M242 Study Guide for E 1 (S. Zhang) .
4 ln(1 2/x)
1/x
4(1 2/x)1 (2x2 )
= lim
x
x2
4(2)
= lim
x 1 2/x
= 8
lim ln F = lim
x
1.
x
Use Newtons method to nd x3 :
2.13
x5 x 1 = 0, x1 = 1
ans:
f (x) = x5 x 1
f (x) = 5x4 1
1
lim
x
xn+1 = xn
f (xn )
f (xn )
x1
f
M242 Q3(c) (S. Zhang) .
Name:
1. Find the volume of the solid obtained by rotating the region bounded by the given cures about the x-axis, by
both (1) rotation method and (2) cylindrical shell method.
x = y 2 , x = 0, y = 1.
ans: Find intersections:
(0,
M242 Q3(b) (S. Zhang) .
Name:
1. Find the volume of the solid obtained by rotating the region bounded by the given cures about the x-axis, by
both (1) rotation method and (2) cylindrical shell method.
x = y, x = 0, y = 1.
ans: Find intersections:
(0, 0),
M242 Q2(b) (S. Zhang) .
Name:
1. Find the limit
2/x
(1) lim (1 + sin x)
x0+
2/x
(2) lim (1 + sin x)
x
ans:
limit(1+sin(x)^(2/x), x =0)
exp(2)
(1) It is of type 1 . Before doing it, we know the answer is between 1 and since the base is bigger than 1.
2/x
M242 Q3(a) (S. Zhang) .
Name:
1. Find the volume of the solid obtained by rotating the region bounded by the given cures about the x-axis, by
both (1) rotation method and (2) cylindrical shell method.
x=
y, x = 0, y = 1.
ans: Find intersections:
(0, 0),
M242 Q2(c) (S. Zhang) .
Name:
1. Find the limit
(1) lim
x
(2) lim
x1
1
1
2x
3
x
3
x
2x
ans:
limit(1-3/x)^(2*x), x =infinity)
exp(-6)
(1) It is of type 1 . Before doing it, we know the answer is between 0 and 1 since the base is less than 1.
F =
1
3
x
2x
M242 Q2(a) (S. Zhang) .
Name:
1. Find the limit
3/x
(1) lim (ln x)
x
3/x
(2) lim (ln x)
x1
ans:
limit( ln(x)^(3/x), x =infinity)
1
(1) It is of type 0 . Before doing it, we know the answer is between 1 and since the base is bigger than 1.
3/x
F = (ln x)
M242 Q1(c) (S. Zhang) .
Name:
1. For nding the root of the equation by doing 2 steps of the Newtons method:
f (x) = x3 3 + x, x1 = 1
ans:
f (x) = x3 3 + x
f (x) = 3x2 + 1
xn+1 = xn
xi
1
1.25
1.214285714
f (xi )
1
0.203125
0.004737609
f (xn )
f (xn )
f (
M242 Study Guide for E 3 (S. Zhang) .
1.
1 1 1 1
1
1
sn = + + . +
3 4 4 5
n+2 n+3
1
1
=
3 n+3
1
= 0.3333
3
Find the convergence by the integral test.
33.23
ln n
n3
n=1
ans: We nd the integral is convergent: (integration by
parts)
1
ln ndn
n3
n2
n2 dn
l
M242 Study Guide for E nal (S. Zhang) .
1.
limit(sin(x)*ln(x), x =0)
0
Use Newtons method to nd x3 :
2.13
3. Find the volume of the solid obtained by rotating the region
bounded by the given cures about the specied line, by
(1) the method of rotation (add
M242 Q1(b) (S. Zhang) .
Name:
1. For nding the root of the equation by doing 2 steps of the Newtons method:
f (x) = x3 1 2x, x1 = 2
ans:
f (x) = x3 1 2x
f (x) = 3x2 2
xn+1 = xn
xi
2
1.7
1.62308845
f (xi )
3
0.513
0.02971350530
f (xn )
f (xn )
f (xi )
10
M242 Study Guide for E 2 (S. Zhang) .
