1. Differentiation Using Limits of Difference Quotients
3.4
OBJECTIVES
Find derivatives and values of derivatives
Find equations of tangent lines
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2. 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.
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3. 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.
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4. 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.
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5. 3.4 Differentiation Using Limits of Difference Quotients
Example 1: For find Then
find and
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6. 3.4 Differentiation Using Limits of Difference Quotients
Example 1 (concluded):
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7. Find the derivative using the difference quotient.
Example 2: Do in Class
Find the derivative using the difference quotient.
Show work.
Answer:
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8. 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.
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9. 3.4 Differentiation Using Limits of Difference Quotients
Example 3 (continued): a)
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10. 3.4 Differentiation Using Limits of Difference Quotients
Example 3 (continued): b)
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11. 3.4 Differentiation Using Limits of Difference Quotients
Example 3 (concluded): c)
Thus,
is the equation of the tangent line.
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12. 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
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13. 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.
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14. 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

15. 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.
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16. 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.
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17. 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