Chapter 4 The Derivative
4.1 Limits
Example What happens to f (x) = x2 when x becomes very close to 2 (not equal to 2)? Graph:
6

Limit from left is limx2 f (x) = 4
Limit from right is limx2+ f (x) = 4
Then limx2 f (x) = 4
Denition If, by choosing x clos
Chapter 15 Complex Numbers
Complex numbers are an extension of real numbers.
They are important in many areas and applications,
including electrical circuits, electrostatics, magnetism,
gravitation, wave propogation, uid dynamics and
aerodynamics.
15.1 Th
Chapter 2 Linear Functions
2.1 Slopes and Equations of Lines
Graph:
Slope is, taking any points (:61, yl) and (132, yg) on the
line7
m: Changeiny 2%: ygyl
Change in :6 A56 5132 5131
Note slope can be 0 (when Ag 2 0) , i.e. horizontal
line, or slope can
Chapter 3 Nonlinear Functions
3.1 Properties of Functions
Denition A function f is a rule that assigns to
each element from one set X exactly one element from
another set Y. Picture:
Using variable :1: (independent variable) to represent
elements of X a
Chapter 4 The Derivative
4.1 Limits
Example What happens to f (at) 2 3:2 when x be
comes very close to 2 (not equal to 2)? Graph:
Limit from left is limgHg x) 2 4
Limit from right is limmx2+ f (513 = 4
Then limxxg f (m) = 4
Denition If, by choosing :1
Chapter 8 Integration
8.1 Antiderivatives
If F (m) = f (:13) then F (x) is an antiderivative of f l
i.e. reverse of differentiation
Example Find an antiderivative of the following.
<a> f<cc> = 5
(b) f<x> = 2x
If F (:13) and G(a:) are both antiderivatives
Chapter 6 Graphs and the Derivative
6.1 Increasing and Decreasing Functions
Denition A function f is increasing on an interval
if
an) < f(:132) Whenever 61 < :62
f is decreasing on an interval if
f(;z:1) > f(:z:2) Whenever 331 < x2
Example Where is the gr
Chapter 15 Complex Numbers
Complex numbers are an extension of real numbers.
They are important in many areas and applications,
including electrical circuits, electrostatics, magnetism,
gravitation, wave propogation, uid dynamics and
aerodynamics.
15.1 Th
Chapter 3 Nonlinear Functions
3.1 Properties of Functions
Definition A function f is a rule that assigns to
each element from one set X exactly one element from
another set Y . Picture:
Using variable x (independent variable) to represent
elements of X an
Chapter 9 Further Techniques and
Applications of Integration
9.1 Integration by Parts
d(uv)
From product rule d1; = togZ + 03: integrate both
sides to get
d
m; = fudv das+/ vu d9: (write) 2 fudv+fvdu
d9: dCE
So
fudv=uv/vdu
Need to differentiate one funct
Revision
Example Find 3: for the following.
1. y I V 2 + 3331113: f ' cfw_L $14
sin(3a:+2e5m)
4x+1
2. y: Example Find the following integrals.
1. f(2:c1 3)(a:2 +1)da:
\
z
3
e
r J i W J W
w x V W C 7? w M 1 mi; (law/:9:
1
J
RM 4) w
M J, . / a, J
Chapter 7 Applications of the Derivative
7.1 Absolute Extrema
Let f be dened on an interval and C E interval.
0 f (c) is the absolute maximum of f on the
interval if f (as) S f (c) for every :6 in the interval.
0 f (c) is the'absolute minimum of f on the
Chapter 7a Exponential and Logarithmic
Functions
3.4 Exponential Functions
An exponential function with base a is dened as
f<$> 2 am, Where a > O and a #1.
Exponential growth When a > 1
Exponential decay when 0 < a < 1
Important in applications.
Example G
Chapter 1 Algebra Reference
1.1 Polynomials
Real numbers
Order of operations: B (brackets), I (indices),
MD (multiplication / division) , AS (addition / subtraction)
Properties
1a+b=b+a
ab 2 ba (commutative properties)
2. (a+b)+c=a+(b+c)
(ab)c = a(bc) (as
Chapter 9 Further Techniques and
Applications of Integration
9.1 Integration by Parts
dv
du
From product rule d(uv)
=
u
+
v
dx
dx
dx integrate both
sides to get
dv
du
uv = u dx+ v dx (write) = u dv+ v du
dx
dx
So
u dv = uv v du
Need to dierentiate one fun
Chapter 11 Dierential Equations
A dierential equation is an equation involving
an unknown function y(x) and a nite number of its
derivatives.
Very important in many discipline areas.
11.1 Solutions of Elementary and Separable
Dierential Equations
dy
Simpl
Chapter 7 Applications of the Derivative
7.1 Absolute Extrema
Let f be dened on an interval and c interval.
f (c) is the absolute maximum of f on the
interval if f (x) f (c) for every x in the interval.
f (c) is the absolute minimum of f on the
interval
Chapter 7a Exponential and Logarithmic
Functions
3.4 Exponential Functions
An exponential function with base a is dened as
f (x) = ax, where a > 0 and a = 1.
Exponential growth when a > 1
Exponential decay when 0 < a < 1
Important in applications.
Example
Chapter 6 Graphs and the Derivative
6.1 Increasing and Decreasing Functions
Denition A function f is increasing on an interval
if
f (x1) < f (x2) whenever x1 < x2
f is decreasing on an interval if
f (x1) > f (x2) whenever x1 < x2
Example Where is the grap
Chapter 8 Integration
8.1 Antiderivatives
If F (x) = f (x) then F (x) is an antiderivative of f
i.e. reverse of dierentiation
Example Find an antiderivative of the following.
(a) f (x) = 5
(b) f (x) = 2x
If F (x) and G(x) are both antiderivatives of f (x)
Chapter 1 Algebra Reference
1.1 Polynomials
Real numbers
Order of operations: B (brackets), I (indices),
MD (multiplication/division), AS (addition/subtraction)
Properties
1. a + b = b + a
ab = ba
(commutative properties)
2. (a + b) + c = a + (b + c)
(ab)
THINGS You NEED To KEViQE
As 03 W r2; Malta 6 Mg 132
W mg: M 6:19:15 mfg/Was (See, mmqufm
u'a "Wu. Unit WWI/7. I/m gfdug m
8m 0?an 47060,; cfw_SW9 Wm. mot +2 ram/92,
hm gm Aguzz,
1. M W amt W who; Km.
cram 06 OfW.
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5. 9M
Chapter 5 Calculating the Derivative
5.1 Techniques for Finding Derivatives
Notation for deriVative: ,
m I 39 iwn D W]
 dzc da: 3"
Constant Rule If f (51:) = k for any real number k,
then f (:13) = 0 7
Power Rule If f (:13) = :13 for any real number n,
t