# The Derivative Chapter 3:. What is a derivative? A mathematical tool for studying the rate at which one quantity changes relative to another.

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The Derivative Chapter 3:

What is a derivative? A mathematical tool for studying the rate at which one quantity changes relative to another

What are some of the applications of the derivative? Slope of the tangent line to a curve The instantaneous velocity of an object In general, you can use the derivative to see how a change in one variable causes a change with another variable –The volume of liquid in a tank with respect to time if the liquid is leaking out of a hole –Cost effectiveness in construction and design

How does a derivative relate to a limit? Since the most basic definition of the derivative is the slope of the tangent line to a curve at a given point, let’s begin with that. What is a tangent line? Why might we have a problem finding the slope of a tangent line?

Consider the graph of a line: y = 2x What is the slope of this line? Does the slope of the line change from when x = 0 to when x = 4?

Consider the graph of a curve: y = x² What is the slope of this curve? Does the slope of the line change from when x = 0 to when x = 4? x = -4?

Definition of the Derivative:

What’s the difference?

Example 1 Use the definition to find the slope of the tangent line to the parabola at the point (2, 4).

Example 1A Use the derivative to find the equation of the tangent line at x = 2.

Example 2 Find the equation of the tangent line to the curve y=2/x at the point (2, 1) on this curve.

Example 2 Find the equation of the tangent line to the curve y=2/x at the point (2, 1) on this curve.

Example 3 Find the slopes of the tangent lines to the curve at

Example 3a Find the slopes of the tangent lines to the curve at

Example 3b Find the slopes of the tangent lines to the curve at

Example 3b Find the slopes of the tangent lines to the curve at

Example 3b Find the slopes of the tangent lines to the curve at

Example 3b Find the slopes of the tangent lines to the curve at

Example 3b The slopes of the tangent lines are: at x = 1 at x = 4 at x = 9

Example 4 (as presented in the book): a. Find the average rate of change of y with respect to x over the interval

Example 4 (as presented in the book): a.Find the average rate of change of y with respect to x over the interval b.Find the instantaneous rate of change of y with respect to x at an arbitrary value of x

Example 4 (as presented in the book): a.Find the average rate of change of y with respect to x over the interval b.Find the instantaneous rate of change of y with respect to x at an arbitrary value of c.Find the instantaneous rate of change of y with respect to x at the specified value for

Example 4 (as presented in the book): a.Find the average rate of change of y with respect to x over the interval b.Find the instantaneous rate of change of y with respect to x at an arbitrary value of c.Find the instantaneous rate of change of y with respect to x at the specified value for d.The average rate of change in part a is the slope of a secant line and the instantaneous rate of change in part c is the slope of a tangent line. Graph y=f(x) and these two lines together.

Practice / Homework: pg. 177 # 9 – 16 (13 – 16 don’t ask for average but do ask for the equation of the tangent line)

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