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**Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter**

Chapter 4 Determining Change: Derivatives Copyright © by Houghton Mifflin Company, All rights reserved.

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**Copyright © by Houghton Mifflin Company, All rights reserved.**

Chapter 4 Key Concepts Numerically Estimating Rates of Change The Four-Step Method Simple Derivative Formulas More Simple Derivative Formulas Chain Rule Product Rule Copyright © by Houghton Mifflin Company, All rights reserved.

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**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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**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. Copyright © by Houghton Mifflin Company, All rights reserved.

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**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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**Copyright © by Houghton Mifflin Company, All rights reserved.**

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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**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) Copyright © by Houghton Mifflin Company, All rights reserved.

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**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) Copyright © by Houghton Mifflin Company, All rights reserved.

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**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) Copyright © by Houghton Mifflin Company, All rights reserved.

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**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) Copyright © by Houghton Mifflin Company, All rights reserved.

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**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. Copyright © by Houghton Mifflin Company, All rights reserved.

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**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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**Copyright © by Houghton Mifflin Company, All rights reserved.**

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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**Chain Rule: Example (Form 1)**

Copyright © by Houghton Mifflin Company, All rights reserved.

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**Chain Rule: Example (Form 2)**

Copyright © by Houghton Mifflin Company, All rights reserved.

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**Copyright © by Houghton Mifflin Company, All rights reserved.**

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) Copyright © by Houghton Mifflin Company, All rights reserved.

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**Copyright © by Houghton Mifflin Company, All rights reserved.**

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) Copyright © by Houghton Mifflin Company, All rights reserved.

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**Copyright © by Houghton Mifflin Company, All rights reserved.**

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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**Copyright © by Houghton Mifflin Company, All rights reserved.**

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 Copyright © by Houghton Mifflin Company, All rights reserved.

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**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 Copyright © by Houghton Mifflin Company, All rights reserved.

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