SECTION 3.2Basic Differentiation Rules and Rates of Change

3.1 Additional Example 1Use the formal definition to find the derivative of the function.

3.1 Additional Example 2aSketch the graph of .

3.1 Additional Example 2bSketch the graph of .

3.1 Additional Example 2cSketch the graph of .

Example 1

Find the derivative of the function.

a.

b.

c. , where is a constant.

Theorem 3.2 The Constant Rule (p. 127)

The derivative of a constant function is 0. That is, if is a real number, then

.

More Differentiation Rules

Example 2

Find the derivative of the function.a.

b.

Example 3

Find the derivative.

a.

b.

Example 4

Find the derivative.

a.

b.

c.

Derivative of the Natural Exponential Function

=

Example 5Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing calculator to confirm your results.

Example 6Calculate.

a.

b.

Example 7Find an equation of the tangent line to the graph of at the given point.

Example 8Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.

a.

b.

Physics Applications

• What is a rate of change that we use to measure the speed of our vehicle?

• The function that gives the position (relative to the origin) of an object as a function of time is called the position function.

Average Velocity

Example 9Find the average rate of the fnc. over the given interval.

Position, Velocity & AccelarationVelocity

Given a position function, , for an object moving along a straight line, the velocity of the object at time is

the instantaneous rate of change of the position function.

Thus, “velocity is the derivative of position”

“acceleration is the derivative of velocity.”

Example 10Position Function of a Free-Falling Object (Neglecting Air-Resistance)

where is the initial height of an object, is the initial velocity of the object, and is the acceleration due to gravity.

(Note: On Earth, or .)

A ball is thrown straight down from the top of a 220-ft building with an initial velocity of -22 ft per second. What is its velocity after 3 seconds? What is its velocity after falling 108 ft.?

Questions???

• Don’t forget to be working the practice problems.

• Study hard!!! Test 1 on Tuesday!