Review sheet for the midterm and HW 5
Homework 5: Questions 6 to 9 are to be handed in on Thursday, 4 November.
However please note that the material in those questions is part of the midterm.
Material covered on midterm:
1st order ODEs and applications:
Math 2400 Final exam, January 2011
Please write all your answers in the booklet provided. You have 3 hours. No calculators are allowed.
1. Set up the Newtons method to determine 31/3 . Starting with an initial guess of x0 = 1, compute
x1 .
2.
(a) Determin
Assmmwi Ll Scmaw
2. The two characteristic polynomials are
4. The two characteristic polynomials are
14 2 0
arm: 0 lA 1
D 1 1/\
lA 1 0 1
=1- e.
( Alihl lAi 2h 14
=(1A)((1A)21):(1A)(A22A)=A3+3A2~2A
- 2A 1 1
030): 0 lA 0
2 0 1,\
_ _ 2). 1
(1
MATH 2040 COURSE NOTES
ROB NOBLE
The following course notes are based closely on [Poole2006].
4. Eigenvalues and Eigenvectors
4.1. Introduction to Eigenvalues and Eigenvectors.
Definition 1 (Eigenvalues and Eigenvectors). A scalar
A2R
nn
2 R is called an
Dalhousie University/Winter 2017 - MATH 2040
1. Let S = cfw_ [
Assignment 2 - Solution
1 2 1 1 2 4
],[
],[
] be a set of vectors in M2,2 .
2 0
0 0 2 0
(a) Find the coordinate vector of each element of S with respect to the basis
B = cfw_Eij 1 i 2 , 1 j 2
Dalhousie University/Winter 2017 - MATH 2040
Assignment 2
Solve the following problems and hand in your solutions to be marked.
Write up your solutions in the order the problems are given and put a box around the final
answer for each question.
Put a cove
Dalhousie University/Winter 2017 - MATH 2040 Assignment 1 Part B
1. Show that
,_ 0 a . _
Vizcfw_[17 0].a,beR and n+260
is a subspace of Mg; and nd a basis for W.
we, ma +0 aux w A doSecl wser claws and SCbJar mm.
6) lei M7006 W ) M: [1 :l ) aybef J (14133
Dalhousie University/Winter 2017 - MATH 2040
Assignment 1
Part A: Practice problems - Do not hand in.
Solve the following problems on your own and check your answers against the posted solutions which will be available Jan. 26.
1. Let
W = cfw_ax3 + b sin(
Supplemental questions for midterm
1.
(a) Can the bisection method be used to nd the zero of the polynomial y = (x 2)2 ?
(b) Set up Newtons method to determine the point x and the number m for which the line y = mx
intersects the curve y = cos(x) tangenti
MATH 2400 Midterm
No calculators or other aids. Write all answers in the booklet provided.
1. Set up Newtons method to determine 3 2. Starting with x0 = 1, compute x1 .
2. The function f (x) is tabulated below.
x
012
f (x) 1 1 0
(a) Suppose that f (x) is
Review sheet.
Topics covered:
Nonlinear ODEs:
Solving: separable, linear (nonhomogeneous), exact, homogeneous ODE
Modelling: gravity, velocity, acceleration; Newtons law of cooling; Toriccellis law;
mixing problems.
Reducing 2nd order ODEs of the type
MATH 2400, Homework 1
Due date: 20 September (Tuesday)
1. Install either Maple, Octave, matlab on your computer.
2. Using a computer program of your choice, plot the functions y = x, y = x2 and y = x on the
same graph, with x [0, 2] and with y [0, 2]. Han
MATH 2400, Homework 2
Due date: 28 September (Tuesday)
1. You are asked to nd the root of
x3 = x2 + x + 1
(a) On the same graph, plot the two sides of this equation nd bounds for the root. Hand in the
resulting graph.
(b) Use Bisection method to nd the ro
MATH 2400, Homework 3
Due: Monday, 10 October
1. Given the data points (x, y ) = (0, 1) , (2, 0), (4, 2), write down the interpolating polynomial through
these points using
(a) Lagranges polynomial
(b) Newtons divided dierences polynomial.
(c) Using eithe
MATH 2400 HW 4
1
0
1. Consider I =
exp x2 dx = 0.7468241328124270.
(a) Estimate I using the Trapezoid rule with n = 4 subintervals.
(b) Estimate I using the Simpsons rule with n = 4 subintervals.
(c) How many subintervals n are needed to estimate I with T
MATH 2400 HW 5
Due date: 11 Nov (Fri)
1. Richardson extrapolation can be used with with ODEs. Here we demonstrate how.
(a) Let A(h) be the solution to y (t) = y (t) with y (0) = 1 at t = 1 using the forward Euler with
stepsize h; so that y (1) = A(0). In
MATH 2400 HW 6
Due date: 7 Dec (wed)
1. Use the multi-variable Newtons method to nd at least one root of the following system of equations:
x2 + y 2 = 1; x = sin(y ).
Note: Please use the multi-variable Newtons method, not simply solving sin2 y + y 2 = 1.
Homework 7
Due 25 November (Thurs)
1. Find the general solution (in terms of real functions) to the system
x = Ax
x
y
where x =
and A is one of the following matrices.
(a) A =
1 4
0 3
(b) A =
1 4
1 3
(c) A =
1 4
5 3
2. Solve the system 1.c with initial co
Math 2120 Homework 8
Due: 2 Dec
1. Find etA where A is one of the following.
(a) A =
1 4
0 3
(b) A =
1 4
1 3
(c) A =
1 1
11
1 0
0
0
1 1 0
0
(HINT: write A = N I .)
(d) A =
1
0 1 0
0
1
1 1
2. (a) Solve the initial value problem
d
x=
dt
x+
et cos t
et s
Math 2120 Homework 9
Not to be handed in. However please note: this material is part of the course and
will be on the nal exam.
1. For each of the matrix A below, sketch the phase portrait of the linear system x = Ax.
(a) A =
1 4
0 3
(b) A =
1 4
1 3
2. Co
6. Let V be a real vector space. Prove that for any vector v V and any scalar r R, the
following hold:
(a) 0v = 0
Using Property 8 of vector spaces: (c + d)v = cv + dv
Add 0v to both sides:
(0 + 0)v = 0v + 0v
0v = 0v + 0v
0v + 0v = 0v + 0v + 0v
Using Prop