MATH 2070: Sample nal problems
Below are some sample problems (from the 2nd half of the course) to guide your revision for the nal
exam. In addition to the problems in the relevant sections of the text book, you can use these questions
as preparation for
Chapter 4: Errors
Accuracy vs. Precision
Accuracy refers to how closely a computed or measured value agrees
with the true value.
Precision refers to how closely individual computed or measured values
agree with each other
Error Denitions
If x approximat
Chapter 17: Polynomial Interpolation
Example: Population data of the US from 1940 to 1990.
Year
1940
1950
1960
1970
1980
1990
Population (thous) 132 165 151 326 179 323 203 302 226 542 249 633
What if we need to know the population in 1965, or 1935?
Can u
1A
Make sure that this examination has 10 pages including this cover
The University of Ontario Institute of Technology
March 3, 2014
Mathematics 2070U
Numerical Methods
Time: 75 mins
Student Number :
Name :
Signature :
Tutorial Instructor :
Tutorial Day /
1B
Make sure that this examination has 10 pages including this cover
The University of Ontario Institute of Technology
March 3, 2014
Mathematics 2070U
Numerical Methods
Time: 75 mins
Student Number :
Name :
Signature :
Tutorial Instructor :
Tutorial Day /
MATH 2070U:
Numerical Methods
Instructor: Greg Lewis
Oce: UA4033
Phone: ext. 2608
Email: Use Blackboard Messages
Text:
Applied Numerical Methods with Matlab for Engineers and Scientists, 3rd Ed., Steven C. Chapra
Course Description:
There are two dominant
2B
Make sure that this examination has 10 pages including this cover
The University of Ontario Institute of Technology
March 3, 2014
Mathematics 2070U
Numerical Methods
Time: 75 mins
Student Number :
Name :
Signature :
Tutorial Instructor :
Tutorial Day /
2A
Make sure that this examination has 10 pages including this cover
The University of Ontario Institute of Technology
March 3, 2014
Mathematics 2070U
Numerical Methods
Time: 75 mins
Student Number :
Name :
Signature :
Tutorial Instructor :
Tutorial Day /
Denite integrals
Trapezoidal rule
Simpsons rules
Quadrature of data
Adaptive quadrature
Numerical Integration
MATH 2070U Numerical Methods
adapted from notes by D. Aruliah
Numerical Integration
MATH 2070U
1 / 51
Denite integrals
Trapezoidal rule
Simpsons
FACULTY OF SCIENCE
Science Final Examination View Request
This form is to be used to request to view a final examination in a Science course for the purpose of calculating
your final numeric grade, to gather information on where you may have had difficult
Review: The Initial Value Problem (IVP)
Find the function y(t) that satises the ordinary dierential equation (ODE)
y (t) = f (t, y(t),
for a t b
subject to an initial condition
y(a) =
Examples:
Many real world applications can be modelled very eectively
MATH 2070: Sample midterm problems
Below are some sample problems to guide your revision for the midterm test on March 3. In addition to the
problems in the relevant sections of the text book, you can use these questions as preparation for the test.
Be aw
Runge-Kutta Methods
Were looking at the approximation of initial-value problems:
y = f (t, y(t),
y(a) = ,
atb
In previous section, we used a rst-order Taylor expansion to derive Eulers
method.
To improve accuracy, maybe we can use higher-order Taylor poly
Formula Sheet for MATH2070 Final
Error denitions: If x approximates a number x (i.e. the true value), then the true error is Et = x x, the absolute error is
|x x|
|Et | = |x x|, and the relative error is t =
, provided x = 0
|x|
Mean Value Theorem. If f C
Methods for Systems of ODEs
We were looking at the approximation of initial-value problems:
y = f (t, y(t),
y(a) = ,
atb
where there was a single unknown function y(t), and only the rst derivative
of y(t) entered the equations, but . . .
