Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All right reserved.

Differential Calculus Chapter 11 Differential Calculus Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.1 Limits Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. f(x) = x2 + x + 1, find Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Find Solution: Quotient property Polynomial limit Note that So the limit of as x approaches 5 is the value of the function at 5: Copyright ©2015 Pearson Education, Inc. All right reserved.

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One-Sided Limits and Limits Involving Infinity Section 11.2 One-Sided Limits and Limits Involving Infinity Copyright ©2015 Pearson Education, Inc. All right reserved.

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Find each of the given limits. Example: Find each of the given limits. (a) Solution: Since is not defined when the right-hand limit (which requires that does not exist. For the left-hand limit, write the square root in exponential form and apply the appropriate limit properties. Exponential form Power property Polynomial limit Copyright ©2015 Pearson Education, Inc. All right reserved.

Find each of the given limits. Example: Find each of the given limits. (b) Solution: Exponential form Sum property Power property Polynomial limits Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Infinite Limits Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.3 Rates of Change Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: If find the average rate of change of with respect to x as x changes from –2 to 3. Solution: This is the situation described by the expression with The average rate of change is Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Example Exercise Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. The rate of change of the cost function is called the marginal cost. Similarly, the rate of change of the revenue and profit function are called the marginal revenue and marginal profit. Copyright ©2015 Pearson Education, Inc. All right reserved.

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Tangent Lines and Derivatives Section 11.4 Tangent Lines and Derivatives Copyright ©2015 Pearson Education, Inc. All right reserved.

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According to the definition, the slope of the tangent line is Example: Let a be any real number. Find the equation of the tangent line to the graph of at the point where Solution: According to the definition, the slope of the tangent line is Copyright ©2015 Pearson Education, Inc. All right reserved.

Hence, the equation of the tangent line at the point is Example: Let a be any real number. Find the equation of the tangent line to the graph of at the point where Solution: Hence, the equation of the tangent line at the point is Thus, the tangent line is the graph of Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. 1) 2) 3) Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. If f’(x) is defined, then the function f is said to be differentiable at x. The process that produces f’ from f is called differentiation. Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Examples 1) 2) 3) 4) Label f and f’ correctly: Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Existence of the Derivative Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Select the graph of f’ Copyright ©2015 Pearson Education, Inc. All right reserved.

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Techniques for Finding Derivatives Section 11.5 Techniques for Finding Derivatives Copyright ©2015 Pearson Education, Inc. All right reserved.

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From the function p, the revenue function is given by Example: The demand function for a certain product is given by Find the marginal revenue when units and p is in dollars. Solution: From the function p, the revenue function is given by The marginal revenue is Copyright ©2015 Pearson Education, Inc. All right reserved.

When the marginal revenue is Example: The demand function for a certain product is given by Find the marginal revenue when units and p is in dollars. Solution: When the marginal revenue is or $1.20 per unit. Thus, the next unit sold (at sales of 10,000) will produce an additional revenue of about $1.20. Copyright ©2015 Pearson Education, Inc. All right reserved.

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Derivatives of Products and Quotients Section 11.6 Derivatives of Products and Quotients Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Derivative of product of 3 functions: [u(x)v(x) w(x)]’ = u’(x)v(x) w(x) + u(x)v’(x) w(x) + u(x)v(x) w’(x) Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Proof [f(x)g(x)]’ Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Solution: Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Find the derivative of Solution: Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example Given that: (sin(x))’ = cos(x) and (cos(x))’ = -sin(x) Find derivative of Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Complete the table without using the Quotient Rule Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The cost in dollars of manufacturing x hundred items is given by: C(x) = 4x2 + 6x + 5 Find the average cost Find the marginal average cost Find the marginal cost Find the level of production at which the marginal average cost is zero. Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Exercises Use quotient rule to prove the Power Rule for negative integers n. i.e. (xn)’ = nxn-1 for negative integers n. Find derivative of g(x). y = 1 Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Identify each graph. Sketch the graph of f’ Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Review f(x) = x2 Write equation of the tangent line to the graph of function f(x) = x2 at (1, 1) Write equation of the tangent line to the graph that passes through the point (-1, -1) Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.7 The Chain Rule Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Composite function Let f and g be functions. The composite function, or composition, of f and g is the function whose values are given by f[g(x)] for all x in the domain of g such that g(x) is in the domain of f. Example: find functions f and g so that f[g(x)] are the followings: f(x) = g(x) = Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: find functions f and g so that f[g(x)] are the followings: Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Proof c is in the domain of h Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Use the chain rule to find Solution: Copyright ©2015 Pearson Education, Inc. All right reserved.

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Summary of Differentiation Rules Copyright ©2015 Pearson Education, Inc. All right reserved.

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Exercises: find derivative of functions below. Find the equation of the tangent line to the graph of f at the given point. Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Label the graph as f or f’: Copyright ©2015 Pearson Education, Inc. All right reserved.

Derivatives of Exponential and Logarithmic Functions Section 11.8 Derivatives of Exponential and Logarithmic Functions Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Proof: Example: y = e2x – 1. Find y’. Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Let Find Solution: Use the quotient rule: Exercise: for a > 0, y = ax. Find y’. Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Examples: find the derivatives Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. If y = ln|u|, then y’ = Example: y = loga(x) Copyright ©2015 Pearson Education, Inc. All right reserved.

Continuity and Differentiability Section 11.9 Continuity and Differentiability Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Is the function in the following graph continuous on the given x-intervals? (a) Solution: The function g is discontinuous only at Hence, g is continuous at every point of the open interval which does not include 0 or 2. (b) Solution: The function g is not defined at so it is not continuous from the right there. Therefore, it is not continuous on the closed interval Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Is the function in the following graph continuous on the given x-intervals? (c) Solution: The interval contains a point of discontinuity at So g is not continuous on the open interval (d) Solution: The function g is continuous on the open interval continuous the right at and continuous from the left at Hence, g is continuous on the closed interval Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Match each function with each graph Copyright ©2015 Pearson Education, Inc. All right reserved.