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 rig
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 alge
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 minimiz
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 c
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. Ex
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 tha
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
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 el
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 tha
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
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 mathemati
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 a
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 arr
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.
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 gr