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Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter Chapter 4 Determining Change: Derivatives

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

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Copyright © by Houghton Mifflin Company, All rights reserved. Numerically Estimating Rates of Change The slope of the tangent line is the limiting value of the slopes of nearby secant linesThe 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 quotientSlopes of piecewise continuous graphs at discontinuities may be estimated with symmetric difference quotient

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Copyright © by Houghton Mifflin Company, All rights reserved. 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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Copyright © by Houghton Mifflin Company, All rights reserved. Estimating Rates: Exercise 4.1 #3 Numerically estimate the limit of slopes of secant lines on the graph of h(x) = x x between x = 2 and close points to the right of x = 2. xh(x) h'(2) = 20

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Copyright © by Houghton Mifflin Company, All rights reserved. The Four-Step Method To find f '(x),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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Copyright © by Houghton Mifflin Company, All rights reserved. The Four-Step Method: Example Calculate f '(3) for f(x) = x 2 - x. 1. (3, f(3)) = (3, 6) 2. (3 + h, f(3 + h)) = (3 + h, (3 + h) 2 - (3 + h)) = (3 + h, h 2 + 5h + 6)

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Copyright © by Houghton Mifflin Company, All rights reserved. Four-Step Method: Exercise 3.2 #11 Calculate f '(x) for f(x) = 3x (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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Copyright © by Houghton Mifflin Company, All rights reserved. Four-Step Method: Exercise 3.2 #13 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 2 + 6xh + 3h 2 )

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Copyright © by Houghton Mifflin Company, All rights reserved. Simple Derivative Formulas Constant RuleConstant Rule –If f(x) = b, f '(x) = 0. Linear Function RuleLinear Function Rule –If f(x) = ax + b, f '(x) = a. Simple Power RuleSimple Power Rule –If f(x) = x n, f '(x) = nx n-1. Constant Multiplier RuleConstant Multiplier Rule –If f(x) = k g(x), f '(x) = k g'(x). Sum RuleSum Rule –If f(x) = g(x) + h(x), f '(x) = g'(x) + h'(x)

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Copyright © by Houghton Mifflin Company, All rights reserved. Simple Derivatives: Examples Constant RuleConstant Rule –If f(x) = 5, f '(x) = 0. Linear Function RuleLinear Function Rule –If f(x) = -3x + 4, f '(x) = -3. Simple Power RuleSimple Power Rule –If f(x) = x 4, f '(x) = 4x 3.

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Copyright © by Houghton Mifflin Company, All rights reserved. Simple Derivatives: Examples Constant Multiplier RuleConstant Multiplier Rule –If f(x) = 4x 3, f '(x) = 4(3x 2 ) = 12x 2 –If f(x) = -3(2x - 1), f '(x) = -3(2) = -6 Sum RuleSum Rule –If f(x) = 4x 3 - 3(2x - 1), f '(x) = 12x 2 - 6

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Copyright © by Houghton Mifflin Company, All rights reserved. Simple Derivatives: Exercises 4.3 #7, Calculate y' for y = 7x x + 13 Using the Sum Rule, Power Rule, Constant Rule, and Constant Multiple Rule we get y' = 14x Calculate y' for y = 3x -2 Using the Power Rule and Constant Multiple Rule we get y' = -6x -3

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Copyright © by Houghton Mifflin Company, All rights reserved. More Simple Derivative Formulas Exponential RuleExponential Rule –If f(x) = b x with b > 0, f '(x) = ln(b) b x e x Rulee x Rule –If f(x) = e x with b > 0, f '(x) = e x Natural Log RuleNatural Log Rule –If f(x) = ln(x), f '(x) = x -1 for x > 0

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Copyright © by Houghton Mifflin Company, All rights reserved. More Simple Derivatives: Example Calculate y' for y = 7x x Using the Sum Rule, Power Rule, Exponential Rule, and Constant Multiple Rule we get y' = 14x - (ln3)3 x Calculate y' for y = e x + 4ln(x) Using the e x Rule and Natural Log Rule we get y' = e x + 4x -1

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Copyright © by Houghton Mifflin Company, All rights reserved. More Derivatives: Exercise 4.4 #11, Calculate y' for y = 100,000( /12) 12x 14. Calculate y' for y = lnx + 3.3(2.9) x

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Copyright © by Houghton Mifflin Company, All rights reserved. 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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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, thenForm 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)) thenForm 2: If f(x) = (h g)(x) = h(g(x)) then

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Copyright © by Houghton Mifflin Company, All rights reserved. Chain Rule: Example (Form 1)

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Copyright © by Houghton Mifflin Company, All rights reserved. Chain Rule: Example (Form 2)

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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 investors gold increasing? (Use Form 1)

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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 investors gold increasing? (Use Form 2)

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Copyright © by Houghton Mifflin Company, All rights reserved. Product Rule Product RuleProduct Rule If f(x) = g(x) h(x) then f '(x) = g'(x) h(x) + g(x) h'(x) Example:Example: f(x) = (x 3 + 1)(2 x ) f '(x) = 3x 2 2 x + (x 3 + 1) (ln2)2 x

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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

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Copyright © by Houghton Mifflin Company, All rights reserved. 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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