chapter 3: the derivative
DESCRIPTION
Chapter 3: The Derivative. JMerrill, 2009. Review – Average Rate of Change. Find the average rate of change for the function from 1 to 5. Review – Tangent Lines. Find the tangent to the curve f(x) = 3x 2 – 2 at x = 1 y = 6x - 5. - PowerPoint PPT PresentationTRANSCRIPT
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Chapter 3: The Derivative
JMerrill, 2009
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Review – Average Rate of Change
Find the average rate of change for the function from 1 to 51f(x) x
15
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Review – Tangent Lines
Find the tangent to the curve f(x) = 3x2 – 2 at x = 1
y = 6x - 5
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3.5 – Graphical Differentiation
Given a graph of a cost function, how can we find the graph of the marginal cost function?
Sometimes, all we have is a graph so it’s easier to find the derivative graphically
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Graphing the Derivative
When graphing the derivative, you are graphing the slope of the original function.
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Graphing the Derivative
When x < -2, the slope is 1 When -2 < x < 0, the slope is -1 At x = -2 and x = 0 the derivative does not
exist—why?
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Graphing the Derivative
For x > 0, the derivative is positive—estimated to be a slope of 1 at x = 1
As x approaches 0 from the right, the derivative becomes larger
As x approaches infinity, the derivative approaches 0.
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Graphing
Which is the f(x) and which is f’(x)?The derivative is 0 (crosses the x-axis)
wherever there is a horizontal tangentY1 = f(x)Y2 = f’(x)
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Chapter 4 – Calculating the Derivative
4.1Techniques for Finding the Derivative
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The Derivative
The process of finding the derivative is changing. But the interpretation will not change—it is still the taking the limit as h approaches 0.
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Notation
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Constant Rule
If f(x) = 4, then f’(x) = 0
If f(x) = , then f’(x) = 0
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Power Rule
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Power Rule – Examples
If f(x) = x6, find DxyDxy = 6x6-1 = 6x5
If
If
0
dyf(x) x,fi nddxdy 1x 1dx
3dy1y fi nddxx
3
3
44
1 must be rewrittenxxdy 33xdx x
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Power Rule Examples
Example 1: Given f(x) = 3x2, find f’(x). f’(x) = 6x Example 2: Find the first derivative given f(x) = 8x. 8x0 = 8
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Sum or Difference Rule
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Sum/Difference Examples
The Sum/Difference rule can be used on each term in a polynomial to find the first derivative.
Find f’(x), given f(x) = 5x4 – 2x3 – 5x2 + 8x + 11
f’(x) = 20x3 – 6x2 – 10x + 8The derivative of a constant is 0 because
11 is the same as 11x0, which is (0)11x-1
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Sum/Difference Examples
Find p’(t) given
Rewrite p(t):
4 5p(t) 12t 6 t t
14 12p(t) 12t 6t 5t
13 22
32
p'(t) 48t 3t 5t3 5p'(t) 48t t t
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Applications
Marginal variables can be cost, revenue, and/or profit. Marginal refers to rates of change.
Since the derivative gives the rate of change of a function, we find the derivative.
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Application Example
The total cost in hundreds of dollars to produce x thousand barrels of a beverage is given by
C(x) = 4x2 + 100x + 500Find the marginal cost for x = 5
C’(x) = 8x + 100; C’(5) = 140
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Example Continued
After 5,000 barrels have been produced, the cost to produce 1,000 more barrels will be approximately $14,000
The actual cost will be C(6) – C(5): 144 or $14,400
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The Demand Function
The Demand Function, defined by p = D(q), related the number of q units of an item that consumers are will to purchase at the price, p.
The total revenue R(q) is related to price per unit and the amount demanded
The total revenue is R(q) = qp = qD(q)
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Demand Function Example
The demand function for a certain product is given by
Find the marginal revenue when q = 10,000 units and p is in dollars.
50,000 qp 25,000
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Demand Function Example
The revenue function is R(q) = qp 50,000 qp 25,000
2
2
R(q) = qp 50,000 q=q 25,000
50,000q q25,000
12q q25,000
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Example
The marginal revenue is
212q q25,0002R'(q) 2 q25,000
Forq 10,0002R'(10,000) 2 (10,000) 1.225,000
$1.20 per unit
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4.2
Derivatives of Products and Quotients
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Product Rule
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Product Rule - Example
Let f(x) = (2x + 3)(3x2). Find f’(x)
=(2x + 3)(6x) + (3x2)(2) =12x2 + 18x + 6x2 = 18x2 + 18x
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Power Rule
Find f’(x) given that
2f (x) x 3 x 5x
1 1
22 21x 3 2x 5 x 5x x2
3 12 25 15x 6x x 152 2
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Quotient Rule
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Quotient Rule Example
Find f’(x) if 2x 1f(x) 4x 3
2
4x 3 (2) 2x 1 44x 3
210
4x 3
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Product & Quotient Rules
Find
2
27x 9 3 4x (5) (5x 1)( 4) (3 11x 20x )(7)
(7x 9)
x
3 4x 5x 1D 7x 9
x x2
7x 9 D 3 4x 5x 1 3 4x 5x 1 D 7x 9(7x 9)
22
140x 360x 120(7x 9)