MA129 Mock Final
Name:
Time Allowed: 120 minutes
Total Value: 85 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 sheet
MA129 Week 12 - Lagrange Multipliers; Antiderivatives
1.
Lagrange Multipliers (9.4)
Method For Finding Relative Extrema Subject To A Constraint
Consider the functions f (x, y) and g(x, y). To nd any relative extrema of f (x, y) subject to the constraint
g
MA129 Week 13 - Substitution; Denite Integrals
1.
The Substitution Rule (7.2)
Used to express the integrand in a form where the general antiderivative is known.
substitution, say u = g(x), where both g(x) and g (x) are factors in the integrand.
For exampl
MA129 Week 9 - Second Derivatives; Curve Sketching
1.
Second Derivatives and Graphs
(a) Concavity (5.3):
Suppose f is dierentiable for x (a, b) .
- If f (x) > 0 for all x (a, b), the graph of f is concave up on (a, b) (i.e. the curve looks like a smile!).
MA129 Week 10 - Absolute Extrema
1.
Absolute Extrema (6.1)
(a) Absolute Max/Min:
Suppose f is dened on some interval I.
- If f (x) f (c) for all x I then f attains an absolute maximum at (c, f (c).
- If f (x) f (c) for all x I then f attains an absolute m
MA129 Week 4 - Exponential and Logarithmic Functions
1.
Exponential Functions
Function of the form f (x) = ax = expa (x), where a > 0, a = 1 is constant.
Note the dierence between f (x) = xn , n R, which is a power function and f (x) = ax , a > 0 which
MA129 Week 8 - Dierentiation; Graphs and Derivatives
1.
Rules of Dierentiation (4.1,4.2)
Constant Function Rule:
d
[k] = 0, k R
dx
Constant Multiple Rule:
d
[k f (x)] = k f (x), k R
dx
Sum/Dierence Rule:
Power Rule:
Product Rule:
Quotient Rule:
2.
d
= [f
MA129 Week 2 - Pre-Calculus Topics
1.
Interval Notation
When interval notation is used, we assume that the real numbers are being considered. A square [closed]
bracket indicates that the endpoint value is included whereas a round (open) bracket means that
MA129 Week 6 - Applications of Limits
Limits
1.
Denition
To determine lim f (x) we are asking the question: as x gets closer and closer to the number a, which
xa
single value, if any, does f (x) approach?.
Notes:
(a) If f (x) does not approach a unique fi
MA129 Week 3 - Functions
1.
Interval Notation
When interval notation is used, we assume that the real numbers are being considered. A square [closed]
bracket indicates that the endpoint value is included whereas a round (open) bracket means that the endpo
MA129 Week 11 - Multivariable Functions
1.
Multivariable Functions (9.1)
A function,say z, may be dened in terms of 2 (z = f (x, y) or more (z = f (x1 , ., xn ) variables. Such a
function assigns to each ordered pair (for a function of 2 variables), or n-
MA129 Week 5 - Linear Systems of Equations and Matrices
1.
Matrices
A matrix is a rectangular array of numbers where the numbers individually are called entries. If Amn is a
matrix, then it has m rows, n columns, and m n entries.
2 4
Example: A = -1 6 is