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.
Related rate
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 the graph of f x
x 1
1
where m tan .
x 1
2
THE GENERAL
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 2 5 x 5
2
5 x 3 x 2 x 1
2
Example: f x 3 x 4 5 x 3
4d
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 indicated limit: lim
3)
Construct a table and find the indi
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, maximizing profit, minimizing surface area, or minimizing co
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 function. That is, sometimes a graphing calculator ma y
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 asymptote.
Sometimes this end behavior may follow the pa
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 (increasing at a decreasing rate
) and where a function i
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.
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 not continuous?
What if f is not differentiable?
Exampl
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. y 2 x
d. y
dy d
d 1
2
2 x 1 2
x 2 1x 2 2
dx dx
dx
x
2
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 has slope =
Solution:
m=
Example:
The cost of leasing
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
6 y 4x 1
When finding the derivative of an implicitly 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 easier, way would be to use the Chain Rule. This rule is us
Higher Order Derivatives
Consider a function where the independent variable is time (t) and the dependent variable
is distance, (s) so that s = f ( t ) .
ds rate of change in distance
=
= v( t )
dt
with respect to time
where v( t ) is the velocity functio
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.
The
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,
d
C 0
dx
This
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
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 written like this:
lim f ( x ) = L
xc
and is read the lim
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
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
1990
248.7
1992
255.4
1994
260.7
1995
263.0
Find the avera
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