V t
y
tH t
y
ctH t
t
5
H t
V t
t
5
V t
H t
V t
t
0
V t
H t
0
1
if
t
0
if
t
0
51.
Use the given graphs of
and
to evaluate each expression,
or explain why it is undefined.
(a)
(b)
(c)
(d)
(e)
(f)
52.
Use the given graphs of
and
to estimate the value of
for
. Use these estimates to
sketch a rough graph of
.
A stone is dropped into a lake, creating a circular ripple that
travels outward at a speed of
.
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