R E F E R E N C E PA G E S
Cut here and keep for reference
ALGEBRA
GEOMETRY
Arithmetic Operations
Geometric Formulas
a
c
ad " bc
" !
b
d
bd
a
a
b
d
ad
! ) !
c
b
c
bc
d
a!b " c" ! ab " ac
a"c
a
c
! "
b
b
b
Formulas for area A, circumference C, and volume V
Math 10b
Homework for Section 6.2
Show all your work.
1. In each of the following, nd the volume of the solid S.
(a) The base of S is the region in the x-y plane bounded by the graph of y = ex and
the xaxis over the interval [0, ln 3]. Each cross-section
Math 10b
Homework for Sections 5.9 and 5.10
I. Section 5.9. Note: Youll nd it helpful to use a scientic calculator for this assignment.
1. Read pages 401-404 in the textbook.
1
2. Consider the integral
0
4
dx.
1 + x2
(a) Use the Midpoint Rule with n = 3 t
Math 10b
Homework for Sections 5.6 and 5.7
Show all your work on all homework assignments.
I. Section 5.6.
1. Do the following problems on pages 387: # 7, 12, 14, 201 , 262 , 43.
2. Find the following:
e
a.
x4 ln x dx
x sec2 x dx
b.
c.
sin(ln x) dx
1
3. R
Math 10b
Homework for Sections 7.3 and 5.5
Show all your work on all assignments.
I. Section 7.3.
1. Do the following problems on page 514:
# 2, 4, 5, 6, 11, 13, 16
2. Find the general solution to the following separable dierential equation:
dy
+ exy = 0.
Math 10b
Homework for Section 8.2
Show all your work.
1. Do the following problems on pages 572573. Show all your work in every problem.
#1113, 15, 16, 19, 23, 28, 41, 42.
2. Suppose that the nth partial sum of a series
an is
n=1
sn =
3n + 1
.
4n + 5
What
Math 10b
Solutions to Practice Exam for Exam 1 (Problems 12 - 25)
12. Evaluate the following:
1
(a) sin(arctan( 3 ) = sin( ) =
6
1
2
(b) arcsin(sin 2) = sin1 (0) = 0
13. Find the domain of f (x) = ln(arcsin(x).
The domain of arcsin(x) is [1, 1]. Since f (
Math 10b
Review Sheet for Exam 2
The rst Math 10b midterm will be on Thursday, November 6, from 7 9 p.m.
Locations:
Gerstenzang 122: Sections 2 & 4 (Yan and Angelica)
Gerstenzang 121: Sections 1, 3 & 5 (Becci, Shahriar and Yiting)
NOTE: These location
Math 10b
Review Sheet for Exam 1
The rst Math 10b midterm will be on Thursday, October 2, from 7 9 p.m.
Locations:
Gerstenzang 122: Sections 1 & 2 (Becci and Yan)
Gerstenzang 121: Sections 3, 4 & 5 (Shahriar, Angelica and Yiting)
The exam will cover
y = 0
Math 10b
Solutions for First Review for Exam 1 (Problems 1-11)
1. Evaluate the following:
(a) sin1
3
)
2
=
3
1
(b) tan(sin1 6 )
To nd tan(sin1 1 ) , start by letting = sin1 1 ). Note that is in the
6
6
fourth quadrant. We get the following triangl
Math 10b
Solutions to Practice Problems for Exam 1 (Problems 26 - 37)
26. In each of the following, nd f (x).
(a) f (x) = x2 esin
1 (3x)
f (x) = 2xesin
1 (3x)
+ x2 esin
1 (3x)
1
3
1 (3x)2
(b) f (x) = ln( arctan x)
1
1
1
1
f (x) =
(arctan x) 2
1 + x2
ar
Math 10b
Solutions to Review Sheet for Exam 2
The rst Math 10b midterm will be on Thursday, November 6, from 7 9 p.m.
Locations:
Gerstenzang 122: Sections 2 & 4 (Yan and Angelica)
Gerstenzang 121: Sections 1, 3 & 5 (Becci, Shahriar and Yiting)
NOTE: T
Math 10b
Self-Quiz on Section 5.2
1. Let f (x) be the function drawn below.
(a) Estimate the area between f (x) and the x-axis over [0, 3] by computing R3 .
(b) Estimate the area between f (x) and the x-axis over [0, 3] by computing L3 .
6
f (x) dx by com
Math 10b
Homework for Section 5.4 and 5.3
Show all your work on all assignments.
I. Section 5.4.
1. Do the following problems on page 37273: # 3, 4ad, 8, 9, 12, 13, 15, 17, 21 and 22.
2. Find a function F (x) that satises both of the following conditions:
Math 10b
Homework for Section 6.4
Show all your work on all homework assignments.
