Section 2.1: Using First Derivatives to Find Maximum and Minimum Values and
Sketch Graphs
OBJECTIVE
Find relative extrema using the First-Derivative Test.
Sketch graphs of continuous functions.
DEFINITION: A function f is increasing over I if, for every a
3.2 Logarithmic Functions
OBJECTIVE
Solve exponential equations.
Solve problems involving exponential and logarithmic functions.
Differentiate functions involving natural logarithms.
DEFINITION:
A logarithm is defined as follows:
log a x = y
means
a y = x
3.1: Exponential Functions
OBJECTIVE
Graph exponential functions.
Differentiate exponential functions.
DEFINITION:
Exponential Functions
For a > 0 , a 0 , and all real numbers x
() =
Defines the exponential function with base a
5
4
3
2
1
0
-4
-2
Graphing
Section 2.2: Using Second Derivatives to Find Maximum and Minimum Values and
Sketch Graphs
OBJECTIVE
Find the relative extrema of a function using the Second-Derivative Test.
Graph a continuous function in a manner that shows concavity.
We will tackle the
2.4: Using Derivatives to Find Absolute Maximum and Minimum Values
OBJECTIVE
Find absolute extrema (also called global extremes)
Example 1:
() = 2 4 + 5
Find the critical points and relative max/min points.
Find any inflection points.
Now, sketch the grap
2.5: Maximum-Minimum Problems; Business and Economics Applications
OBJECTIVE
Solve maximum and minimum problems using calculus.
A Strategy for Solving Maximum-Minimum Problems
1. Read the problem carefully. List the variables involved. If relevant, make a
1.5: Differentiation Techniques: The Power and Sum-Difference Rules
OBJECTIVE
Differentiate using the Power Rule or the Sum-Difference Rule.
Differentiate a constant or a constant times a function.
Determine points at which a tangent line has a specified
1.8: Higher Order Derivatives
OBJECTIVE
Find derivatives of higher order.
Given a formula for distance, find velocity and acceleration.
Consider the function given by = () = 5 3 4 +
Its derivative f is given by = () = 5 4 12 3 + 1
The derivative function
1.7: The Chain Rule
OBJECTIVE
Find the composition of two functions.
Differentiate using the Extended Power Rule or the Chain Rule.
THEOREM 7: The Extended Power Rule
Suppose that g(x) is a differentiable function of x. Then, for any real number k,
d
k
k-
1.6: Differentiation Techniques: The Product and Quotient Rules
OBJECTIVE
Differentiate using the Product Rule.
THEOREM 5: The Product Rule
Let () = () (). Then
() =
[(). ()]
() = [
()] . () + [ ()] ()
() = ()() + ()()
Note: The derivative of a product
1.4: Differentiation Using Limits of Difference Quotients
OBJECTIVE
Find derivatives and values of derivatives
Find equations of tangent lines
DEFINITION: The slope of the tangent line at (, () is
= lim
0
(+)()
This limit is also the instantaneous rate o
1.3: Average Rates of Change
OBJECTIVE
Compute an average rate of change.
Find a simplified difference quotient.
DEFINITION: The average rate of change of y with respect to x, as x changes from x1 to x2, is the
ratio of the change in output to the change