PROBLEM SET 9
EXERCISES ON APPLYING THE DEFINITE INTEGRALL
PROBLEMS: 3, 5, 6, 11, 20, 22
Problem 3
We must be careful to do everything in yards. The density function is given by
30 x
3
where x is the distance from the goal line in yards. We want the numbe
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
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
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-
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
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
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).
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 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, 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
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
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
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
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
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
Power Series Representations of Functions
Theorem. If the power series
X
ak (x c)k = a0 + a1 (x c) + a2 (x c)2 + has radius of convergence
k=0
R where R > 0 or R = , then the function f (x) =
X
ak (x c)k = a0 + a1 (x c) + a2 (x c)2 +
k=0
is differentiabl
More Series Problems
1. (a) Give a power series representation of ex . For what x is it valid? (How do we know that the power
series is actually equal to ex ?)
(b) Use (a) to find a series whose sum is e.
(c) Explain why the series in (b) converges.
(d) S
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
Problem Set 20 The Ratio Test
1. In this problem, well look at the series
X
72k+1
k=1
(3k)!
. Let ak =
72k+1
.
(3k)!
(a) Write the series in notation.
Solution.
(b) What is
75
77
73
+
+
+ .
3!
6!
9!
a3
a2 ?
Solution. This is equal to
(c) What is
77
9!
75
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 centered at 0 is
X
ak xk , or a0 +a1 x+a2 x2 + ,
k=0
where th
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 of friction; the coefficient of x (9 in this case) mea
Problem Set 30 - Introduction to Systems of Differential Equations
1. In each part, you are given a system of differential equations modeling a predator-prey system. Answer
the following questions:
Which of x(t) and y(t) represents the prey population (a
Problem Set 32 - Systems and Shapes of Trajectories
We have now learned a few ways of studying a system, each giving us slightly different information:
We can look directly at the solutions.(1)
We can use phase plane analysis to describe which direction
Problem Set 31 - Phase Plane Analysis
1. Let x(t) be the number of hundreds of beasts of species X at time t and y(t) be the number of hundreds
of beasts of species Y at time t. In this problem, youll analyze the system
dx
= 0.1x 0.025xy
= 0.025(4x xy)
dt
Problem Set 28 - x00 + bx0 + cx = 0
1. Solve the following differential equations for x(t); give the general solution in each. (Be careful! These
are not all of the same form, so dont go on auto-pilot.)
(a) x00 9x0 = 0.
(c) x00 9 = 0.
(e) x00 + 6t + 9 = 0