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MATH 244 Name: AMWU
Spring 2014
Exam 2
04/09/2014
Time: 80 lVIinutes _ _ ' Section
This exam contains 8 pages (including this cover page) and 8 problems. Check to see if any
pages are missing. Enter all requested information on the top of this page,
Math 244 Exam 1 Review
Material Covered: 1.1 to 1.3, 2.1 to 2.6, 3.1 and 3.2 (Gravity problems, 2.7, 8.1-3 omitted).
Format of the Exam and Material to Review:
Seven problems: First one is a non-computational one to test your overall understanding and is
MATH 244
Spring 2014
Exam 1
02/26/2014
Time: 80 Minutes
Name:
Section
This exam contains 8 pages (including this cover page) and 7 problems. Check to see if any
pages are missing. Enter all requested information on the top of this page, and put your
initi
MATH244
FINAL EXAM Practice Problems
SPRING 2014
These are just practice problems. The final exam will look
different and may contain different material. Please see the
review document posted in sakai for a guide to final exam format
and contents.
1. Find
Math 244 Exam 2 Practice Problems
1. Consider the initial value problem
( )
( )
(a). Find the solution.
( )
(b). Change the second initial condition to
and find
the solution as a function of
Then find the critical value
of that separates solutions that al
CALCULUS BC
WORKSHEET 1 ON LOGISTIC GROWTH
NAME _
Do not use your calculator.
1. Suppose the population of bears in a national park grows according to the logistic differential
dP
5 P 0.002 P 2 , where P is the number of bears at time t in years.
equatio
:2m
2
Section _si_
_. cfw_37 4" vg WWW m
* : iiis m,
lx/EATH 244 (13), Dr. Z. , FINAL EXAM, Friday, Dec. 20, 2013, 4:007:00pm,
PHllS .
NAME: (print!) MUQ
EMail address: ME? \L. 1:13
No Calculators! No Cheatsheets! Write the lial answer to each problem
Dr. Z.s Calc4 Lecture 8 Handout:
Homogeneous Second-Order Differential Equations With Constant Coefficients
By Doron Zeilberger
The general form of a general (usually non-linear) second-order diff.eq. is
y 00 (t) = F (t, y(t), y 0 (t)
.
Most diff. eq. can
Dr. Z.s Calc4 Lecture 7 Handout: Numerical Solutions of Ordinary Differential Equations
By Doron Zeilberger
Important Method: (Eulers method for solving a first-order ode)
For the initial value problem
y 0 = f (x, y) ,
y(x0 ) = y0
,
with mesh-size h, you
Dr. Z.s Calc4 Lecture 6 Handout
By Doron Zeilberger
[Version of Sept. 23, 2013, correcting a typo, thanks to Jonathan Chang]
Some first order diff.eqs. are neither linear nor separable, but you can still solve them exactly, at
least implicitly, (i.e. find
Dr. Z.s Calc4 Lecture 5 Handout: Autonomous Equation
By Doron Zeilberger
An autonomous first-order diff.eq. is a special case of a separable equation that has the form
y 0 (t) = f (y) ,
where the function on the right side only depends on y. Of course we
Dr. Z.s Calc4 Lecture 1 Handout: Introducing Differential Equations
By Doron Zeilberger
Section 1: Direction Fields
A general first order differential equation looks like
dy
= f (y, t) ,
dt
where f (t, y) is (usually) a function of both t (usually time),
MATH 244 (1-3), Solutions to Dr. Z. , Exam II, Tue., Nov. 26, 2013, 8:4010:00am, PH-115
1. (10 pts.) Solve the initial value problem
4 y(t) + y 00 (t) = 3 sin t
y 0 (0) = 3 y(0) = 1.
,
Ans.:
y(t) = cos 2t + sin 2t + sin t
,
We must first rewrite it in the
MATH 244 (1-3), Dr. Z. , Solutions to Exam I, Mon., Oct. 14, 2013, 8:4010:00am, PH-115
Version of Oct. 16, 2013, 9:55am (thanks to Amar Patel, correctiong a typo in problem
6 (in the answer box)
1. (10 pts.) Find the general solution to the following diff
Answers to Dr. Z.s Math 244 (Calc4) Homework assignments (when applicable)
Disclaimer: not responsible for any errors. The first finder of any error will get $1.
