Math 142: Test 3 Review Sheet

Math 142: Test 3 Review Sheet
This document tells you everything that you will need to know in order to do well on your third test. Each
bullet is a requirement that you should ensure that you satisfy prior to taking the test. Not all bullets
have problems following them, but these concepts are just as important.
11.8
• Find the radius and interval of convergence of a given power series.
Power Series 1: Find the radius of convergence and the interval of convergence of the given series.
∞
∞ n n
∞
X
X
X
√
2 x
(x − 2)n
n
n
2.
2 n (x + 3)
1.
3.
n!
nn
n=0
n=0
n=1
4.
7.
∞
X
(x − 1)n
n=0
∞
X
n=0
5.
2n n2
n
n (2x + 3)
n
8.
∞
X
n
n! (x − 2)
n=0
∞
X
(x + 1)n
3n
n=0
6.
9.
∞
X
(−1)n−1 xn
n=1
∞
X
n3
(−x + 6)n
4n n!
n=0
11.9
• Express a given function as a power series and find the radius and interval of convergence using geometric
techniques.
Power Series 2: Use geometric techniques to find the power series that converge to the following
functions, and find the radius and interval of convergence.
x
1
3x
2. f (x) =
3. f (x) =
1. f (x) =
2
2
4−x
9+x
1+x
• Use differentiation and integration along with the geometric techniques to express even more functions
as power series, and find the radius of convergence.
Power Series 3: Use integration and differentiation along with geometric techniques to find the power
series representation of the following functions function, and find the associated radius of convergence
of the power series.
1
3. f (x) = tan−1 (x2 )
1. f (x) = ln (1 + x)
2. f (x) =
2
(1 − x)
11.10
• Use the Taylor Series formula to write the MacLaurin series for a given function. You do not need to
prove that the function is equal to its Taylor series expansion.
MacLaurin Series: Use the Taylor Series formula to find the MacLaurin series of the following functions.
1. f (x) = e−3x
2. f (x) = (1 + x)−1
3. f (x) = xex
• Use a known MacLaurin Series to obtain a MacLaurin Series for a more complicated function.
MacLaurin Series 2: Use a MacLaurin Series in the Table to obtain a MacLaurin series for the given
functions.
2
1. f (x) = x2 cos (πx/2)
2. f (x) = 2e−x
3. f (x) = tan−1 (x2 )
1
6.1
• Find the area of a region bounded by given curves.
Area Between Curves: Find the area of the region bounded by the following curves.
1. y = x2
2. y = 9 − x2
3. y = x2
y=x
y =x+3
y = 4x − x2
x = −2
x=1
√
2
4. y = 1 − x
5. y = ln x
6. y = x
y = x2
y = x − 1, y ≥ −1
y=1
y = −x − 1, y ≥ −1
y=2
x=0
1
7. y =
8. y = −x
9. x = y 2 − 1
x
2
x=y −2
x = 1 − y2
1
y= 2
x
x=1
x=e
6.2, 6.3
• Find the volume of a solid obtained by rotating a region bounded by given curves around a given
horizontal or vertical line. You will need to be able to use the disk, washer, and shell methods, and you
will expected to use each at least once on the test.
Volume: Find the volume of the solid obtained by rotating the region bounded by the given curves
around the
√ given horizontal or vertical line. 2
3. y = ln x
2. y = x
1. y = x
2
2
y=1
y = 4x − x
y=x
Rotate: x-axis
y=2
Rotate: x-axis
x=0
Rotate: x-axis
4.
7.
y = x3
y = x (x ≥ 0)
Rotate: x-axis
√
y= x
y = x2
Rotate: y-axis
5.
y = cos x
y = sin x
x=0
π
x=
4
Rotate: x-axis
6.
8.
y = x2
y = 4x − x2
Rotate: y-axis
9.
2
1
y=
x
y=1
y=2
x=0
Rotate: x-axis
y = ln x
y=1
y=2
x=0
Rotate: y-axis
10. y = x3
y = x (x ≥ 0)
Rotate: y-axis
√
13. y = x
y = x2
Rotate: y = −1
16. y = x3
y = x (x ≥ 0)
Rotate: x = −3
2
11. y = cos x
y = sin x
x=0
π
x=
4
Rotate: y-axis
12. y = e−x
y=0
x=0
x=1
Rotate: y-axis
14. y = ln x
y=1
y=2
x=0
Rotate: y = 3
1
15. y =
x
y=1
y=2
x=0
Rotate: y = 1
17. y = x2
y = 4x − x2
Rotate: x = 2
18. y = cos x
y = sin x
x=0
π
x=
4
Rotate: x = −2
6.5
• Find the average value of a function on an interval.
