MATHS 132 CLASS NOTES 9.1
Part one
Limits
The notion of limit is a central theme in calculus.
Lets consider the function
f ( x) x 2 2 x 5
for the values of x near 1.
The following table gives values of f (x ) for values of x close to 1, but not equal to 1

MATHS 132 CLASS NOTES 9.1
LIMIT PROPERTIES
Limit Laws
Given: lim f ( x) L and lim g ( x) K exist. Then
x a
x a
1.
lim[ f ( x) g ( x)] lim f ( x) lim g ( x)
2.
lim[ f ( x) g ( x)] lim f ( x) lim g ( x)
3.
lim[cf ( x )] c lim f ( x ) (scalar multiple)
4.
li

MATHS 132 CLASS NOTES 9.2
Continuous functions; limits at infinity
Definition of continuity
Let c be a number in the interval (a,b), and let f be a function whose domain contains the
interval (a,b). The function f is continuous at the point c if the follo

Maths 132 Class notes 9.3
The Derivative
In this section, we will address the two basic problems of calculus:
1. Find the equation of the tangent line to a curve.
2. Find the instantaneous velocity of a falling object.
3.
Difference quotient can be repres

Maths 132 Class notes 10.5 Rational functions: Curve Sketching Techniques
Limits at Infinity for Power Functions:
k
1. lim p 0
x x
p
2. lim kx
x
Limits at Infinity and Horizontal Asymptotes for Rational Functions
am x m am 1 x m 1 . a1 x a0
f ( x)
bn x

Maths 132 Class notes 9.4
Derivatives of Constants, Power Forms, and Sums
Derivative notation:
y
dy
dx
f ( x) all represent the derivative of f at x .
Shortcut rules for differentiation:
1. The derivative of a constant is zero.
Example: y 6 y 0
f ( x) x n

Maths 132 CLASS NOTES 13.2
Definite Integral
FUNDAMENTAL THEOREM OF CALCULUS
The Fundamental Theorem of Calculus is named because it demonstrates a connection
between differential calculus (finding the tangent to a curve) and integral calculus
(finding th

Maths 132 Class notes 9.5
Derivatives of Products and Quotients
The derivative of the product of two functions is the first function times the derivative of
the second function plus the second function times the derivative of the first function.
f ( x) u

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Maths 132 quiz 10 pts.
Whittern
Find the derivative of the following functions by using the definition.
1a.
f ( x ) 4 x 2 7 x 5
b. find
f (1) ?
a._
2. a)
f ( x) 3x 2 2 x 4
b) find the equation of the tangent line to the function at x

Maths 132 CLASS NOTES-10.2
Concavity and the second-derivative test
First Derivative Test
Suppose that c is a critical number of a continuous function f .
a) If f changes from positive to negative at c, then f has a local maximum at c.
b) If f changes fro

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1.
f ( x)
Maths 132 9.4 quiz 10 pts.
Whittern
Find the equation of the line tangent to the graph of
at x = -1.
7 2
x 3x 1
2
1._
2. Find the derivative:
f ( x)
3
3 x 45
2x2
2._
3. Find the coordinates of point where the graph of f(x

Maths 132 Class notes 9.6
The Chain rule and generalized Power Rule
Generalized power rule
y f ( x) [u ( x )]n
y f ( x) n[u ( x)]n 1 ( x )
u
Example 1
Example 2
y 3(
f ( x) (5 x 2 3)6
f ( x ) 6(5 x 2 3) 5 x) 60 x(5 x 2 3) 5
(10
y
3
3
x x2
3( x x )
2
1
3

Maths 132 Class notes 9.8 and 9.9
9.8 Higher order derivatives
The derivative of a first derivative is called the second derivative.
Notation:
d2 y
f ( x) 2
y
dx
Higher order derivatives:
d3 y
y f ( x) 3
x
Application:
p. 615 #36
Suppose that a particle

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Maths 132 9.5-9.6 quiz
Whittern 10 pts.
Show all work and simplify answers!
1. Find the equation of the line tangent to the graph of
f ( x)
x2 2
at (2,1)
x4
1._
2. Find y given y 2 x3 (3 x 2 4) 2
(simplify into factored form)
2._
3.

