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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

More Derivatives: Exercise 4.4 #11, 14

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.

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.

Chain Rule: Example (Form 1)

Chain Rule: Example (Form 2)

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.

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.

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.