• Then, find the value c in the interval such that f (c) = fave .
Average Value: Find the average value of the given function on the given interval. Then, find the
number c in the interval such that f (c) = fave .
√
1
1. f (x) = x2 − 1, [−1, 2]
2. f (x) = x, [0, 9]
3. f (x) = 2 , [1, 2]
x
Helpful Rules/Formulas:
1. MacLaurin Series Table(Given on Test 3)
∞
X
1
=
xn = 1 + x + x2 + x3 + ...
1 − x n=0
∞
X
xn
x2 x3
e =
=1+x+
+
+ ...
n!
2!
3!
n=0
x
sin x =
∞
X
(−1)n x2n+1
n=0
cos x =
tan−1 x =
2.
(2n + 1)!
∞
X
(−1)n x2n
n=0
∞
X
(2n)!
=x−
=1−
x3 x5 x7
+
−
+ ...
3!
5!
7!
x2 x4 x6
+
−
+ ...
2!
4!
6!
(−1)n x2n+1
x3 x5 x7
=x−
+
−
+ ...
2n
+
1
3
5
7
n=0
R =1
R =∞
R =∞
R =∞
R =1
The radius of convergence and interval of convergence of a power series are found by determining the
values of x for which the series converges. This is usually done by the ratio or root test, followed by
checking the endpoints individually.
3
∞
X
cn (x − a)n , where |x − a| <
3.
Taylor Series: If f has a power series expansion at a, that is, if f (x) =
4.
f (n) (a)
R, then the coefficients are given by the formula cn =
.
n!
∞
X
MacLaurin Series (Taylor Series centered at 0): If f (x) =
cn xn , where |x| < R, then the coefficients
n=0
n=0
f (n) (0)
.
are given by the formula cn =
n!
b
Z
5.
6.
f (x) − g (x) dx.
Za d
f (y) − g (y) dy. (Note:
The area between f (y) and g (y) (f (y) ≥ g (y) on [c, d]) is given by A =
The area between f (x) and g (x) (f (x) ≥ g (x) on [a, b]) is given by A =
c
7.
f (y) ≥ g (y) means that f (y) is to the RIGHT of g (y).)
Disk Method: Let R be a region bounded by y = f (x) (f (x) ≥ 0 on [a, b]), y = 0, x = a, and x = b.
Z b
π [f (x)]2 dx.
The volume of the solid generated by revolving R around the x-axis is V =
a
8.
(y-axis version) Let R be a region bounded by x = f (y) (f (y) ≥ 0 on [c, d]), x = 0, y = c, and y = d.
Z d
The volume of the solid generated by revolving R around the y-axis is V =
π [f (y)]2 dy.
c
For these, note that the formula πr2 is used, so if rotating around an axis parallel to the x- or y-axis,
let r be the distance from the axis of rotation to an arbitrary point x or y in your interval.
10. Washer Method: Let R be a region bounded by y = f (x), y = g (x) (f (x) ≥ g (x) ≥ 0 on [a, b]),
x = a, and x = b. The volume of the solid generated by revolving R around the x-axis is V =
Z
b
π [f (x)]2 − [g (x)]2 dx.
9.
a
11. (y-axis version) Let R be a region bounded by x = f (y), x = g (y) (f (y) ≥ g (y) ≥ 0 on [c, d]),
yZ = c, and y = d. The volume of the solid generated by revolving R around the y-axis is V =
d
π [f (y)]2 − [g (y)]2 dy.
c
12. For these, note that the formula π (R2 − r2 ) is used, so if rotating around an axis parallel to the x- or
y-axis, let R be the distance from the axis of rotation to an arbitrary point on the the further curve
and r be the distance to that point on the closer curve.
13. Shell Method: Let R be a region bounded by y = f (x), y = g (x) (f (x) ≥ g (x) ≥ 0 on [a, b]),
x
Z = a, and x = b. The volume of the solid generated by revolving R around the y-axis is V =
b
2πx [f (x) − g (x)] dx.
a
14. (y-axis version) Let R be a region bounded by x = f (y), x = g (y) (f (y) ≥ g (y) ≥ 0 on [a, b]),
yZ = c, and y = d. The volume of the solid generated by revolving R around the y-axis is V =
d
2πy [f (y) − g (y)] dy.
c
15. For these, note that the formula 2πrh is used, so if rotating around an axis parallel to the x- or y-axis,
let r be the distance from the axis of rotation to an arbitrary point x or y in your interval, and let h
be the height of the rectangle at that point.
Z b
1
16. The average value of a function is given by fave =
f (x) dx.
b−a a
4