Calculus Problems
Math 504 – 505
Jerry L. Kazdan
1. Sketch the points
(
x
,
y
)
in the plane
R
2
that satisfy

y
−
x
 ≤
2.
2. A certain function
f
(
x
)
has the property that
Z
x
0
f
(
t
)
dt
=
e
x
cos
x
+
C
. Find both
f
and the constant
C
.
3. Compute lim
x
→
0
parenleftBig
cos
x
cos2
x
parenrightBig
1
/
x
2
.
4. Sketch the curve that is defined implicitly by
x
3
+
y
3
−
3
xy
=
0. Calculate
y
′
(
0
)
.
5. Calculate
∞
∑
n
=
1
1
4
n
2
−
1
.
6. Determine the indefinite integral
Z
log
(
1
+
x
2
)
dx
.
7. Let
f
(
x
)
be a smooth function for 0
≤
x
≤
1. If
f
′
(
x
) =
0 for all 0
≤
x
≤
1, what can
you conclude? Prove all your assertions.
8. Solve the initial value problem
(
1
+
e
x
)
yy
′
=
e
x
with
y
(
1
) =
1
.
9. Let the continuous function
f
(
θ
)
, 0
≤
θ
≤
2
π
represent the temperature along the
equator at a certain moment, say measured from the longitude at Greenwich.. Show
there are antipodal points with the
same
temperature.
10. a)
Let
g
(
x
)
:
=
x
3
(
1
−
x
)
. Without computation, show that
g
′′′
(
c
) =
0 for some 0
<
c
<
1.
b)
Let
h
(
x
)
:
=
x
3
(
1
−
x
)
3
. Show that
h
′′′
(
x
)
has exactly three distinct roots in the
interval 0
<
x
<
1.
c)
Let
p
(
x
)
:
=
parenleftbigg
d
dx
parenrightbigg
4
(
1
−
x
2
)
4
. Show that
p
is a polynomial of degree 4 and that it
has 4 real distinct zeroes, all lying in the interval
−
1
<
x
<
1.
1
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11. In
R
2
, let
Q
1
= (
x
1
,
y
1
)
,
Q
2
= (
x
2
,
y
2
)
, and
Q
3
= (
x
3
,
y
3
)
, where
x
1
<
x
2
<
x
3
.
a)
Show there is a unique quadratic polynomial
p
(
x
)
that passes through these points:
p
(
x
j
) =
y
j
,
j
=
1
,
2
,
3.
b)
If
y
1
>
y
2
and
y
3
>
y
2
and
f
(
x
)
is any smooth function that passes through these
three points, show there is some point
c
∈
(
x
1
,
x
3
)
where
f
′′
(
c
)
>
0. Even better,
for some
c
,
f
′′
(
c
)
≥
p
′′
, so
p
′′
is the optimal constant. [Remark: It is enough
to consider the special case where
x
2
=
0 and
y
2
=
0. Then write
x
1
=
−
a
<
0,
x
3
=
b
>
0].
12. a)
If
f
(
x
)
>
0 is continuous for all
x
≥
0 and the improper integral
R
∞
0
f
(
x
)
dx
exists,
then lim
x
→
∞
f
(
x
) =
0. Proof or counterexample.
b)
If
f
(
x
)
>
0 is continuous for all
x
≥
0 and the improper integral
R
∞
0
f
(
x
)
dx
exists,
then
f
(
x
)
is bounded. Proof or counterexample.
13. Find
explicit
rational functions
f
(
x
)
and
g
(
x
)
with the following Taylor series:
f
(
x
) =
∑
∞
1
nx
n
,
g
(
x
) =
∑
∞
1
n
2
x
n
.
14. a)
Let
x
= (
x
1
,
x
2
)
be a point in
R
2
and consider
Z
R
2
1
(
1
+
bardbl
x
bardbl
2
)
p
dx
. For which
p
does this improper integral converge?
b)
This integral can be computed explicitly. Do so.
c)
Repeat part a) where
x
∈
R
3
and the integral is over
R
3
instead of
R
2
.
15. Compute
ZZ
R
2
1
[
1
+(
2
x
+
y
+
1
)
2
+(
x
−
y
+
3
)
2
]
2
dxdy
.
16. Let
v
(
x
,
t
)
:
=
Z
x
+
2
t
x
−
2
t
g
(
s
)
ds
, where
g
is a continuous function. Compute
∂
v
/
∂
t
and
∂
v
/
∂
x
.
17. Let
H
(
t
)
:
=
Z
b
(
t
)
a
(
t
)
f
(
x
,
t
)
dx
, where
a
(
t
)
,
b
(
t
)
, and
f
(
x
,
t
)
are smooth functions of their
variables. Compute
dH
/
dt
.
18. a)
Let
p
(
x
)
:
=
x
3
+
cx
+
d
, where
c
, and
d
are real. Under what conditions on
c
and
d
does this has three distinct real roots? [A
NSWER
:
c
<
0 and
d
2
<
−
4
c
3
/
27].
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 Calculus, Derivative, Continuous function, Let, Smooth function

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