Bulleted (
·
) steps toward solving the problems.
Use calculus, not your calculator, to compute the slope of tangent line to the curve y = e
-
2x
cos(2x) at the point where x = 0 .
Use product rule.
Use chain rule (twice).
Use e
0
= 1
Use cos(0) = 1 ; while sin(0) = 0 .
Evaluate the derivative at x = 0 .
Use calculus, not your calculator, to compute the value of the derivative of the function
y = ln
æ
ç
è
æ
Ö
2x + 1
3x+ 1
ö
÷
ø
at x =
-
1/6 .
First use properties of logarithms:
y = ln
æ
ç
è
æ
Ö
2x + 1
3x+ 1
ö
÷
ø
=
1
2
ln(2x+1)
-
1
2
ln(3x+1)
Use the chain rule (twice).
Evaluate the derivative at x =
-
1/6
Use the tables in our textbook, not your calculator, to integrate
y =
ó
õ
x
7
cos(x
4
) dx
There is no formula involving cos(x
4
) in the tables in our book.
Try the substitution u = x
4
along with the corresponding differential substitution (1/4)du = x
3
dx .
Use the formula for
ò
u cos(u) du .
Use calculus, not your calculator, to carry out the integration
ó
õ
x
3
e
2x
dx
There is no integration formula for
ò
u
3
e
au
du in our book.
There is an integration formula for
ò
u
2
e
au
du .
Use integration by parts to lower the exponent on x .
Start with u = x
3
and dv = e
2x
dx .
Get v = (1/2)e
2x
. The chain rule forces the fraction 1/2 .
Use the integration formula for
ò
u
2
e
au
du .
Does the function
y(x,t) = 2sin(3x) cos(4t)
satisfy the
wave equation
¶
2
y
¶
t
2
= a
2
¶
y
2
¶
x
2
for some constant a ? If yes, what is the value of a ? If no, explain why not.
When computing the second partial derivative of y with respect to t, the multiplicative factor cos(4t) is a
multiplicative constant
.
"The derivative of a constant times a function is constant times the derivative."
When computing the second partial derivative of y with respect to x , the multiplicative factor sin(3t) is a
multiplicative constant
.
¶
2
y
¶
t
2
=
-
9y
and
¶
y
2
¶
x
2
=
-
16y
Given
t =
ó
õ
1
(4
-
x)(2
-
x)
dx
Find find the equation relating t and x so that t = 0 minutes when x = 0 grams.
Prepare for integration:
1
(4
-
x)(2
-
x)
=
A
4
-
x
+
B
2
-
x
After you have found the numbers A and B , add the two fractions to make sure you have not made any mistakes.
Integrate to find a formula for t in terms of x up to an additive constant C .
Find C so that t = 0 when x = 0 .
Find the area of the infinite region bounded by the curves y = 1/x
2
, y = 0 , and x = 1 .
First interpretation
ó
õ
1
0
1
x
2
dx =
lim
a
®
0
+
ó
õ
1
a
1
x
2
dx =
¥
Second interpretation
ó
õ
¥
0
1
x
2
dx =
lim
b
®
¥
ó
õ
b
1
1
x
2
dx = 1
Let
f
0
= f
s
æ
è
v + v
0
v
-
v
s
ö
ø
where v , v
s
, and f
s
are constants.
Decide whether or not the equation
f
s
¶
f
0
¶
v
s
= f
0
¶
f
0
¶
v
0
is true. Defend your answer.
When computing the partial of f
0
with respect to v
s
the multiplicative factor f
s
(v + v
0
) is a multiplicative constant.
"The derivative of a constant times a function is a constant timees the derivative of the function."
When computing the partial of f
0
with respect to v
0
the multiplicative factor
f
s
v
-
v
s
is a multiplicative constant.
¶
f
0
¶
v
s
= f
s
v + v
0
(v
-
v
s
)
2
and
¶
f
0
¶
v
0
= f
s
1
v
-
v
s
File translated from T
E
X by
T
T
H
, version 3.72.
On 6 Mar 2006, 14:00.