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18.01 Single Variable Calculus
Fall 2006
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View Full Document 18.01 EXERCISES
SUnit
.D
iffej i
tiaiti oi•7.
......
....................
...............
1A.
Graphing
1A1
By
completing the square, use translation and change of scale to sketch
a)
y=z
'
2

1
b)
y
= 3X
2
+
6z
+ 2
1A2
Sketch, using translation and change of scale
a)y
=
1+
I
+
21
b)
y
=

(Z

1)2
1A3
Identify each of the following as even, odd
,
or neither
z
3
+
3z
a)
+
b)
sinz
tan z
c)
tan
d) (1
+
z)
4
1
+ X
2
e)
Jo(z ),
where
Jo(z)
is a function you never heard of
1A4 a) Show that every polynomial is the sum of an even and an odd function.
b)
Generalize part (a) to an arbitrary function
f(z)
by
writing
f(s)
+
f
()
f(
C)
+
f(z)
A )
2
2
Verify this equation, and then show that the two functions on the right are respectively even
and odd.
c)
How would you write
as the sum of an even and an odd function?
z+a
1A5.
Find the inverse to each of the following
,
and sketch both
f(z)
and the inverse
function
g(z).
Restrict the domain if necessary. (Write
y
=
f(z)
and solve for
y;
then
interchange
x and
y.)
a)
21
)
b)z
2
+ 2
2T
+3
1A6
Express in the form
A
sin
(x
+
c)
a) sinx
+
Vcos
x
b) sinx

cos
1A7 Find the period , amplitude., and phase angle,and use these to sketch
a) 3 sin (2

7r)
b)
4cos
(x
+
r/2)
1A8
Suppose
f(z)
is odd and periodic. Show that the graph of
f(z)
crosses the xaxis
infinitely often.
@Copyright David Jerison and MIT 1996, 2003
E. 18.01
EXERCISES
1A9 a) Graph the function
f
that consist of straight line segments
joining the points
(1,
1),
(1,
2),
(3, 1),
and
(5,2).
Such a function is called piecewise linear.
b)
Extend the graph of
f
periodically. What is its period?
c) Graph the function g(r)
=
3f((z/2)

1) 
3.
1B. Velocity
and
rates of change
1B1 A
test tube is knocked off a tower at the top of the Green
building. (For the purposes
of this experiment the tower is 400 feet above the ground, and all the air in the vicinity of
the Green building was evacuated, so as to eliminate wind resistance.) The test tube drops
16t
2
feet in t seconds. Calculate
a) the average speed in the first two seconds of the fall
b)
the average speed in the last two seconds of the fall
c) the instantaneous speed at landing
1B2
A
tennis ball bounces so that its initial speed straight upwards
is
b
feet per second.
Its height s in feet at time
t seconds is.given
by
a
=
bt
 16t
a) Find the velocity v
=
ds/dt
at time t.
b)
Find the time at which the height of the ball is at its maximum height.
c) Find the maximum height.
d)
Make
a
graph of
v
and directly below it
a
graph of
a
as a function of time. Be sure
to mark the maximum of
s
and the beginning and end of the bounce.
e) Suppose that when the ball bounces a second time it rises to half the height of
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This note was uploaded on 10/10/2011 for the course MATH 31 taught by Professor Blake during the Spring '11 term at MIT.
 Spring '11
 Blake
 Math, Calculus, The Land

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