PRODUCT AND QUOTIENT RULE
FIND THE EQUATION OF THE LINE TANGENT TO f x
1. f x x 3 1 x 2 2 at x 2
2. f x x 3 2 x 2 x 2 at x 1
3. f x
at x 2
4. f x
at x 2
5. Find the point(s) on the graph of f x
where m tan .
Before graphing calculators, Calculus was needed to sketch an accurate graph of a
function. Today, we use Calculus to locate any hidden behaviors that might exist on
the graph of a function. That is, sometimes a graphing calculator ma y
Functions can be defined explicitly (when y is defined explicitly in terms of x) or
Explicit form of a function
Implicit Form of a Function
6 y 4x 1
When finding the derivative of an implicitly d
More Differentiation Rules
If possible, always try to rewrite the function so that you can use the Power Rule. When
it is not possible or if rewriting is too burdensome, there are some other rules that we can
use to find the derivative of a function.
Basic Differentiation Rules
Instead of having to always use the limit process to find the derivative of a function, there
are some basic differentiation rules that we can use.
The Constant Rule:
The derivative of a constant is zero. That is,
The Chain Rule
Suppose f x x 2 1 and you were asked to find f x . Certainly you could expand
1 and use the Power Rule, but the expansion would be quite burdensome. An
alternative, and much easier, way would be to use the Chain Rule. This rule is us
Limits and Continuity
What is a limit? A limit is a value that a function's value gets arbitrarily close to as its
independent variable "goes" towards a certain number.
In general a limit is written like this:
lim f ( x ) = L
and is read the lim
Calculus 1 Lecture Notes
Review of Linear Functions
y mx b or f x mx b
Finding the equation of a line:
Case 1: m and b are known
and y intercept 0,5 .
Find the equation of a line that has slope =
The cost of leasing
Average Rates of Change:
Consider the following data:
1900 to 1995
Year In Millions
Find the avera
The Tangent Line Problem and the Derivative
Recall the Tangent Line Problem: Finding the slope of the line tangent to f x at a
given value of x.
We started by looking at f x x 2 and found that the slope of the tangent line at x = 1
was equal to 2 or, mtan
Related Rate Problems
Often, one encounters problems in which two or more variables are functions of time.
An example of this is an ice-cube melting. The volume, weight, and dimensions of the
ice cube are all continuously changing over time.
Rolles Theorem and the Mean Value Theorem
If f is continuous on a, b and differentiable on a, b
and f a f b ,
then there must be at least one c in a, b such that f c 0 .
What if f is not continuous?
What if f is not differentiable?
THE DERIVATIVE: SOLVED PROBLEMS
Cx n Cnx n 1
b. f t
d 4 2 4 d 2
5 t 5 dt t 5 2t 5 t
2 x 2 2 x 2 x 2
c. y 2 x
2 x 1 2
x 2 1x 2 2
SIMPLIFYING BY FACTORING OUT LEAST POWERS
Example: f x x 3 x 2
f x x 3
x 23 x 23 d x 32
x 3 3 x 2 x 2 2 x 3
x 3 x 2 3 x 3 2 x 2
Factoring Out Least Power
x 3 x 2 5 x 5
5 x 3 x 2 x 1
Example: f x 3 x 4 5 x 3
Applied problems that involve finding the maximum or minimum value of a function are
called optimization problems. Examples might include maximizing the volume of a
geometric solid, maximizing profit, minimizing surface area, or minimizing co
Concavity and the Second Derivative Test
The derivative can also tell us where a function is concave up (increasing at an increasing
) or decreasing at a decreasing rate
concave down (increasing at a decreasing rate
) and where a function i
Limits at Infinity
Limits at infinity or lim f x described the end behavior of a function. Sometimes
this end behavior follows the path of a constant in which case we say that f x has a
Sometimes this end behavior may follow the pa
Increasing and Decreasing Functions and the First Derivative Test
The derivative can tell us where a function is decreasing, increasing, or turning.
Notice the slopes of the tangent lines.
When mtan s 0 , f is increasing. When mtan s 0 , f is decreasing.
Factor as indicated:
(f) sin x
(b) 2 x
(b) 2 x
1) Use the table to find the indicated limit: lim x 2
f x x 2
Construct a table and find the indicated limit: lim
Construct a table and find the indi
Extrema on an Interval
The derivative can be very useful in describing the behavior of a function. Where the
function is increasing or decreasing, how the function is increasing or decreasing, where
the function reaches a maximum or minimum value and whet
The Tangent Line Problem
How can you find the slope of a tangent line?
Recall: The difference between and tangent line and a secant line.
Finding the slope of a tangent line, mtan , presents a problem because there is only one
point. To determine the slop