PRODUCT AND QUOTIENT RULE
I.
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
x2
at x 2
x 1
4. f x
3x
at x 2
x 2
2
5. Find the point(s) on t
1
Curve Sketching
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 f
Implicit Differentiation
Functions can be defined explicitly (when y is defined explicitly in terms of x) or
implicitly.
Explicit form of a function
Implicit Form of a Function
1
x
2
1
y x
3
6
xy 1
y
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 tha
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 d
The Chain Rule
Suppose f x x 2 1 and you were asked to find f x . Certainly you could expand
x
5
5
1 and use the Power Rule, but the expansion would be quite burdensome. An
alternative, and much easi
Limits and Continuity
Limits
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 wr
Calculus 1 Lecture Notes
Review of Linear Functions
I.
y mx b or f x mx b
Finding the equation of a line:
Case 1: m and b are known
1
and y intercept 0,5 .
3
Example:
Find the equation of a line that
Average Rates of Change:
Consider the following data:
U.S. Population
1900 to 1995
Year In Millions
1900
76.0
1910
92.0
1920
105.7
1930
122.8
1940
131.7
1950
151.3
1960
179.3
1970
203.3
1980
226.5
199
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 slo
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
Rolles Theorem and the Mean Value Theorem
Rolles 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
THE DERIVATIVE: SOLVED PROBLEMS
d
Cx n Cnx n 1
dx
Function
a. y
Derivative
2
x
b. f t
4t 2
5
2x
e. y
d 4 2 4 d 2
4
8
5 t 5 dt t 5 2t 5 t
dt
1
3
f t
dy d
1
1
1
1 1
2 x 2 2 x 2 x 2
dx dx
x
2
c.
SIMPLIFYING BY FACTORING OUT LEAST POWERS
Example: f x x 3 x 2
2d
f x x 3
x 23 x 23 d x 32
dx
dx
2
2
3
1
x 3 3 x 2 x 2 2 x 3
2
3
x 3 x 2 3 x 3 2 x 2
2
Product Rule
Factoring Out Least Power
x 3 x
Optimization
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, maxim
Concavity and the Second Derivative Test
Concavity
The derivative can also tell us where a function is concave up (increasing at an increasing
rate
) or decreasing at a decreasing rate
concave down (i
Limits at Infinity
Limits at infinity or lim f x described the end behavior of a function. Sometimes
x
this end behavior follows the path of a constant in which case we say that f x has a
horizontal a
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
Algebra Review
0.1
Monomial Factors
Factor as indicated:
(a) 3x4
x
(c) e
(e)
(g)
4x3
x
xe
x
2
x2
x2
2x2e x
x
e
(d) x
x
2
6x2
1
1
(f) sin x
4x
x2
2
2
x
tan x
2x
x
1
x
1
sin x
1
2x
4x3
6x3
(b) 2 x
2x2
S
Calculus 1
Limit Exercises
1) Use the table to find the indicated limit: lim x 2
x2
x
f x x 2
1.99
3.960
1.999
3.996
1.9999 2.0001
3.9996 4.0004
2.001
4.004
1
x2
2)
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
t
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