Math 152 (21-23): Workshop 1
September 11, 2015
1-1. Sketch the region R defined by 1 x 2 and 0 y 1/x3 .
a) Find (exactly) the number a such that the line x = a divides R into two parts of equal area.
b) Then find (to 3 places) the number b such that the
Math. 152, Additional review problems for exam 2.
1. Find the limit of each of the given the sequences.
(
(a) an = 1 +
3
2n
)4n
sin2 n
n3
(b) bn =
2. Find the exact sum of each of the given series.
(a)
(
n=4
(c)
1
1
(n + 1)2 (n + 3)2
2n + 3n
5n
n=8
)
(b)
640:152 Calculus II, Midterm Exam #2, Spring 2014
Department of Mathematics, Rutgers University
NAME (PLEASE PRINT):
SIGNATURE:
INSTRUCTOR: Dr. Nikolaev
SECTION:
Instructions:
Turn off and put away all mobile phones, computers, iPods, etc.
Show answers
640:152 Calculus II, Spring 2012
Final Examination
Department of Mathematics, Rutgers University
NAME (PLEASE PRINT):
SIGNATURE:
RUTGERS ID #:
INSTRUCTOR & SECTION:
Instructions:
Turn off and put away all mobile phones, computers, iPods, etc.
Show answe
Algebra / Trig / Calculus I Review
Critically important: You absolutely must be able to do the following types of problems in order
to succeed in this course. If you are unsure of any of these, you should study the appropriate
sections in your current tex
Sample Homework Write-Up - Excellent
Problem: A fa.rmerhas 200 feet of fencingto make a rectangula,rpen. What are the length and width
of the largestarea pen that she can make with her fencing?
Let x:
Lurd-th pen, t^ cfw_t
"\
.f ?c^, t^ cfw_t
y - te,ryi\t
Challenge Problems Exam 1
1. (a) State the Mean Value Theorem, making sure you clearly state the hypotheses and the conclusion.
(b) Show that if F (x) and G(x) are antiderivatives of f (x), then F (x) G(x) = C, where C is a constant
(does not depend on x)
Economics Applications Homework
1. Suppose a company can produce a product for $500 a piece. They have 20 potential customers,
each interested in buying a single item. There are 6 that are willing to pay $500, 3 willing to pay
$750, 4 willing to pay $900,
Series Problems Fun Pack !
I. Do the series below converge or diverge? What test would you use? Find the sum of each series when reasonable.
X
n
3
n +1
n=1
13.
X
(1)n n
2.
n+1
n=1
14.
1.
X
cos(3n)
3.
1
+ (1.2)n
n=1
4.
X
2
2 + 4n + 3
n
n=1
X
4n+1
5.
5n
n=0
Integration Problems Fun Pack !
I. Evaluate the integrals below, clearly noting which integration technique(s) you use in your solution. If the
integral is improper, say so, and either give its value or say that the integral is divergent. You may only use
Summary of Convergence and Divergence Tests for Series
Test
nth -term
Series
Convergence or Divergence
Comments
X
Diverges if lim an 6= 0
Inconclusive if lim an = 0
an
Geometric
Series
n
n
(i) Converges with sum S =
X
arn1
a
if |r| < 1
1r
(ii) Diverges if
152 Review problems for exam 2
Try to do all the problems before you watch the video solution.
1. (a) Find the 4th Maclaurin polynomial of the function f (x) = e2x .
(b) Use (a) to approximate e0.2
(c) Find the remainder of the approximation in (b).
2. Fi
Volumes
A. Consider the region bounded by the graphs of
and
Find the volume of the solid obtained by rotating the above region about the following axes. Find
the volume twice each time: first using the cross-sections method and then again using the shells
Final Exam Review Questions Part 1.
Area between curves- Volumes- Average Value-integration by parts-improper integral- parametric-polardifferential equation
1. The base of a solid is the region inside the ellipse
Each cross section of the
solid perpendic
MATH 152 REVIEW
Area between curves: Find intersection point (unless a, b given), find top function f(x), bottom function g(x)
function is with respect to y, find right function f(y), and left function g(y)
Example:
Volume by cross section
(OR if
Volume o
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TECHNIQUES OF INTEGRATION
Due to the Fundamental Theorem of Calculus
7
(FTC), we can integrate a function if we know
an antiderivative, that is, an indefinite integral.
TECHNIQUES OF INTEGRATION
We summarize the most important integrals
we have
Math 152 Workshop 6
1
Z
xk (1 x)l dx.
Problem 1. Let k, l be nonnegative integers, and let I(k, l) =
0
a) Show that I(k, l) = I(l, k).
b) Find I(k, 0) and I(0, k)
c) For k > 0, prove that I(k, l) =
k
I(k 1, l + 1)
l+1
d) Find I(1, 1) and I(3, 2).
e) Show
Math 152(01-03): Workshop 4
October 19, 2016
4-1. Suppose f (x) = x ln x .
a) Verify that lim+ f (x) = 0 and lim f (x) = 0. Graph f on the interval [0, 10].
x
x0
x2
b) A remarkable result of third semester calculus is that e
dx = . Assume that this
resul
Beginning Differential Equations Homework
1. Verify that the function y(x) = e3x is a solution to the DE y 00 + 2y 0 3y = 0
2. Verify that the function y(x) =
ln x
is a solution to the DE x2 y 00 + 5xy 0 = 4y
x2
3. Determine for what value(s) of r the fun