Problem Set 01 Density and Approximation
1. Eric is using his printer to print a rectangular photograph measuring 8 inches by 10 inches. Unfortunately, the printer starts running out of ink as soon as Eric puts the paper in, so the density of ink on
the f
Homework 03 Area and Volume
1. Let R be the region enclosed by the curves y = x2 +1 and y = 4x+1. Find the area of R two ways, once
by slicing vertically and once by slicing horizontally. For each method, please draw a picture showing
the rectangle you us
Homework 02 Density and the Definite Integral
1. Evaluate lim
n
n
X
sin xk x, where xk = + kx and x =
k=1
.
3n
Solution. The expression we are evaluating is a limit of Riemann sums, so we should be able to
write it as a definite integral. We can see that
Problem Set 12 Taylor Approximation, Continued
1. In this problem, you will look numerically at Taylor polynomial approximations for the function f (x) =
ex centered at 0.
(a) Find the degree n Taylor polynomial approximation Pn (x) for f (x) = ex centere
Problem Set 14 Definition of Convergence
1. In this problem, youll investigate the series
X
k=1
1
.
k2 + k
(a) Write out the first 5 terms of the series.
1
1
k2 +k , which gives 2 .
1 1
1
1
1
2 , 6 , 12 , 20 , and 30 .
Solution. To find the first term, we
Problem Set 11 Taylor Approximation
1. (a) Use a degree 2 Taylor polynomial approximation to approximate 3 995. (Its up to you to pick a
suitable function to approximate and a suitable center for the approximation.) Please simplify the
coefficients of you
Problem Set 07 Numerical Methods and Error
Most of your solutions on this problem set will be pretty short. Please use complete English sentences, but
there is no need to explain in very great detail usually a brief sentence that hits on the main point is
Problem Set 08 Error in Numerical Methods
1. One of your friends is doing his mathematics homework on the run. He managed to squeeze in one
problem between lunch and his expository writing class. He used his calculator to find Ln , Rn , Tn , and
Mn to app
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 following cases? (Please draw pictures!)
(a) if f is incr
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 + a4 + . Because this is different from sums weve
dealt w
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 anything about whether the following integrals converge or d
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 function f (x),
Z
a
Z
3. Does the improper integral
1
a
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 sum?
3. (a) Does the series 1 +
1 1 1
+ + + converge o
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 in terms of a, b, and n.)
2. There is a similar formula
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 suggests that 4 15 is also pretty close to 2.
This is a perfec
Some Methods for Solving Differential Equations
1. Find the general solution of the differential equation
dM
= 2.4 0.2M .
dt
dy
t
= using separation of variables. (Previously, we solved by drawdt
y
ing the slope field, guessing the solution, and checking
Ratio Test
Definition. A series
convergent.
Theorem. If a series
P
P
ak is called absolutely convergent if the series of absolute values
P
|ak | is
ak is absolutely convergent, then it is convergent.
The opposite (converse) of the theorem is not true: the
Definition of Convergence
1. Does the series 1 + 1 + 1 + converge or diverge? If it converges, what is its sum?
Solution. The n-th partial sum sn is the sum of the first n terms, which is just n. Since lim sn =
n
lim n does not exist, the series diverges
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 following cases? (Please draw pictures!)
(a) if f is incr
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 this up as a sum of two improper integrals. We may as wel
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 polynomial.
If we were just finding the degree n Taylor poly
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
b
1
= lim x
b
1
1
= lim + 1
b
b
1
As b , 1b 0, so th
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 ak .
Solution. Lets call the Taylor polynomial Pn (x).
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 flavoring syrup in the cup varies with
height above the t
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 in terms of a, b, and n.)
Solution. In the sketch, we se
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 region about the x-axis?
1
-1
- 12
1
y Ix3 + 1M
2
1
2
1
y2x-
Differential Equations
1. Mr. Moneybags decides to open a bank account with an opening deposit of $1000. Suppose that
the account earns a nominal annual interest rate of 6%, compounded annually.(1) Assuming Mr.
Moneybags completely ignores the account aft
Problem Set 21 Power Series
1. (a) Find the radius of convergence of the following power series.
i.
X
(x 2)2k
k=1
k 2 9k
.
Solution. As usual, we try the Ratio Test first:
(x2)2(k+1)
(x 2)2 k 2
(k+1)2 9k+1
lim (x2)2k = lim
k 9(k + 1)2
k
2
k
k 9
Problem Set 18 Asymptotics
1. Arrange the following expressions in order of relative growth rate. In other words, arrange these with
signs in between them: k!, ln k, k 0.3 , k 1000 , k k , 0.99k , 1.01k , 500k , k. (Hint: ln k is not the one
with the sma
Problem Set 10 Improper Integrals, Continued
1. Which of the following integrals are improper? Why? (You need not evaluate any of the integrals.)
Z
3
(a)
1
1
dx.
3x 6
Solution. This integral is improper because
vertical asymptote at x = 2)
Z
1
(b)
0
1
3x6
Area and Volume
1. Find the area under the curve y = ln x from x = 1 to x = e.
Solution. Here is the area we are interested in:
1
e
1
Z
e
ln x dx, but that integral is a little hard to compute (you may not know
We know that this area is
1
how to do it yet
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.
Solution. See Weekly Problem TI 3. b).
2. In each part, decide which method of integ
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.
Solution. The solid looks like a football.
To find its volu
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 sprinkled on each wedge so that the
density of cheese is gi
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 population of Cambridge
in the year 2000?
Solution. Th