Few physical mod
Direct Methods for Systems of Linear Equations
Formal solution of Ax = b
Formal solution of Ax = b where A is n n is
x = A1 b
If A1 exists, A is nonsingular
If A1 does not exists, A is singular
In practise: Never compute the solution of Ax = b by dire
Midterm Formula Sheet for MATH2070
Error denitions: If x approximates a number x (i.e. the true value), then the true error is Et = x x, the absolute
|x x|
error is |Et | = |x x|, and the relative error is t =
, provided x = 0
|x|
Mean Value Theorem. If f
MATH 2070U
Midterm examination
Page 3 of 8
1. Assume the variables A, B, C, and D have been defined in a M ATLAB session as follows.
A = [-1,4;-3,4]
C = [1,3;0,-5;-4,3]
B = [-2,0;2,2]
D = [-1,-3,-4;-1,-3,3]
With these values, what results do the following
MATH 2070U
Midterm examination
Page 3 of 8
1. (a) Assume the variables A, B, C, and b have been defined in a Matlab session as follows.
A = [ -1, 3 ; 5, 4 ];
C = [ 5, -2 ; -3, -4 ; 7, -3 ];
B = [ 6, 1 ; -2, -4 ];
D = [ -5, -3, -1 ; -1, 4, 5 ];
With these
MATH 2070U
Midterm examination
Page 3 of 8
1. (a) Assume the variables A, B, C, and b have been defined in a Matlab session as follows.
A = [ -1, 3 ; 5, 4 ];
C = [ 5, -2 ; -3, -4 ; 7, -3 ];
B = [ 6, 1 ; -2, -4 ];
D = [ -5, -3, -1 ; -1, 4, 5 ];
With these
MATH 2070U
Midterm examination
Page 3 of 8
1. (a) Assume the variables A, B, C, and b have been defined in a Matlab session as follows.
A = [ -1, 3 ; 5, 4 ];
C = [ 5, -2 ; -3, -4 ; 7, -3 ];
B = [ 6, 1 ; -2, -4 ];
D = [ -5, -3, -1 ; -1, 4, 5 ];
With these
Exam solutions (Yellow cover sheet)
MATH 2070U
Midterm examination
Page 3 of 7
1. [8 marks] Write in the space provided the output you would expect in an interactive M ATLAB session
as a result of entering the given statements.
For numerical values retur
Input & output
Working with arrays
Functions
Plotting
M ATLAB Tricks
MATH 2070U Numerical Methods
adapted from notes by D. Aruliah
M ATLAB Tricks
MATH 2070U
1 / 36
Input & output
Working with arrays
Functions
Plotting
M ATLAB Tricks
1
Input and output in
Scripts
M ATLAB functions
Control ow
Programming in M ATLAB
MATH 2070U Numerical Methods
adapted from notes by D. Aruliah
Programming in M ATLAB
MATH 2070U
1 / 38
Scripts
M ATLAB functions
Control ow
Programming in M ATLAB
1
M ATLAB scripts
The M ATLAB wo
BVPs
1D Poisson
2D Poisson
Boundary-Value Problems
MATH 2070U Numerical Methods
adapted from notes by D. Aruliah
Boundary-Value Problems
MATH 2070U
1 / 31
BVPs
1D Poisson
2D Poisson
Key questions
What is a boundary-value problem (BVP)?
What is meant by a
InteractiveM ATLAB
Numeric data
logical operators
Other data types
Basic M ATLAB
MATH 2070U Numerical Methods
adapted from D. Aruliah
Basic M ATLAB
MATH 2070U
1 / 44
InteractiveM ATLAB
Numeric data
logical operators
Other data types
Basic M ATLAB
1
Intera
Chapter 5: Roots
In this chapter were interested in looking for solutions of equations that look
like
f (x) = 0
where x is real.
The solutions to such equations are called roots of f (x) = 0. They are also
called zeros of f (x).
What does it mean if x is