Math 247 Winter 2002
Solutions
Quiz 5
1. Use separation of variables to solve the following dierential equations:
dy
a)
= cos(2x), y (0) = 3
dx
Solution: We separate variables:
dy
= cos(2x)
dx
=
dy =
=
dy = cos(2x) dx
=
cos(2x) dx
y=
1
sin(2x) + C
2
Now w
Math 247 Winter 2002
Solutions
Quiz 4 (Worksheet)
b
ex dx for b = 1, b = 10 and b = 1000.
1. Compute
0
b
b
ex dx = ex + C , so
Solution: Well,
ex dx = ex
0
= eb e0 =
0
1 eb .
For b = 1 this is 1 e1 0.632.
For b = 10 this is 1 e10 0.999954.
For b = 1000 th
Math 247 Winter 2002
Solutions
Quiz 3
1. Use integration by parts to evaluate the indenite integrals:
a)
x cos(x) dx
Solution: Let u = x and v = cos(x), so u = 1 and v = sin(x). Integration by parts says,
x cos(x) dx
= x sin(x)
1 sin(x) dx
= x sin(x)
si
Math 247 Winter 2002
Solutions
Quiz 2
1. Compute the average value of f (x) = 4 x2 on the interval [2, 2].
1
Solution: The average value of a function f on the interval [a, b] is
ba
Here f (x) = 4 x2 , a = 2 and b = 2, so the average value is
1
2 (2)
=
1
Math 247
Solutions
Midterm 1
3
1. Write the sum in expanded form:
k=1
3
1
1
1
1
=
+
+
k (k + 1)
1(1 + 1) 2(2 + 1) 3(3 + 1)
Solution: Well,
k=1
3
2. Given that
1
k (k + 1)
5
5
f (t) dt = A,
1
f (t) dt = B and
f (t) dt = C , compute the following:
1
2
2
a)
Math 247
Solutions to Review for Final
10
sin(t2 + 1) dt. Compute F (x).
1. Let F (x) =
x2
h(x)
f (t) dt, then F (x) = f (h(x)h (x)
Solution: By Leibnizs Rule, if F (x) =
g (x)
f (g (x)g (x). We have
F (x) = sin (10)2 +1 (10) sin (x2 )2 +1 (x2 ) = sin (1
Math 247
Review for Final
10
sin(t2 + 1) dt. Compute F (x).
1. Let F (x) =
x2
2. Compute the average value of f (x) = ln(x) over the interval [1, e2 ].
3. Compute the area between the curves y = x2 + 1 and y = x + 3.
4. Evaluate the integrals: (you may wa
Math 247
Review for Exam #2
Solutions
1. Use integration by parts to evaluate the integrals:
a)
x sec2 (x) dx
Solution: Let u = x and v = sec2 (x). Then u = 1 and v = tan(x). We get
x sec2 (x) dx = x tan(x)
b)
1 tan(x) dx = x tan(x) ln | sec(x)| + C
xe3x
Math 247
Solutions to Review for Exam #1
4
1. Write the sum in expanded form:
k (k + 1)
k=1
4
Solution: Well,
k (k + 1) = 1(1 + 1) + 2(2 + 1) + 3(3 + 1) + 4(4 + 1).
k=1
2
4 x2 dx
2. Use geometry to compute the denite integral:
0
Solution: The integral rep