I. Section 6.4.
1. Find the length of each of the following curves on the given interval:
(a) 4y 5x = 7 on the interval [3, 1]
4
(b) y = x3/2 on the interval [0, 3 ]
(c) y =
Math 10b
1.
Solutions to Practice With Integrals II
5x 2
dx
3
x
Z 8
1
Z 8
1
Z 8
Z 8
2
1
5x 2
5x
2
dx
=
dx
=
5x 3 2x 3 dx
3
3
3
x
x
x
1
1
= 5
= 5
2.
Z 8
2
3
x dx 2
Z 8
x
13
dx = 5
1
1
3
(32
5
1) 2
3
(4
2
8
3 35
x
5
1
2
8
3 23
x
2
1
1) = 93 9 = 84.
Z 1
Solutions to Self-Quiz on Sections 5.9 and 5.10
1. (a)
Z
1
x
8
4/3
(b)
0
x
8
1
x
8
1
t
Z
So
Z
dx = lim
Zt
4/3
4/3
dx = lim
t
1
3x 3
it
8
= lim
t
3
3
t
+
3
2
=0+
3
2
= 32 .
dx converges to 23 .
t
Z
it
1
1
dx = lim
dx = lim ( ln |x + 1| ) = lim ( ln |t +
Math 10b
1. (a)
Z /2
Solutions to Self-Quiz for Section 5.6
x cos(2x) dx. Let u = x and dv = cos(2x) dx. Then du = dx and v =
0
1
sin(2x). The result is
2
Z
So
1 Z
1
1
1
x sin(2x)
sin(2x) dx = x sin(2x) + cos(2x).
2
2
2
4
x cos(2x) dx =
Z /2
x cos(2x) dx
Math 10b
Solutions to Self-Quiz on Section 6.1
1. Set x3 4x2 + 3x = 0, getting x = 0, x = 1 and x = 3. So the graph of f (x) =
x3 4x2 + 3x intersects the x-axis at the points x = 0, x = 1 and and x = 3. A sign
analysis shows that f (x) > 0 on (0, 1) and f
Math 10b
Solutions to Extra Practice for Final Exam
1. Let n(t) be the number
Z of inmates in country prisons at time t, where t is measured in
years. Then n(t) =
5
Z
t
t
t
300e 5 dt. Use substitution, with u = e 5 and du = 51 e 5 , getting
u
u
e du, whic
MATH 10b
Solutions to Homework on Taylor Series
1. Let f (x) = ln(x + 1).
(a) Find the Taylor series for f (x) centered at x = 0. Simplify the series that there
are no factorials in your answer, then write the series using sigma notation.
Solution. Note t
Math 10b
Solutions to Review Sheet for Exam 2
Note: On the review sheet, there are many definite integrals that require u-substitution. For
these solutions, we sometimes integrate, resubstitute, and then evaluate using the original
limits of integration;
Math 10b
1.
Z10
Solutions to Practice With Integrals I
x
e 10 dx. Use substitution, with u =
x
10
and du =
1
10
x
dx. Get 10e 10
i10
0
= 10(e 1).
0
Note: could also change limits of integration, getting 10eu
Z12
Z12
i1
0
= 10(e 1).
!
Z12
1
sin(3x)
sin(3x)
Math 10b
Solutions to Self-Quiz on Section 5.5
1. (a) (1 + sin t)9 cos t dt. Let u = 1 + sin t; then du = cos t dt. The resulting integral is
u10
(1 + sin t)10
9
u
du,
which
equals
+
C.
Resubstituting
gives
+ C.
10
10
1
1
1
(b)
dx. Let u = 2x 5; then du
Math 10b
1.
Z b
n
X
f (x) dx = lim
n
a
Solutions to Final Review Sheet
f (xi )x, where x =
ba
n
and xi is the right endpoint of the ith
i=1
subinterval. So xi = a + i x. In this integral, a = 1, b = 3 and f (x) = 2x2 + 1. So
x = n2 and xi = 1 + i x = 1 +
Math 10b
Solutions to Self-Quiz on Sections 6.2 and 6.4
y = ( 3 x)
1. The base of the solid S is shown below. (Its not necessary to draw the base, but a
0 < y < ( 3 x if 0 < x < 8 )
picture can be helpful.)
x = y3
0 < x < ( y3 if 0 < y < 2 )
Each crosssec
Math 10b
Solutions to Extra Practice for Exam 2
tan( x)
1
1. (a)
dx. We use substitution, with u = x and du = dx, so we
2 x
Zx
sin u
get 2 tan u du. We integrate this by writing tan u =
and doing another
cos u
substitution, with w = cos u and dw = sin u d
Math 10b
Self-Quiz on Section 5.1
1. Let f (x) = x2 x + 1 over the interval [0, 2]. The graph of f (x) is shown below.
y = x2 ! x + 1
(a) Approximate the area under the graph of f (x) over [0, 2] by computing R4 . Sketch
the rectangles you use on the grap