Version of Sept. 21, 2016: Thanks to Vinay Chang (correcting a typos in #4, problem 2).
Previ
Dr. Z.s Calc4 Lecture 2 Handout: Method of Integrating Factors for First-Order Linear Equations
By Doron Zeilberger
(Version of Sept. 6, 2013, correcting a typo found by Meredith Taghon )
Theory: After possibly dividing the coefficient of y 0 (t), any fir
Dr. Z.s Calc4 Lecture 3 Handout: Separable Differential Equations
By Doron Zeilberger
A general first-order diff.eq. has the form
dy
= f (x, y) ,
dx
where f (x, y) is a function of both x and y, and x and y are intertwined. If you are extremely
lucky, the
Dr. Z.s Calc4 Lecture 4 Handout: Existence and Uniqueness of First-Order Diff.Eqs.
By Doron Zeilberger
Version of Sept. 19, 2016 (correcting an omission pointed out by Emily Davis)
There are two important theorems that tells when you are guaranteed to hav
Math 251, Midterm Two Review Questions
1. Find the max and min values of f (x, y, z) = x+y+2z on the ellipsoid x2 +4y 2 +9z 2 = 1.
2. Consider the cone frustum show below
Figure 1: Dont get FRUSTrated!
Compute the volume of this cone by parameterizing the
Math 251, Quiz #0 Solutions, September 9, 2014
1. Parametrize all vectors perpendicular to v = [1, 2, 3]. Describe this set of points. Find a
vector in this set that is also perpendicular to w = [3, 1, 4].
Solution: If the vector [r, s, t] is perpendicula
Math 251, Quiz #1 Solutions, September 23, 2014
1. A particle travels along the curve r(t) = (cos(t), sin(t), t) as t ranges from 0 to 8. Describe this path in words. How far did the particle travel? Compare this to the particles
net displacement over thi
Math 251, Quiz #5 Solutions, November 4, 2014
1. Evaluate
ZZZ
(x2 + y 2 + z 2 )dV
E
where
pE is the region bounded by the xy-plane and the hemispheres z =
z = 9 x2 y 2 .
p
1 x2 y 2 and
Solution: We will parameterize E in spherical coordinates and then com
Math 251, Quiz #2 Solutions, October 7, 2014
1. At which of the following three points on the surface z = x2 + 3y2 is the terrain steepest?
(1, 1, 4), (0, 2, 12), (2, 1, 7)
Solution: For the surface z = f (x, y), the steepness can be found by computing f
640:251:0103
REVIEW PROBLEMS FOR FINAL EXAMPage 1
FALL 2014
This set of problems concentrates primarily on material from Chapters 16 and 17 of Rogawski, with
some questions material from earlier chapters. To review for the final you should study also the
Math 251, Quiz #6 Solutions, November 11, 2014
1. Let D be the parallelogram spanned by vectors h7, 2i and h4, 4i anchored at the origin.
Let G(u, v) = h7u + 4v, 2u + 4vi.
a) Show that G maps the square [0, 1] [0, 1] to D.
Solution: We know linear maps se
Math 251, Quiz #4, October 28, 2014
1. Compute the average value of f (x, y) = xy over the square [0, 1] [0, 1]. Recall that the mean
value theorem for integrals says that (since f is continues and the region is closed, bounded and connected) that this av
Math 251, Quiz #4 Solutions, October 21, 2014
1. Use Lagrange multipliers to find the maximum value of f (x, y, z) = xyz subject to the
condition x + 2y + 3z = 9.
Solution: The question is misworded. There is no maximum value. To see this, we will
interse
Math 251, Quiz #7 Solutions, December 9, 2014
1. Let F(x, y, z) = (xyz, x2 , y2 + z).
a) Show that H = F = (2y, xy, 2x xz).
i
j
k
= (2y 0, (0 xy), 2x xz) = (2y, xy, 2x xz).
Solution: F = x
y
z
xyz x2 y 2 + z
ZZ
H dS where S is the hemisphere x2 +