Factoring and Binomial Formulae to remember:
2
( x + a ) = x 2 + 2ax + a 2
2
( x a ) = x 2 2ax + a 2
3
( x + a ) = x3 + 3ax 2 + 3a 2 x + a 3
3
( x a ) = x3 3ax 2 + 3a 2 x a3
x2 a2 = ( x + a ) ( x a )
x3 a 3 = ( x a ) ( x 2 + ax + a 2 )
x3 + a 3 = ( x + a
Chapter 16
Aqueous Ionic Equilibria
Sections 16.2-16.6, 16.8
Objectives
Describe buer capacity and buer range
Use the Henderson-Hasselbalch equation with buer solutions
Calculate the pH at several points during a pH titration
Write the equilibrium exp
SOLUTIONS TO MATHEMATICS 113/114
MIDTERM I
Instructions:
Answer all questions. No calculators allowed!
Value
1.
(8)
3
.
x
Use the definition of the derivative to calculate f(x).
Let f ( x ) = 2 x +
(a)
1
f '( x ) = lim ( f ( x + h) f ( x)
h 0 h
1
3
3
= li
MATHEMATICS 113/114
MIDTERM I (Amended)
February 8, 2002
Instructions:
Answer all questions. No calculators allowed!
Value
(8)
(a)
3
.
x
Use the definition of the derivative to calculate f(x).
(8)
(b)
Find the equation of the tangent line to the graph of
Math 0099 Practice Final Questions
1.(a) Graph the polynomial P ( x ) = x 4 5 x 2 + 4 . Show clearly all intercepts.
(b)
Find the amplitude, period, and phase shift of the graph of the function
U[_
. and use this information to sketch one full period of t
M8H Homework 12 Solutions
tan 2 u sin 2 u = tan 2 u sin 2 u
sin 2 u sin 2 u
=
cos 2 u
1
sin 2 u cos 2 u sin 2 u
69.
=
cos 2 u
cos 2 u
sin 2 1 cos 2 u
=
cos 2 u
sin 4 u
sin 2 u
= sin 2 u
= sin 2 u tan 2 u
2
2
cos
cos u
(
)
Sussman
t
Math 0099 (160) Winter 2007
First Midterm Exam
1.
(a)
Evaluate the following:
3 1 8
2
3
(b)
8
2.
Simplify the following expressions as much as possible
(a)
xy 2 z 3
3 4 (your final answer should not include any negative exponents)
2x y
(b)
1
x
1
+
x
Grant MacEwan University
Math 099 : Precalculus
Lecture Notes
Instructor: Adrian Biglands
1
2
1
Fundamental Notions in Mathematics
1.1
The Real Numbers
We recall that the set of all real numbers, which we denote by R, is made up of several dierent
types o