Section 3.2: Simple Rate-of-Change Formulas
1. Constant Rule for Derivatives
dy
=0
If y = b, then
dx
Example: y = 4
Find
dy
dx
2. Derivative of a Linear Function
dy
=m
If y = mx + b, then
dx
Example: y = -3x + 5
Find
dy
dx
3. Power Rule for Derivatives
If
Section 1.5: Polynomial Functions and Models
Quadratic Models:
A quadratic model is one that represents data that is parabolic in nature. A quadratic
model has the form.
f ( x) = ax 2 + bx + c
We can recognize a model is quadratic by looking at the Second
Section 1.4: Logistic Functions and Models
Logistic Function: A logistic model is one that represents an S-Shaped curve. It
usually represents how a total changes over time (i.e. Population, Total Sales, )
A Logistic Model has an equation of the form:
L
f
Section 1.3: Exponential and Logarithmic Functions
Data that can be represented with an exponential function is one that can
be written in the following form:
f (x) = abx
where b is based on the percent change. (b 1)100% = percent. a is based
on the initi
Section 1.1: Models, Functions and Graphs
Functions
A function is a rule that assigns to each value of x one and only one value of y .
Example 1
f (x) = x2 1
f (2) =
f (4) =
Example 2
Pennsylvania sales tax is 6%. This can be written as a linear function
Section 1.2: Linear Functions and Models
Linear Functions
A function f is a Linear Function if it is in the form
f (x) = mx + b (Note, the book writes this as f (x) = ax + b)
where m is the slope of the line and b is the y-intercept.
Slope of a Line
The S
Section 2.2: Instantaneous Rates of Change
In the previous section, we were mostly concerned with how something changes over
time looking at things like Average Rate of Change and Percentage Change. We are now
interested in the Instantaneous Rate of Chang
Section 2.1 Change, Percentage Change, and Average Rate of
Change
Change: The change in the quantity is found by subtracting the rst value
from the second.
Change= n m
Percentage Change: The change divided by the rst number multiplied
by 100.
Percentage C
Section 4.3: Inection Points
Another type of point that we are interested in is called an Inection
Point. These are points on the graph where the function is increasing or
decreasing the fastest.
To nd inection points, the thing we are interested in is th
Section 3.4: The Chain Rule
The chain rule is a rule that lets us take the derivative of a composite function.
Form 1 of the Chain Rule
If C is a function of p, and p is a function of t, then
dC dp
dC
dt = dp dt
Example 1:
C (p) = 2p2 1 and p(t) = ln t
Fi
Section 3.3: Exponential and Logarithmic Rate-of-Change Formulas
1. Derivative of e x
If y = e x , then
dy
= ex
dx
Example: y = 3 x 3 + 4e x
dy
dx
2. Derivative of b x
If y = b x , then
dy
= (ln b)b x
dx
Example: y = 10 x
dy
dx
Example: $1000 dollars is i
Section 2.3: Derivative Notation and Numerical Estimates
Derivative: The derivative of a function describes the instantaneous rate of change of a
function.
df d
Notation: f ' (t ) ;
; [ f (t )]
dt dt
The following all have the same meaning:
-Instantaneous
Section 1.1: Models, Functions and Graphs
Combining Functions
We can perform operations on functions, just like on numbers.
Function Addition
h = f + g Describes the total output of two functions. For example,
Total Cost is the sum of two functions where