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**Differentiation Using Limits of Difference Quotients**

3.4 OBJECTIVES Find derivatives and values of derivatives Find equations of tangent lines Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**Difference Quotient Average Rate of Change (Slope) of linePQ.**

The slope of the secant line PQ Would be close to the slope of a tangent line through P. It would be even closer if Q were closer to P. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

DEFINITION: The slope of the tangent line at (x, f(x)) is This limit is also the instantaneous rate of change of f(x) at x. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

DEFINITION: For a function y = f (x), its derivative at x is the function defined by provided the limit exists. If exists, then we say that f is differentiable at x. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

Example 1: For find Then find and Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

Example 1 (concluded): Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**Find the derivative using the difference quotient.**

Example 2: Do in Class Find the derivative using the difference quotient. Show work. Answer: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

Example 3: Students can look at outside of class. For : a) Find b) Find **Do part c once have done section 4.1.** c) Find an equation of the tangent line to the curve at x = 2. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

Example 3 (continued): a) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

Example 3 (continued): b) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

Example 3 (concluded): c) Thus, is the equation of the tangent line. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**Need 2 things to write equation of line. point (2) slope **

Use the information from example 1 to write an equation of a line tangent to f(x) where x = 4. Need 2 things to write equation of line. point (2) slope Already know x=4 find f(4) Find derivative at x = 4 Already found f’(4) = 8 Equation of Line y - 16 = 8(x – 4) y -16 = 8x – 32 y = 8x - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

3 Cases Where a Function is Not Differentiable: 1) A function f(x) is not differentiable at a point x = a, if there is a “corner” at a. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

Where a Function is Not Differentiable: 2) A function f (x) is not differentiable at a point x = a, if there is a vertical tangent at a. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**3.4 Differentiation Using Limits of Difference Quotients**

Where a Function is Not Differentiable: 3) A function f(x) is not differentiable at a point x = a, if it is not continuous at a. Example: g(x) is not continuous at –2, so g(x) is not differentiable at x = –2. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**1) The demand for an item is **

Come back to these word problems after section 4.1 1) The demand for an item is where p represents the price of the item in dollars. Find the rate of change of demand with respect to price. Find and interpret the rate of change of demand when the price is $10. Answers: (a) D’(p) = -4p-4 (b) D’(10) = -44 Demand is decreasing at rate of 44 when price is $10. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**Find the marginal profit at the following expenditures. **

2)The profit (in thousands of dollars) from the expenditure of x thousand dollars on advertising is given by Find the marginal profit at the following expenditures. $ b)$ c)$12,000 (note use 6, 8, and 12). 3) The revenue, in dollars, generated from the sale of x picnic tables is given by Find the marginal revenue when 1000 tables are sold. Estimate the revenue from the sale of the 1001st table. Answers: 2) (a) $8 thousand (b) $0 (c)$-16 thousand ) (a) $16 (b) $15.998

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