Problem Set 29 - x00 + bx0 + cx = 0, Eulers Formula
1. The differential equations x00 + 10x0 + 9x = 0 and x00 + 6x0 + 9x = 0 both model springs. The difference
between the two situations is the amount
Problem Set 33 - Introduction to Systems of Differential Equations
1. In each part, you are given a system of dierential equations modeling a predator-prey system. Answer
the following questions:
Whi
Numerical Methods and Error 2
Rb
1. Let I = a f (t) dt, and suppose we approximate the integral by Rn and Ln . What did you discover in
Problem Set 07 about the order of I, Rn and Ln in each of the fo
Improper Integrals, Continued
Z
1. Does the improper integral
x
dx converge or diverge? If it converges, to what number does
1 + x2
it converge?
x
y
x2 + 1
-20
-10
10
20
Z
2. True or false: For any f
3-dimensional Density Problems
1. A cone with height 8 inches and radius 6 inches is filled with flavored slush. When the cup is held
upright with the pointed end resting on a table, the density of fl
Volumes of Revolution
1. Here is one loop of the sine curve.
1
(a) If you rotate this region about the x-axis, what shape do you get? What is its volume? You may
leave your answer as an integral.
(b)
Taylor Approximation, Continued
1. If you want to find the degree n Taylor polynomial approximation a0 + a1 (x c) + a2 (x c)2 + +
an (x c)n for f (x) centered at c, write a formula for the coefficient
Density and Approximation
1. According to Wikipedia, the population density in Cambridge in the year 2000 was 15,000 people per
square mile. The area of Cambridge is about 7 square miles. What was the
Definition of Convergence
1. Does the series 1 + 1 + 1 + converge or diverge? If it converges, what is its sum?
2. Does the series 1 + (1) + 1 + (1) + converge or diverge? If it converges, what is its
Density and the Definite Integral
1. (Problem Set 1, #3) Pizza Pythagoras is known for its pizza wedges, which are shaped like right
triangles with sides of 3, 4, and 5 inches. Parmesan cheese is spri
Improper Integrals
Z
1
dx converge or diverge? If it converges, evaluate it.
x2
1
dx converge or diverge? If it converges, evaluate it.
x
1. (a) Does
1
Z
(b) Does
1
2. Using #1, can you conclude anyth
Definition of Convergence / Partial Sum Diagrams
In class, you looked at the definition of convergence of an infinite series; here is a brief recap.
Lets say we have an infinite series a1 + a2 + a3 +
Numerical Methods and Error 2
Rb
1. Let I = a f (t) dt, and suppose we approximate the integral by Rn and Ln . What did you discover in
Problem Set 07 about the order of I, Rn and Ln in each of the fo
Improper Integrals, Continued
Z
x
dx converge or diverge? If it converges, to what number does
1 + x2
1. Does the improper integral
it converge?
x
y
x2 + 1
-20
-10
10
20
Solution. We need to split th
Taylor Series
1. What is the Taylor series expansion of cos x centered at 0? Write it in both sigma and notation.
Solution. Remember that a Taylor series is essentially an infinite-degree Taylor polyn
Improper Integrals
Z
1
dx converge or diverge? If it converges, evaluate it.
2
x
1
Z
Z b
1
1
Solution. We know that
dx
really
means
lim
dx. So,
2
b 1 x2
x
1
1. (a) Does
Z
1
1
dx = lim
b
x2
Z
b
x2 dx
Taylor Approximation, Continued
1. If you want to find the degree n Taylor polynomial approximation a0 + a1 (x c) + a2 (x c)2 + +
an (x c)n for f (x) centered at c, write a formula for the coefficient
3-dimensional Density Problems
1. A cone with height 8 inches and radius 6 inches is filled with flavored slush. When the cup is held
upright with the pointed end resting on a table, the density of fl
Numerical Methods and Errors 1
Rb
1. Suppose we use a left-hand Riemann sum with n slices (Ln ) to approximate a 1 + 3x dx. Find a
formula for the error in your approximation. (Your answer should be i
Area and Volume
1. Find the area under the curve y = ln x from x = 1 to x = e.
2. Find the area enclosed by y = 2x, y = x3 , and y = 1.
1
3. (a) What shape do you get if you rotate the following regio
Taylor Approximation
Suppose we want a
decimal approximation of a number like 4 15. We could eyeball
it and think to ourselves,
Well, 24 = 16, so 4 16 = 2, and 15 is pretty close to 16, which suggest
Numerical Methods and Errors 1
Rb
1. Suppose we use a left-hand Riemann sum with n slices (Ln ) to approximate a 1 + 3x dx. Find a
formula for the error in your approximation. (Your answer should be i
Integration Techniques
1. (Weekly Problem TI 3. b).) Evaluate three of the the following four integrals.
Z
Z
(a)
ln(3x + 2) dx.
(c)
e x dx.
Z
(b)
Z
2
ex dx.
(d)
ln x dx.
2. In each part, decide which
Taylor Series
1. What is the Taylor series expansion of cos x centered at 0? Write it in both sigma and notation.
2. What is the Taylor series expansion of sin x centered at 0? Write it in both sigma
Problem Set 13 Taylor Series
1. Find the Taylor series expansion of ex centered at 0. Write it in sigma notation as well as in
notation.
x
Solution. We know that the Taylor series of f (x) = e center
Executive Information System
Introduction
An Executive Information System (EIS) is a computer-based system intended to
facilitate and support the information and decision making needs of senior execut
Problem Set 22 Power Series Representations of Functions
In any question that asks for a series (either a power series or a series of numbers), please write your final
answer in both sigma and notatio
Problem Set 17 p-series, Comparing Series to Integrals
1. Comparing series to integrals can seem very mysterious at first. In this problem, youll look again at
the idea you saw in class.
X
1
1
1
+ 3/2
Problem Set 19 Alternating Series
1. Determine whether the following series converge or diverge. If the series converges, does it converge
absolutely or does it converge conditionally?
When writing th
Problem Set 20 The Ratio Test
1. When using the Ratio Test, common mistakes are writing down incorrect expressions for ak+1 and
ak+1
incorrectly simplifying the ratio
. These errors can usually be avo
Problem Set 16 Geometric Series
1. In this problem, you will look at the geometric series
X
k=2
5
.
72k+1
(a) Write the series out in notation.
(b) What is the common ratio of this geometric series?
L