MAT132 Midterm I
Spring 2010
Print your name, ID number and section number on your answer sheet.
Do each of the following 6 problems. Show some work or give an explanation when you are
asked to do so. It is not necessary to simplify your answers.
Please d
MATH 132
20 pts
Solutions to Midterm 2
1. Compute the average value of the function sin x on the interval [0, ].
Solution: The average value is
1
0
20 pts
0
1
sin x dx =
cos x
=
0
2
.
2. Let S be the region bounded by the graph of y = x2 + 1, the x-axis,
P
Calculus Applied to
Probability and Statistics
Case Study: Creating a Family Trust
P.1 Continuous Random
Variables and
Histograms
P.2 Probability Density
Functions: Uniform,
Exponential, Normal,
and Beta
You are a nancial planning consultant at a neighb
MAT 132 Spring 2006 Review - Chapter 5
References are to Stewart, Single Variable Calculus - SBU Edition
5.5 Substitution rule. Know how to recognize the "outer" function f and the "inner" function g in
the integrand f(g(x) g'(x) dx, and how to then simpl
MAT 132 Spring 2006 Review - Chapter 6
References are to Stewart, Single Variable Calculus - SBU Edition
6.1 Areas between curves. Understand that if f > g on an interval [a,b] then the area between
the graphs is the area under f minus the area under g [E
MAT 132 Spring 2006 Review - Chapter 7, Appendix I, Notes on 2nd order D.E.s
References are to Stewart, Single Variable Calculus - SBU Edition
7.2 Direction fields. Understand how the direction field for the first-order differential equation y'
= f(x,y) a
MAT 132 Spring 2006 Review - Chapter 8
References are to Stewart, Single Variable Calculus - SBU Edition
8.1 Know what a sequence is, and be able to check convergence/divergence in simple cases: for
rational functions of n as in Example 4 [Exercises 10, 1
SUNY at Stony Brook MAT 132 Calculus II
Exercises on Second Order Differential Equations
For each of the differential equations 1-5, give the general solution.
1. y' + 6 y' + 13 y = 0
2. y' - 4 y' + 4 y = 0
3. y' - 2 y' + 26 y = 0
4. y' - 4 y' + 13 y
Homework 12 Solutions
Section 12.10:
2. (a) The graph is increasing at x = 1 but the c1 Taylor coecient (namely f (1)
is negative.
(b) The graph is concave downward at x = 2 but the c2 Taylor coecient (namely
1
2 f (2) is positive.
7. If f (x) = sin(x) th
MAT132Fall 2011 Topics
5.1, 5.2, 5.3 Review of the definite integral and the Fundamental Theorem of Calculus.
5.3, 5.4 The Fundamental Theorem of Calculus
5.5 Substitution
5.6 Integration by parts
5.7 Trigonometric integrals, Trigonometric substitutions,
Consider the solid obtained by rotating the region bounded by the given curves about the Xaxis.
y = 9 252:2, y = 0
Find the volume V of this solid.
1296
25
v:
J
Sketch the region, the solid, and a typical disk or washer. (Do this on paper. Your instr
Find a power series representation for the function.
n=|E|
Determine the interval of convergence.
SIS
Need Help?
Find a power series representation for the function and determine the interval of convergence.
00
power series representation
Oil leaked from a tank at a rate of )(t) liters per hour. The late decreased as time passed, and values cf the rate at twu hour time intervals are shown in the table. Find luwer and
upper estimates for the total amDunt of Dll that ieaked out.
V = J L
Paper Homework 4 Solution
1. Consider all possible cases and their probabilities that Professor Lee wins.
4 for Lee :
1
.
6
5 5 1
.
6 6 6
non-4 for Lee, non-4 for Sharland, non-4 for Lee, non-4 for Sharland,
5 5 5 5 1
4 for Lee : .
6 6 6 6 6
non-4 for
MAT132 PAPER HOMEWORK 3
DUE IN RECITATION ON 4/2 OR 4/3
Problem 1. We will prove the Generalized Binomial Theorem; that is, for any k and for |x| < 1
(1 + x)k = 1 + kx + +
(i)
(ii)
(iii)
(iv)
(v)
k(k 1) (k (n 1) n
k n
x + =
x .
n!
n
n=0
k1
k
Show that k1
MA 2320: Series Tests
Telescoping
P
Geometric - Given k=0 rk
1
if |r| < 1 then the series converges to 1r
if |r| 1 then the series diverges.
P
Divergence Test - Given
ak .
if lim ak 6= 0 then the series diverges.
P
Integral Test - Given 1 ak where ak
MAT 132 - Calculus II - Practice problems
Below is a list of practice problems. It starts with the table of integrals
that you will have on the actual exam. Remember that in the actual exam
you need to show all work and explain reasoning whenever possible
11.8
3-20 Find the radius of convergence and interval of convergence of the series.
3.
X
xn
.
n
n=1
We will apply the ratio test.
n+1
x
n x n
n + 1 xn = n + 1 |x|
as n .
Hence the radius of convergence is 1. For x = 1, the series is a divergent p-se
MAT 132: Calculus II
Stony Brook,
Fall 2011
About this course: The goal of this course is to develop your understanding of the concepts of integral
calculus, infinite series, and differential equations and your ability to apply them to problems both withi
REVIEW FOR TEST I: FALL 2015
Test I is scheduled for Wednesday September 30 from 8:45pm to 10:15pm.
The room assignments are announced on your lecture website. The test will
cover material in sections 5.1-5.7 and in approximating areas by Riemann
sums a D
Worksheet 8 Solutions, Math 1B
Series Representations of Functions
Friday, March 9, 2012
2. Uses series to evaluate the limit
lim
x0
1 cos x
1 + x ex
Solution Sketch
We notice that the Maclaurin series for cos x begins with constant term equal to 1, and s