1.3: Average Rates of Change
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
3.2 Logarithmic Functions
Solve exponential equations.
Solve problems involving exponential and logarithmic functions.
Differentiate functions involving natural logarithms.
A logarithm is defined as follows:
log a x = y
a y = x
3.1: Exponential Functions
Graph exponential functions.
Differentiate exponential functions.
For a > 0 , a 0 , and all real numbers x
Defines the exponential function with base a
Section 2.2: Using Second Derivatives to Find Maximum and Minimum Values and
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
Find absolute extrema (also called global extremes)
() = 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
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
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
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
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,
1.6: Differentiation Techniques: The Product and Quotient Rules
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
Find derivatives and values of derivatives
Find equations of tangent lines
DEFINITION: The slope of the tangent line at (, () is
This limit is also the instantaneous rate o
Section 2.1: Using First Derivatives to Find Maximum and Minimum Values and
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