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The Derivative Chapter 3:

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

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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

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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?

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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?

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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?

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Definition of the Derivative:

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What’s the difference?

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

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

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

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

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

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

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

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

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

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

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

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

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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

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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

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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.

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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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CHAPTER 2 2.4 Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h 0 ( f(a + h) – f(a))

CHAPTER 2 2.4 Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h 0 ( f(a + h) – f(a))

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