Math 2410Q Section 1.6
Name: January 22, 2016
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1. Sketch the phase line for the differential equation 2;:— = tan(y). Identify
the equilibrium points as sinks, sources, or nodes.
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Math 2410 1
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1. Four differential equations and four slopes are given below. Determine the equation that
corresponds to each slope ﬁeld.
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Math 2410Q
Section 1.5
Name: January 22, 2016
1. (a) Show that y1(t) : t2 and y2(t) : t2 + 1 are solutions to
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1 Problem. Averages Consider the plot below in which we have drawn a continuous function of time and evaluated it at the center of 10 intervals.
a) Write an expression for the average of these 10 points.
b) Generalize this expression to N points and write
1 Problem. Light source For a low light level source, the probability that a single photon is emitted in the time interval [t, t + t] is proportional to t e(t/t0 ) t, where t0 is a characteristic time scale of the emission process and t is assumed to be v
1 Problem. Going up? When a body moves a distance d along a straight line as a result of being acted on by a force of constant magnitude F in the direction of motion, then the work W done by the force is W = F d. We will generalize the denition of work to
1 Problem. Escape velocity As you know, Isaac Newton was trying to gure out how the moon goes around when an apple fell nearby. But do you know what he saw between the branches of the apple tree when he looked up? Youll gure it out. Anyway his train of th
1 Problem. Simple harmonic oscillator In this problem you will use one of the most important conservation laws in Physics, conservation of energy to calculate the period of a simple harmonic oscillator. The gure below displays a mass m, which is attached
Prelim 3 Math 1910 Calculus for Engineers
April 22, 2010
You have 90 minutes to complete this test. The test has 100 points. Calculators are not allowed. You should show
all your work, and explain where calculations are coming from. Write clearly and legi
MATH 1910: Prelim #3
Tuesday, Nov. 29th Answer the following 5 questions. Show all work. Closed book, no calculators; 1-sided (8.5 x 11) "cheat sheet" is allowed - individually and uniquely hand-written (without collaboration) - will be collected along wi
Prelim 2 Math 1910 Calculus for Engineers
April 1, 2010 You have 90 minutes to complete this test. The test has 100 points. Calculators are not allowed. You should show all your work, and explain where calculations are coming from. Write clearly and legib
Prelim 3 Math 1910 Calculus for Engineers
April 22, 2010 You have 90 minutes to complete this test. The test has 100 points. Calculators are not allowed. You should show all your work, and explain where calculations are coming from. Write clearly and legi
Prelim 1 Math 1910 Calculus for Engineers
February 23, 2010 You have 90 minutes to complete this test. The test has 100 points. Calculators are not allowed. You should show all your work, and explain where calculations are coming from. Write clearly and l
1910 - CALCULUS FOR ENGINEERS, F ALL 2011, PRELIM II
7:30-9pm, October 27, 2011.
Answer the following 5 questions.
Show all work.
Closed book, no calculators; 1-sided (8.5 x
11) cheat sheet is allowed - individually and uniquely hand-written (without coll
Prelim 1 Math 1910 Calculus for Engineers
February 23, 2010 You have 90 minutes to complete this test. The test has 100 points. Calculators are not allowed. You should show all your work, and explain where calculations are coming from. Write clearly and l
Final Exam Math 1910 Calculus for Engineers
May 17, 2010 You have 150 minutes to complete this test. The test has 150 points. Calculators are not allowed. You should show all your work, and explain where calculations are coming from. Write clearly and leg
Math 1910 Prelim I, 7:30-9pm, September 29, 2011 Answer the following 5 questions. Show all work. Closed book, no calculators; 1-sided (8.5 x 11) "cheat sheet" is allowed individually and uniquely hand-written (without collaboration) will be collected alo
Math 4310
Homework 10 Solutions
1. Let F = C, and let m, n N with 0 < m n. Prove that there exists a polynomial p(x) F[x]
of degree n with exactly m distinct roots. What goes wrong if F is a nite eld?
S OLUTION .
One can choose m distinct complex numbers
Math 4310
Name:
Homework 10
Collaborators:
Due 11/30/2012
Please print out these pages. I encourage you to work with
your classmates on this homework. Please list your collaborators
on this cover sheet. (Your grade will not be affected.) Even if you
work
Math 4310
Homework 9
Due 11/21/2012
Exercises.
1. In R4 , let
U = S(1, 1, 0, 0), (1, 1, 1, 2).
Find u U such that |u (1, 2, 3, 4)| is as small as possible.
Answer. We want to nd the orthogonal projection of (1, 2, 3, 4) onto U. To do this, we rst
nd an or
Math 4310
Name:
Homework 9
Collaborators:
Due 11/21/2012
Please print out these pages. I encourage you to work with
your classmates on this homework. Please list your collaborators
on this cover sheet. (Your grade will not be affected.) Even if you
work i
Math 4310
Homework 8 Solutions
1. Suppose that u, v V are in an inner product space, and
|u| = 3, |u + v| = 4, and |u v| = 6.
What number must |v| equal?
S OLUTION . The computation done in exercise 2 shows that
u+v
Hence 2 v
2
2
2
+ uv
= 42 + 62 2 32 = 3