2.2 Basic Differentiation Rules and Rates of Change
Differentiability of a Function
A function is differentiable at any point in its domain except in three cases
1.
2.
3.
The Power Rule
Evaluating Limits Analytically
Dividing out/ Cancellation Technique
x3 1
lim
x1 x 1
Remember this limit from yesterday. We examined it yesterday and determined
x3 1
lim
=3
that x1 x 1
Now lets see why.
Complete
5.5 Derivatives of Bases other than e
Recall that
d x
e = e x
dx
Now if base is a which is anything other than e
d x
[a ] =
dx
Now recall that
d u
e = eu du
dx
So for a base other than
1.2 Finding Limits Graphically and Numerically
Limit: As a function approaches a given x value what y value does it look like the
graph is getting closer and closer to
Example:
x3 1
lim
x1 x 1
We read thi
Continuity and One Sided Limits
Definition of Continuity
Continuity at a Point: A function F is continuous at c if the following three conditions
are met.
1. f(c) is defined
2. lim f (x) exists
xc
3. lim f (x)
5.2 Integrating with natural logs
First to review take the derivative of f (x) = ln(3x 2 + 4x)
Notice how we have the entire original inside function on the bottom and the
derivative of that in
2.1 Definition of Derivative
Definition of Tangent Line with Slope m
y
f (c + x) f (c)
lim
= lim
= m
x0 x
x0
x
The line passing through (c,f(c) with slope m is the tangent line to f at the point
(c
5.4 Exponential Functions: Derivatives and Integration
Definition of the Natural Exponential Function
e is the inverse of ln, that is ln(e x ) = x and eln x = x
They undo each other.
Using this property
Name
Relative Extrema Fizz
1. Use the 1st derivative test to find all relative extrema of
f (x) = 2x3 + 9x2 + 24 You must produce a chart to receive full credit.
, Per
rm: MM.th (369mg gemomw a x
, 2'4 x:( {:(f
t 98 +49} K I I
O ~<» (x 3) +"(~n>~¢<~u*
Name_
Relative Extrema Fizz
1. Use the 1st derivative test to find all relative extrema of
f (x) = 2x 3 + 9x 2 + 24 You must produce a chart to receive full credit.
2. Use the se
2.4 Chain Rule
Composite Function: A composite function is a function that is made up of one
function inside another function
Example
y = (3x 2 + 2)3
f (x) = x 2 + 1
y = sin(2x + 1)
Rewrite the above