=
sin5 x cos2 xdx.
1. Find
Check the answer by taking derivative.
ans: u = cos x
sin5 x cos2 xdx =
=
5. Find
(1 u2 )2 u2 (du)
x2
x2
dx
4x + 5
ans:
1
2
1
cos3 x + cos5 x cos7 x + c
3
5
7
x2 4x + 5 = x2 4x + 4 + 1 =
M242 Q1(a) (S. Zhang) .
Name:
1. For nding the root of the equation by doing 2 steps of the Newtons method:
f (x) = x4 1 x, x1 = 1
ans:
f (x) = x4 1 x
f (x) = 4x3 1
xn+1 = xn
xi
1
1.33333333
1.142450142
1.265404666
f (xi )
1
0.827160493
0.43892312
0.298
to get
M242 Hw10 (S. Zhang) 11.9: 4-7, 11, 15-16, 23-24, 38
11.10: 5-9,14-15,18-19,21-22,30-33, 36-37, 47-48, 51, 55-57, 61-66
.
1
= 1 + 2x + 3x2 + . + (n + 1)xn + .
(1 x)2
1. (11.9:4)
Find the power series representation and its interval of convergence
R
Next, we determine further if the series converges absolutely
or conditionally, after knowing it converges. Consider the
corresponding
|an | = bn series:
M242 Hw9 (S. Zhang)
11.5: 4-9, 14-17, 23, 27, 32-33.
11.6: 4-8, 12-14, 20-23, 27-28, 31
11.8: 6-9, 16
f (4) (z)
= 4(1 + 2z)4
4!
M242 Hw11 (S. Zhang) 11.11: 5-6, 15(ab)-18(ab)
10.1: 1, 6-8, 11-13, 19 .
1.
41.41
(11.11:15)
Find the Taylor polynomial Tn (x) for the
function f at the number a. And bound the error for Tn (x)
by Rn on the given interval .
T3 =
1
<5 104
4n4
n4 >10000/20
M242 Hw8 (S. Zhang)
11.3: 6-13, 18-20, 34-37, 42
11.4: 3-6, 11-17, 27-32.
1. (11.3:6)
n >5001/4 4.72
Find the convergence by the integral test.
n=1
We need to compute the rst 5 terms.
1
n+4
Again, it is enough to try directly as
M242 Hw2 (S. Zhang)
6.2: 6-13, 21-26, 49-51
6.3: 4-5, 12-17, 29-32, 39-41
6.5: 7-10, 19-20 .
(6.2:9)
Find the volume of the solid obtained by rotating the region bounded by the given cures about the
specied line. Sketch the region, the solid and a typical
M242 Hw4 (S. Zhang) .
1. (7.3:28)
2. (7.3:29)
Find
Find
x 1 x4 dx
x2 + 1
dx
(x2 2x + 2)2
ans: It is of type a2 x2 . Here we can do a change
of variable rst u = x2 . But it is better to expand the idea
above, letting
ans:
x2 2x + 2 = (x 1)2 + 1
x2 = sin
1
2 y dy
y
1
2 dy
y
= 2 y ln y 2 y + C
ln y
dx = 2 y ln y
y
= 2 y ln y
M242 Hw3 (S. Zhang)
7.1: 2-4, 7-10, 15, 17-18, 22-23, 27-29, 33-34, 47-48, 51-52.
7.2: 3-5, 8-11, 13-14, 21-24, 29-30, 33-34, 43-46, 57, 62 .
1.
(7.1:10)
Find
11.31
9
sin1 x dx
4
l
f (x) = 100x99
M242 Hw1 (S. Zhang)
4.8: 5-7, 11-12, 35-36
4.4: 5-7, 10-11, 24-26, 40-42, 49-55
6.1: 5-10, 31-32, 50-51.
xn+1 = xn
f (xn )
f (xn )
1. (4.8:5)
If we choose x1 = 1 or some other numbers, the iteration would diverge. We need to choose the ini