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

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Chapter 12 Applications of the Derivative

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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 12.1 Derivatives and Graphs

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Determine from the graph whether f has a local minimum on (a, b)

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Slide Find the critical numbers of the given function. Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: We have so exists for every x. Setting shows that Therefore −3 and 4 are the critical numbers of f; these are the only places where local extrema could occur. The graph at the right shows that there is a local maximum at and a local minimum at

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. f is continuous on [a, b].

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Find the local extrema of function:

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Find the local extrema of function:

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Review: Find the local extrema of function

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Review: given the graph of function f(x) = -x and two points A(-1, 4) and B (2, 1) on the graph. Find equation of the tangent line to the graph that is parallel to AB.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Review: given the graph of function f(x) = 5 – 4/x (x>0) and two points A(1, 1) and B (4, 4) on the graph. Find equation of the tangent line to the graph that is parallel to AB.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 12.2 The Second Derivative

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. The second derivative of function f is the derivative of the first derivative of f, denoted f’’(x).

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Slide Find the second derivative of the given functions. Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Here, The second derivative is the derivative of Solution: Using the product rule gives Differentiate this result to get (a) (b) f(x) = 8x 3 – 9x 2 + 6x + 1

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. f’ f’’ f’ f’’ f’ f’’ f’ f’’

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Find a, b, c, d so that the graph of function f(x) = ax 3 +bx 2 +cx+d satifies the conditions below: Local maximum: (3, 3) Local minimum: (5, 1) Inflection point: (4, 2)

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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 12.3 Optimization Applications

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Slide Absolute Extrema Let f be a function defined in the closed interval [a, b]. Let c be a number in the interval. f has absolute maximum on the interval [a, b] at c if f(x) f(c) for all x [a, b] f has absolute minimum on the interval [a, b] at c if f(x) f(c) for all x [a, b] Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Find absolute extrema of the function on indicated interval.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Consider the function Without graphing, show that f has an absolute minimum on the interval (0, 2). Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The derivative is which is defined everywhere, so the critical numbers are the solutions of The only critical number in the interval (0,2) is Use the second-derivative test to determine whether there is a local extremum at Hence, f has a local minimum at Therefore, by the critical- point theorem, the absolute minimum of f on the interval (0, 2) occurs at

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Example: A land has the shape of a right triangle with dimension 8m by 10m. We want to build a rectangle house with a corner on the hypotenuse and the opposite corner is a the right angle. What is the dimension of the house that has maximum area? 8m 10m

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Example: An open box is to be made by cutting a square from each corner of a 12 12 cm piece of metal and then folding up the sides. The finished box must be at least 1.5cm deep, but not deeper than 3 cm. What size square should be cut from each corner in order to produce a box of maximum volume? x=2, V(2) = 128 cm 3

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. x = 700m, y = 350m

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. 9 x 6 in.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. x = 0

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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 12.4 Implicit Differentiation

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Implicit form

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Slide Find given that Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Step 1 Take the derivative of both sides of the equation with respect to x: Simplify.

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Slide Find given that Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Step 2 Solve the last equation for Move terms involving to the left side. Move terms without to the right side. Factor out Divide both sides by

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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Section 12.5 Related Rates

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Slide A cancerous tumor in the shape of a sphere is undergoing radiation treatment. The radius of the tumor is decreasing at the rate of 2 millimeters per week. How fast is the volume of the tumor changing when the radius of the tumor is 10 millimeters? The volume (in cubic millimeters) of a sphere of radius r (in millimeters) is given by Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The relationship between the volume and radius of the tumor was given: To find differentiate both sides of the given equation with respect to time (remember to use the chain rule since r is a function of time): Chain rule Simplify.

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Slide Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The volume of the tumor is decreasing at a rate of about 2513 cubic millimeters per week. Substitute and simplify.Substitute the given values of (since the radius is decreasing by 2 mm per week) into the previous equation to find that A cancerous tumor in the shape of a sphere is undergoing radiation treatment. The radius of the tumor is decreasing at the rate of 2 millimeters per week. How fast is the volume of the tumor changing when the radius of the tumor is 10 millimeters? The volume (in cubic millimeters) of a sphere of radius r (in millimeters) is given by

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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 12.6 Curve Sketching

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Continued on next slide

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Continued from previous slide

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Slide Graph Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Divide both sides by 6. Factor. The y-intercept is Step 1 To find the x-intercepts, we must solve the equation There is no easy way to do this by hand, so skip this step. Since f(x) is a polynomial function, the graph has no asymptotes, so we can also skip Step 3. Step 2 The first derivative is and the second derivative is Step 4 The first derivative is defined for all x, so the only critical numbers are the solutions of Step 5

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Slide Graph Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Using the second-derivative test on the critical number we have, Hence, there is a local maximum when that is, at the point Similarly, so there is a local minimum when (at the point ). Next, we determine the intervals on which f is increasing or decreasing by solving the inequalities Step 5

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Slide Graph Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The critical numbers divide the x-axis into three regions. Testing a number from each region, as indicated below, we conclude that f is increasing on the intervals and decreasing on The second derivative is defined for all x, so the possible points of inflection are determined by the solutions of Step 5 Step 6

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Slide Graph Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Determine the concavity of the graph by solving Therefore, f is concave upward on the interval and concave downward on Consequently, the only point of inflection is Since f is a third-degree polynomial function, we know that when x is very large in absolute value, its graph must resemble the graph of its highest degree term, that is, the graph must rise sharply on the right side and fall sharply on the left. Step 6 Step 7

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Slide Graph Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Combining this fact with the information in the preceding steps, we see that the graph of f must have the general shape shown below. Step 7

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Slide Graph Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Now we plot the points determined in Steps 1, 5, and 6, together with a few additional points to obtain the graph below. Step 8

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Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

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