Math 142-2, Final
Your name:
Student ID:
Instructions
This exam is closed book. No notes, books, electronic devices, or other resources are permitted on this exam.
Be sure to write your name and student ID number at the top of this page. Scratch paper wil
Math 142 Homework 0 Solution
32.11 The easier way: (slightly unrigorous, but its ne for this class)
Suppose the bank compounds the interest n times a year, and let t := 1/n. With the
additional deposit (or withdrawl) D (t), the balance at time t + t is gi
MATH-142 | FINAL EXAM | WINTER 2017 | NAME:
1
Figure 1.
Problem 1 (15pt). Suppose that a mass m were attached between two walls a distance d apart (see
Fig. 1). Let l1 and l2 be the unstretched lengths of the left and right springs, respectively, and let
Math 142-2, Homework 1
Solutions
April 7, 2014
Problem 34.4
Suppose the growth rate of a certain species is not constant, but depends in a known way on
the temperature of its environment. If the temperature is known as a function of time, derive
an expres
Homework Assignment, Math 142 - Spring, Week 1 - HW 1 Due
Friday 4/8/16 in class at 2:00/3:00 pm
Name:
UID:
Pages in total:
Date:
(Please print this page and staple it as a cover page to your homework paper)
Problem 1: - Modeling Process
Pick one of the f
Selected solution to HW3, Math 142 16S
Chuyuan Fu
Disclaimer: For reference only. There might be typos and mistakes. Please send any suggestions or corrections to
[email protected]
Problem 1
(a)
f (x) = tanh(x)
2ex 2
)
1 + e2x
f 00 (x) = 2 sech(x)(
Selected solution to HW1, Math 142 16S
Chuyuan Fu
Disclaimer: For reference only. There might be typos and mistakes. Please send any suggestions or corrections to
[email protected]
Problem 2. Haberman p161, 39.2
(a)
It suffices to show that
1+(
a b
Homework Assignment, Math 142, Week 5 - HW 5 - Due Monday
11/2/15 in class at 11:00am
Name:
UID:
Pages in total:
Date:
Problem 1: - Exponential growth - Discrete Models
Solve the following problems:
(a) Haberman: p.126, Problem 32.4.
(b) Haberman: p.127,
MA 142 Week 2 4/8/2016
Linear versus non-linear ODE
Equation for pendulum
2
= sin
2
2
=
2
Generic form*
Class
+ () = 0
Non-linear, homogeneous
+ = 0
linear, homogeneous
*a and b are const. coefficients
Non-dimensional ODE
Check
correct
variable
Math 142 Homework 1 Solution
32.3 (a) Use N (nt) to denote the population at time nt. Then for any n > 0, we
have
N (nt) = (1 + (b d)t)N (n 1)t) + 1000.
(b) Given that N (0) = N0 , we hope to solve the equation above and get the general
form for N (nt). T
Math 142 Homework 1 Solution
34.5 (a) If the growth rate is 0, the discrete model becomes
Nm+1 Nm = tfm ,
And the solution is
Nm = N0 + t(f0 + f1 + fm1 ),
which is analogous to integration, since let t = mt, then
t
t(f0 + f1 + fm1 )
f (s)ds
0
for t small
Math 142 Homework 2 Solution
34.16 We will rstly proceed similarly as 34.14(c). We assume the theoretical growth
has the form N0 eR0 t , and we hope to gure out which N0 and R0 t the data best. 34.14(b)
suggests that N0 should be equal to the initial popu
Math 142-2, Homework 2
Solutions
April 7, 2014
Problem 35.3
Consider a species in which both no individuals live to three years old and only one-year olds
reproduce.
(a) Show that b0 = 0, b2 = 0, d2 = 1 satisfy both conditions.
All two year olds die befor
Math 142-2, Quiz 5
Solutions
Problem 1
Consider an infinite number of cars, each designated by a number . Assume the car labeled starts
from x = ( > 0) with zero velocity, and also assume it has a constant acceleration . Determine the
velocity field u(x,
Math 142-2, Quiz 1
Solutions
Problem 1
A weight (of unknown mass) is placed on a vertical spring (of unknown spring constant) compressing it
by x. What is the natural frequency of oscillation of this spring-mass system?
Let m be the mass and k be the spri
Math 142-2, Quiz 6
Solutions
Problem 1
Construct a traffic following model with a density-velocity relationship u
() such that drivers try to
maintain a fixed following time T . (For example, if T = 2s, drivers always try to stay two seconds
behind the ca
Math 142-2, Project 1
Solutions
Introduction
This project looks at the equation for the undamped nonlinear pendulum
m + k sin = 0
(0) = 0
(0) = 0
using a mixture of analytic and numerical techniques.
Problem 1
Show that this problem can be reduced to the
Math 142-2, Quiz 4
Solutions
Problem 1
Assume that the forces acting on a mass are such that the
potential energy is the function of x shown in the figure.
Sketch the solutions in the phase plane.
x
1
Problem 2
r
A wheel with radius r spins freely and wit
MA 142 Week 2 4/8/2016
Method of Characteristics to solve linear
(quasi-linear) first order PDEs:
characteristics (e.g. contours)
vector
describing
characteristics
vector describing the orientation of
your PDE solution (u(x,t) = f(x-at)
Additional Refer
Math 142-2, Group work 1
Solutions
Problem 1
This group work activity is based around exploring the second order ordinary differential equation
tx + x + tx =
1
sin t.
4
Write this second order ODE as a first order ODE of the form z = f (z, t) by introduci
Math 142-2, Group work 2
Solutions
Problem 1
The Schrodinger equation is
ih
h2 2
=
+ U
t
2m
where m is its mass and i is the imaginary number. Deduce the units of the other variables (, h, U ).
Based on the units of U , what do you think it represents?
Math 142-2, Group work 3
Solutions
Problem 1
For each of the following, assume that u(t) and v(t) are vector valued functions of t. Assume f (t) is
a scalar function of t. The vectors r and s as well as the scalars a and b are constants. Simplify each
exp
Math 142-2, Group work 5
Solutions
Problem 1
Five energy levels for a system are shown in the phase plane below. (a) List the energy levels (red,
orange, green, blue, violet) in order from lowest energy to highest energy. (b) Mark all stable () and
unstab
MATH-142 | MIDTERM | WINTER 2017 | NAME:
Instructions:
You may use your personal notes (lecture notes, homeworks, etc). Digital devices are
not allowed. The textbook is not allowed.
Show all necessary steps. Answers given without supporting work may rec
Math 462: Mathematical Modeling
Instructor: Dr. Patrick Nelson
Homework 1: Due Monday 2/11
1: Classify the fixed points of the following systems of equations and state the type of
stability for each one.
1a:
dx
= 2x 3y
dt
dy
= y 2x
dt
1b:
dx
= y 2 3x + 2
Math 462 Homework 5
1: Verify the following system has a non-hyperbolic fixed point at the origin
for some value of . Sketch three phase portraits with qualitatively different
behaviors. x 1 = x1 x2 ,x 2 = x1 + x2
2
+ x = A sin t,
2: Consider the forced V
Math 462: Mathematical Modeling
Instructor: Dr. Patrick Nelson
Homework 1: Due Friday 2/1
1: All problems are from the course pack by Meerschaert
Section 1.4: 1, 2
Section 2.4: 1, 2, 3