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)(
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
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 - 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
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 142-2, 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
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 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 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 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 142-2, 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, Group work 8
Solutions
Problem 1
For each row of the table, explain how you can obtain the wanted item from the quantities that are
given to you. If you require initial condition or boundary conditions, state what you need. For example,
for (d
Math 142-2, Quiz 2
Solutions
Problem 1
Consider a mass m located at x = xi + yj, where x and y are unknown functions of time. The mass is
free to move in the x y plane without gravity (i.e., it is not connected to the origin via the shaft of
a pendulum) a
Math 142-2, Quiz 3
Solutions
Problem 1
Consider a spring-mass system with a nonlinear restoring force satisfying
m
d2 x
= kx x3 ,
dt2
where > 0 and k > 0. Which positions are equilibrium positions? Are they stable?
f (x) = kx + x3 . The equilibria occur w
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
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, 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
Jeffrey Wong
Math 142 (Partial) Solutions
Homework 4, Fall 2015
Note: Ive included computer-generated cobweb diagrams even though sketches should be by hand.
Problem 1
The cobweb diagram (starting at a0 = 0.8 and a0 = 3.2) looks like:
2.4
2.2
2
1.8
1.6
1.
Math 142-2, Quiz 7
Solutions
Problem 1
Assuming nearly uniform, but heavy traffic, show that in general it is impossible to prescribe the traffic
flow at the entrance to a semi-infinite highway. (An intuitive explanation for why this is so is sufficient;
Math 142-2, Quiz 8
Solutions
Problem 1
If u = umax (1 /max ), then what is the velocity of a traffic shock separating densities 0 and 1 ?
(Simplify the expression as much as possible.) Show that the shock velocity is the average of the density
wave veloci
Math 142-2, Group work 9
Solutions
Problem 1
A long road has an initial uniform traffic density (x, 0) = max
3 . At t = 0, a traffic accident occurs at
3
umax max . Determine the traffic
x = 0, which effectively limits the flowrate pastx = 0 to q(0, t) =
Math 142-2, Group work 10
Solutions
Problem 1
Solve the PDE for f (x, t) given the initial conditions f (x, 0) = s(x)
x f
f
(x, t) +
(x, t) = f (x, t) + t + x
t
t + 1 x
Use the method of characteristics.
f
f
d
f (z(t), t) =
(z(t), t) + z (t) (z(t), t)
dt
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 Mathematical Modelling, Winter 2017
Traffic Flow Example
Question: Consider traffic moving along a single road that changes surface at x = 0.
For x < 0, the surface is a good paved road, but for x > 0 the road is unsealed gravel, and
the maximum
MATH 142: Mathematical Modelling Winter 2017
Instructor David Arnold
Email [email protected]
Office MS 7360
Office hours TBA
TA Adam Haque
Email [email protected]
Office hours TBA
Lectures MWF 1212:50pm MS 5147
Discussion Tuesday 1212:50pm MS
MATH 142 Mathematical Modelling, Winter 2017, Homework 5
Due: Friday, February 24, 12:00pm
Two marks awarded for overall clarity and detail of explanations and presentation.
Question 1: (Haberman 10.10) Consider an object, initially at x = 0, a distance H
MATH 142 Mathematical Modelling, Winter 2017, Homework 4
Due: Friday, February 17, 12:00pm
2 marks awarded for explanation and presentation.
Question 1: (4 marks) An oscillator with weak damping can be modelled by
y (t) + y (t) + y(t) = 0,
where t > 0 and
MATH 142 Mathematical Modelling, Winter 2017
Midterm Exam 1 Practice Problems
1: Consider a quadratic equation,
at2 + bt + c = 0
where t represents time, and a, b and c are dimensional constants ([a] = LT 2 , [b] = LT 1 ,
[c] = L). Use dimensional analysi