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 tex
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 value
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
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 :
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 :
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., Stev
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 :
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 :
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 2
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
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
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
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 impr
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
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
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
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 err
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
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
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
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
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 ent
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
F
Denitions
Matrix algebra
M ATLAB specics
Matrix Algebra
MATH 2070U Numerical Methods
adapted from notes by D. Aruliah
Matrix Algebra
MATH 2070U
1 / 25
Denitions
Matrix algebra
M ATLAB specics
Matrix A
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
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
lo
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