1. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter
Chapter 4
Determining Change: Derivatives
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Chapter 4 Key Concepts
Numerically Estimating Rates of Change
The Four-Step Method
Simple Derivative Formulas
More Simple Derivative Formulas
Chain Rule
Product Rule
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3. Numerically Estimating Rates of Change
The slope of the tangent line is the limiting value of the slopes of nearby secant lines
Slopes of piecewise continuous graphs at discontinuities may be estimated with symmetric difference quotient
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4. Estimating Rates: Example
The number of Comcast employees (in thousands) x years after 1990 may be modeled by
The graph is discontinuous at 4 so the derivative at x = 4 does not exist. However, we can estimate the rate of change.
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5. Estimating Rates: Exercise 4.1 #3
Numerically estimate the limit of slopes of secant lines on the graph of h(x) = x2 + 16x between x = 2 and close points to the right of x = 2. x
h(x)
2.01
2.001
2.0001 2
36.2
36.02
36.002 36
h'(2) = 20
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The Four-Step Method
To find f '(x),
Begin with a point (x, f(x))
Choose a close point (x + h, f(x + h))
Write the formula for the slope of the secant line between the two points
Evaluate the limit of the slope as h nears 0
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7. The Four-Step Method: Example
Calculate f '(3) for f(x) = x2 - x.
1. (3, f(3)) = (3, 6)
2. (3 + h, f(3 + h))
= (3 + h, (3 + h)2 - (3 + h))
= (3 + h, h2 + 5h + 6)
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8. Four-Step Method: Exercise 3.2 #11
Calculate f '(x) for f(x) = 3x - 2
1. (x, f(x)) = (x, 3x - 2)
2. (x + h, f(x + h))
= (x + h, 3(x + h) - 2)
= (x + h, 3x + 3h - 2)
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9. Four-Step Method: Exercise 3.2 #13
Calculate f '(x) for f(x) = 3x2
1. (x, f(x)) = (x, 3x2)
2. (x + h, f(x + h))
= (x + h, 3(x + h)2)
= (x + h, 3x2 + 6xh + 3h2)
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10. Simple Derivative Formulas
Constant Rule
If f(x) = b, f '(x) = 0.
Linear Function Rule
If f(x) = ax + b, f '(x) = a.
Simple Power Rule
If f(x) = xn, f '(x) = nxn-1.
Constant Multiplier Rule
If f(x) = k g(x), f '(x) = k g'(x).
Sum Rule
If f(x) = g(x) + h(x), f '(x) = g'(x) + h'(x)
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11. Simple Derivatives: Examples
Constant Rule
If f(x) = 5, f '(x) = 0.
Linear Function Rule
If f(x) = -3x + 4, f '(x) = -3.
Simple Power Rule
If f(x) = x4, f '(x) = 4x3.
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12. Simple Derivatives: Examples
Constant Multiplier Rule
If f(x) = 4x3, f '(x) = 4(3x2) = 12x2
If f(x) = -3(2x - 1), f '(x) = -3(2) = -6
Sum Rule
If f(x) = 4x3 - 3(2x - 1), f '(x) = 12x2 - 6
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13. Simple Derivatives: Exercises 4.3 #7, 12
7. Calculate y' for y = 7x2 - 12x + 13
Using the Sum Rule, Power Rule, Constant Rule, and Constant Multiple Rule we get
y' = 14x - 12
12. Calculate y' for y = 3x-2
Using the Power Rule and Constant Multiple Rule we get
y' = -6x-3
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14. More Simple Derivative Formulas
Exponential Rule
If f(x) = bx with b > 0, f '(x) = ln(b) bx
ex Rule
If f(x) = ex with b > 0, f '(x) = ex
Natural Log Rule
If f(x) = ln(x), f '(x) = x-1 for x > 0
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15. More Simple Derivatives: Example
Calculate y' for y = 7x2 - 3x
Using the Sum Rule, Power Rule, Exponential Rule, and Constant Multiple Rule we get
y' = 14x - (ln3)3x
Calculate y' for y = ex + 4ln(x)
Using the ex Rule and Natural Log Rule we get
y' = ex + 4x-1
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16. More Derivatives: Exercise 4.4 #11, 14
11. Calculate y' for y = 100,000( /12)12x
14. Calculate y' for y = lnx + 3.3(2.9)x
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17. More Derivatives: Exercise 4.4 #19
The weight of a laboratory mouse between 3 and 11 weeks of age can be modeled by the equation
w(t) = ln(t) grams
where the age of the mouse is (t + 2) weeks after birth (thus for a 3-week old mouse, t = 1.) How rapidly is the weight of a 9-week old mouse changing?
Note: 9 weeks implies t = 7
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Chain Rule
Form 1: If C is a function of p and p is a function of t, then
Form 2: If f(x) = (h g)(x) = h(g(x)) then
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19. Chain Rule: Example (Form 1)
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20. Chain Rule: Example (Form 2)
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Chain Rule: Exercise 4.5 #3
An investor buys gold at a constant rate of 0.2 ounce per day. The investor currently has 400 troy ounces of gold. If gold is currently worth $ per troy ounce, how quickly is the value of the investor’s gold increasing? (Use Form 1)
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Chain Rule: Exercise 4.5 #3
An investor buys gold at a constant rate of 0.2 ounce per day. The investor currently has 400 troy ounces of gold. If gold is currently worth $ per troy ounce, how quickly is the value of the investor’s gold increasing? (Use Form 2)
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Product Rule
Product Rule
If f(x) = g(x) • h(x)
then f '(x) = g'(x) • h(x) + g(x) • h'(x)
Example:
f(x) = (x3 + 1)(2x)
f '(x) = 3x2 • 2x + (x3 + 1) • (ln2)2x
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Product Rule: Example
A music store has determined from a customer survey that when the price of each CD is $x, the number of CDs sold monthly can be modeled by
N(x) = 6250 ( )x CDs
Find and interpret the rate of change of revenue when the CDs are priced at $10.
R(x) = N(x) • x = 6250 ( )x • x
R'(x) = 6250 ln( )( )x • x ( )x • 1
= x( )x ( )x
R'(10) = 823 means revenue is increasing by $823 per $1 of CD price when the price is $10 per CD
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25. Product Rule: Exercise 4.6 #28
A music store has determined that the number of CDs sold monthly can be modeled by
N(x) = 6250 (0.9286)x CDs
where x is the price in dollars. Find the rate at which the revenue is changing when x = $20.
R(x) = N(x) • x = 6250 (0.9286)x • x
R'(x) = 6250 ln(0.9286)(0.9286)x • x (0.9286)x • 1
= x(0.9286)x (0.9286)x
R'(20) = means revenue is decreasing by $ per $1 of CD price when the price is $20 per CD
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