Chapter 3.1

Tangents and the Derivative at a Point

Review

• Chapter 2 started with finding slope of a curve at a point, and how to measure the rate at which a function changes

• Finding a Tangent to the Graph of a Function– Calculate slope of secant through a Point

P (x0, f(x0)) and a nearby point Q(x0+h, f(x0+h))

Example

• Find slope of curve at any point.

Derivative at a Point

• Started with the difference quotient

• When adding the limit piece this becomes the definition of the derivative function f at a point x0 and written f ’(x0)

• Difference quotient is the average rate of change• Derivative is the instantaneous rate of change with

respect to x at the point x = x0

h

xfhxf )()( 00

Example: Linear Derivative

Application

• What is the rate of change of the volume of a ball with respect to the radius when the radius is r = 2?