MA 131 Lecture Notes
Chapter 4
Calculus by Stewart
4.1) Maximum and Minimum Values
4.3) How Derivatives Affect the Shape of a Graph
A function is increasing if its graph moves up as x moves to the right and is decreasing if its graph
moves down as x moves
MA 131 Lecture Notes
Exponential Functions, Inverse Functions, and Logarithmic Functions
Exponential Functions
We say that a function is an algebraic function if it is created by a combination of algebraic processes
such as addition, subtraction, multipli
MA 131 Lecture Notes
Calculus by Stewart
Optimization
One of the most important applications of the derivative is optimization. It involves finding a value of x
where we can either maximize or minimize our corresponding y value. These will be real world
a
MA 131 Lecture Notes
Calculus by Stewart
4.9) Antiderivatives
We begin a new process called antidifferentiation that is considered the inverse application to
differentiation.
If given a function, we can find the derivative. For example, find
d 3
x .
dx
MA 131 Lecture Notes
Implicit Differentiation
We say that a function is in explicit form if it is of the form y=f(x). In other words, on variable is
5 xe x
explicitly defined in terms of the other. Examples include y 2 x 4 or y
.
x 1
Some functions are n
MA 131 Lecture Notes
Related Rates
First we give special attention to notation. When we say
dy
d
or
, we are saying the derivative with
dx
dx
respect to the variable x. It means that we understand that x is the variable and we can treat all other
letters
MA 131 Lecture Notes
Derivative of Trig Functions and the Chain Rule
Derivatives of Trigonometric Functions
We are also interested in finding the derivatives of trig functions. Let us study the graph of f(x)=sinx.
Draw the graph and estimate the slope of
MA 131 Lecture Notes
Sections 1.1 and 1.2
Functions
Definition of a Function
A function f from a set A to a set B is a rule of correspondence that assigns to each element x in the set
A exactly one element y in the set B. The set A is the domain (or set o
MA 131 Lecture Notes
Product and Quotient Rules
The Product and Quotient Rules
We continue to find short cuts to finding derivatives without using the limit definition of the derivatives.
Consider that we want to find the derivative of a product. First we
Tangent Lines and Derivatives
The Derivative and the Slope of a Graph
Recall that the slope of a line is sometimes referred to as a rate of change. In particular, we are
referencing the rate at which the variable y changes with respect to the change in th
MA 131 Lecture Notes
Continuity
Consider the following graph of the function f(x). Use it to evaluate the limits.
lim f ( x)
x 3
lim f ( x)
lim f ( x)
x 1
x 4
lim f ( x)
x 2
An important mathematical concept when learning calculus is the concept of co
Limits
An important concept in the study of mathematics is that of a limit. It is often one of the harder concepts
to understand. A limit is a bound, it is a value that we approach (but often do not achieve.) One example
is carrying capacity. Consider a c
MA 131 Lecture
Calculus, Early Transcendentals by Stewart
Differentiation Rules
Derivatives of Polynomials and Exponential Functions
In this section we will learn important rules that will help us arrive at the derivative of a function easily.
We will use
MA 131 Lecture Notes
Calculus
Sections 1.5 and 1.6 (and other material)
Algebra of Functions
Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions with overlapping domains. Then for all x common to both
domains, the sum, differe
MA 131 Lecture Notes
Sections 1.3
Here are a few more basic (library) functions. We will discuss exponential, logarithmic, and
trigonometric functions in detail later.
f ( x) sin x
f ( x) cos x
The graph of y cos x is a horizontal shift of the graph of y