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
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
Solutions to Self-Quiz on Sections 4.24.4
1. (a) f (x) = 3x4 + 4x3 , so f 0 (x) = 12x3 + 12x2
f 0 (x) = 0 when x = 0 and x = 1. So f (x) has critical numbers x = 0 and
x = 1. Note that f 0 (x) is defined everywhere so there are no other critical
numbers.
Math 10a
Solutions to Self-Quiz on Section 3.5
1. (a) Differentiating 3x3 + 4y 3 + 8 = 0 implicitly with respect to x gives
9x2 + 12y 2
dy
dy
9x2
3x2
dy
=0
=
=
.
dx
dx
12y 2
dx
4y 2
(b) Differentiating cos y = x2 + y 2 tan x implicitly with respect to x
Solutions to Self-Quiz on Section 4.5
1. lim
x0
tan x
0
. This limit is an indeterminate form of Type , so apply lHopital, getting
x
0
lim
x0
sec2 x
sec2 (0)
1
tan x
= lim
=
=
= 1.
x0
x
x
1
1
x2
0
. This limit is an indeterminate form of Type , so apply l
Math 10a
1. (a) f 0 (x) = x4
Solutions to Self-Quiz on Section 3.7
1
4x5 ln x
x
(log3 x)(cos x + 1) (sin x + x)
(b) f 0 (x) =
(c) g 0 () =
1
x ln 3
(log3 x)2
1
sec tan
sec
(d) f 0 (x) = 3 sec2 (log5 (2x)
1
2
2x ln 5
(e) f 0 (x) =
1
1
cos x
ln(sin
Math 10a
Solutions to Selfquiz on Section 3.8
20, 000
1. N (t) =
+ 21, 000.
2 + .2t
1
(a) Its easiest to write the function as N (t) = 20, 000(2 + .2t) 2 + 21, 000. Then
use the chain rule to differentiate, getting
2000
3
1
N 0 (t) = 20, 000 (2 + .2t) 2
Math 10a
1. In the following, find
Self-Quiz on Section 3.5
dy
by implicit differentiation:
dx
a. 3x3 + 4y 3 + 8 = 0
b. cos y = x2 + y 2 tan x
2. Find the equation of the line tangent to x4 + 16y 4 = 32 at (2, 1).
3. Show that there are no points on the g
Self-Quiz on Section 4.5
Find the following limits:
tan x
x
1. lim
x0
2. lim
x2
x2
x+22
3. lim
1 cos x
sin(x2 )
4. x
lim
x
x cos x
x0
x2 6
5. lim
x+ 2x2 15x
ln x
ex
6. x
lim
7. lim
x+
8. lim
x1
x2
e2x x
ln x
sin(x)
9. lim+ x2 ln x
x0
10. lim+
x1
1
1
x1
l
Math 10a
Self-Quiz on Section 3.7
1. Find the derivative of each of the following functions. You do not have to simplify
your answers.
sin x + x
log3 x
a. f (x) = x4 ln x
b. f (x) =
d. f (x) = 3 tan(log5 (2x)
e. f (x) = ln ( ln(sin x)
c. g() = ln(sec )
f.
Self-Quiz on Sections 4.2 and 4.4
1. For each of the following functions, find the critical numbers of f (x), the interval(s)
on which f (x) is increasing/decreasing, and the x- and y-coordinates of all the local
maxima and minima of f (x). Hint: Be caref
Math 10a
Selfquiz on Section 3.8
1. The total student enrollment t years from now in the Continuing Education division of a
20, 000
small local university is modeled by the function N (t) =
+ 21, 000. Note that
2 + .2t
N (t) measures the number of studen
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
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
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
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
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
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
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 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
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
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 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 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 10a
1. (a)
(b)
Solutions to Selfquiz on Section 2.2
lim f (x) = 1
x3
lim f (x) = 1
x3+
(c) lim f (x) = 1
x3
(d) f (3) doesnt exist
(e)
(f)
lim f (x) = 2
x2
lim f (x) = 2
x2+
(g) lim f (x) doesnt exist since the one-sided limits disagree
x2
(h) f (2)
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
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 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