Page 6 of 6
MA104 Midterm Test
Antiderivatives
f (u) du denotes the general
antiderivative of f (u).
If
f (u) du = F (u) + c
then
dF (u)
= f (u).
du
un du =
un+1
+ c, n = 1
n+1
1
du = ln |u| + c
u
eu du = eu + c
au du =
au
+ c, 1 = a > 0
ln a
Trigonometri
MA104 Mock Midterm
Name:
Time Allowed: 80 minutes
Total Value: 85 marks
Number of Pages: 7
Instructions:
Non-programmable, non-graphing calculators are permitted. No other aids allowed.
Check that your test paper has no missing, blank, or illegible pages.
MA104 Mock Final Exam
Name:
Time Allowed: 120 minutes
Total Value: 110 marks
Number of Pages: 9
Instructions:
Cheat Sheet:
One 8:5" 11" page of study notes (both sides) is allowed as a reference
while completing the mock test. Please note, that the cheat
AMATH 251
!
First-order ODEs
Existence-Uniqueness Theorem
THEOREM. (PICARDS THEOREM): Consider the rst-order initial
value problem
y 0 = f (x, y)
y(x0 ) = x0 .
If f and
@f
@y
are continuous functions in an open rectangle
n
o
R = (x, y) : a < x < b; c < y
AMATH 251
!
Exact First-order ODEs
!
There is no general method for solving
nonlinear DEs. There are classes of
nonlinear DEs that can be solved with
various techniques:
separable DEs
exact DEs
inexact DEs if you can make them exact
some equations can
AMATH 251
!
Introduction to
Dierential Equations
(Advanced Level)
Introduction
Basic Classication I
Basic Classication II
Examples
Examples
linear; 1st order
nonlinear; 2nd order
linear; 2nd order
nonlinear if n != 0,1; 1st order
nonlinear; 4th order
Part
AMATH 251
Mathematical Modelling
Mathematical Modelling requires:
1) A problem statement requiring a detailed,
thorough description of the problem;
2) Use of relevant guiding principles to relate
various quantities resulting in equations that need
to be s
AMATH 251
!
The Brachistochrone
!
Consider a bead sliding along a frictionless
wire under the inuence of gravity. Along
what curve is the travel time between two
xed points A and B minimized?
g
!
This problem was posed in 1696 by Johann
Bernoulli. It was
October 19, 2012
MA104 Midterm Test
Page 1 of 2
[10 marks]
1. Evaluate
sin5 x cos3 xdx.
[10 marks]
2. Evaluate
1sin x
1+sin x dx.
[10 marks]
3. Use a suitable trigonometric substitution to evaluate
[10 marks]
4. Find an explicit expression of y = f (x) wh
February 10, 2012
MA104 Midterm Test
Page 1 of 2
1. Evaluate each of the following integrals. Show all of your work.
[4 marks]
(a)
[2 marks]
(b)
[3 marks]
(c)
[3 marks]
(d)
[2 marks]
(e)
/2
cos3 (x) sin2 (x)dx
0
/4
tan2(t)dt
0
x dx
x2 4
/4
/4 |1 sec(x)|d
Winter 2012
MA104 - Final Examination
Page 1 of 2
1. Evaluate each of the following integrals, using the method of improper Riemann integration.
Proper form must be used.
[5 marks]
(a)
[4 marks]
(b)
ye2y dy
0
1
1
dx
0
1x2
1
1
tan(x3 )dx = 0.
[2 marks]
2.
Fall 2012
MA104 - Final Examination
[5 marks]
Page 1 of 2
tan5 xdx.
1. Evaluate
2. A disk D of radius 1 has centre at the point (2, 0) in the xy-plane.
[2 marks]
(a) Sketch D and write an equation for the boundary circle of D.
[5 marks]
(b) The disk D is
AMATH 251
Non-dimensionalization
and
Dimensional Analysis
Non-dimensionalization
Goal: Introduce scaled nondimensional variables to
reduce the number of parameters in a problem and
hence to simplify it.
Dimensional analysis
Goal: gain insight into how va