Review for Maths 132 Ex I
1. Problem like 13 on p. 632
Find the limit (if it exists). If not, state why.
x3
2. lim 2
x 3 x 2 x 15
3. lim
x 3
100 points
9.1-9.6,9.8-9.9
| x 3|
x3
4. Use the definition of the derivative to find the derivative, f ( x) .
f (

Maths 132 Class Notes 10.1
Relative Maxima and Minima
Increasing and decreasing functions, Extrema and the first-derivative test
Let f
1.
2.
3.
be differentiable on the interval (a,b)
If f ( x ) 0 for all x in (a,b), then f is increasing on (a,b).
If f (

Maths 132 Class Notes 10.4
Applications of Maxima and Minima
Optimization Problems
Guidelines for solving optimization problems
1. Identify all given quantities and unknown quantities. Make a sketch, if
helpful.
2. Write an equation for the quantity that

Maths 132 Class notes 10.3
Optimization; Absolute Maxima and Minima
f (c) is an absolute maximum of f if f (c) f ( x ) for all x in the domain of f .
f (c) is an absolute minimum of f if f (c ) f ( x) for all x in the domain of f .
Extreme value theorem:

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M132 quiz 10.1-10.2
Whittern 10 pts.
For problems 1 and 2, use the graphing strategy to find y intercept, first derivative- make
a sign chart and find increasing and decreasing intervals and relative max and/or min.,
second derivative

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Maths 132 q 10.3-10.4
Whittern 10 pts.
Show all work (must be done by calculus) and simplify answers.
1. A private resort rents 300 canoes per day at a rate of $40 per day. For each $1
increase in charge, 5 fewer canoes are rented. At

Maths 132 Class Notes
Exponential functions
Definition:
The exponential function f with base a is denoted by
a 0, a 1, and x is any real number.
f ( x) a x
Properties of Exponential Functions
Let a be a positive real number, and let x and y be real number

Maths 132CLASS NOTES 12.1
Antiderivatives and indefinite integrals
Definition of antiderivative:
A function F is an antiderivative of a function f if for every x in the domain of f , it
follows that F ( x) f ( x)
Indefinite Integrals
We need a convenient

Maths 132 Class notes
Logarithmic Functions
Definition of a logarithmic function
Let a and x be positive numbers such that a 1 . The logarithm of x with
base a is denoted by log a x and is defined as follows:
y log a x
iff
x ay
The function f ( x) log a x

Exam II (100 pts.) over Chapter 10 Friday, March 21
Review for Maths 132 Ex II 10.1-10.5
1. Sketch a graph of a function f with the following properties:
f(0)=8, f(2)=8, f is continuous for all x, except x = 1
x = 1 is a vertical asymptote y = 4 is a hori

Maths 132 Class Notes 11.3
Implicit Differentiation
So far all our functions have expressed in the explicit form y = f(x).
In many relationships y is not expressed explicitly as a function of x, but
only in implicit form. In this section, we will develop

Maths 132 Class Notes 11.1,11.2
Derivatives of Exponential and Logarithmic Functions
The derivative of e x
d x
e ex
dx
The derivative of e f ( x)
d f ( x)
(e ) e f ( x ) f ( x)
dx
example:
e4 x3 x1
3
y (12 x 2 1) 4 x x1
e
The derivative of
d
1
ln x
dx

Maths 132Class Notes 11.4
Related Rates
Related rates: In this section, we are going to study of rates of change with
respect to time. As an example, suppose the radius r of a circle is allowed to
expand at some know rate, say 2 cm/min. Since A r 2 , it o

Maths 132 Class notes 12.2
Integration by substitution and the general power rule
General power rule for integration
1
u du n 1 u
n
n 1
C , n 1
Example 1
(x
2
1)3 (2 x) dx
let u x 2 1
du
2x
dx
du 2 xdx
x2
1
3x 2
Ex 2 3 2 dx 3 2 dx
( x 1)
3 ( x 1)
du

Maths 132 Class Notes Section 13.3
Area between two curves
Area between two curves
Let y f ( x) and y g ( x) be two continuous functions with f ( x) g ( x) on
[a,b]. Then the area between the graphs of the two curves on [a,b] is given
by the definite inte