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Calculus - Mr Santowski 5/2/2015Calculus - Santowski1

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Lesson Objectives 1. Use Calculus methods to determine the absolute and relative extrema of a continuous & differentiable function 2. State the extreme value theorem 3. Apply concepts of increase, decrease and critical numbers and absolute extrema to a real world problem 5/2/2015Calculus - Santowski2

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Fast Five 1. Determine the x coordinates of the critical point(s) of f(x) = x 4 - 18x 2 + 1 2. On the restricted domain of [0,4], find the function values at the end points if 3. Evaluate 4. If, determine whether or not the function has an extreme point. 5/2/2015Calculus - Santowski3

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(A) Terms Given a function, f(x), that is defined on a given interval and let c be a number in the domain f(c) is the ABSOLUTE or GLOBAL maximum of f(x) on the interval if f(c) > f(x) for every x in the interval Now, sketch an example of what has just been described. 5/2/2015Calculus - Santowski4

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(A) Terms Given a function, f(x), that is defined on a given interval and let c be a number in the domain f(c) is the ABSOLUTE or GLOBAL minimum of f(x) on the interval if f(c) < f(x) for every x in the interval Now, sketch an example of what has just been described. 5/2/2015Calculus - Santowski5

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(A) Terms A FUNCTION is said to have an absolute extremum (or extrema) at x = c if it either has either an absolute maximum or an absolute minimum at x = c 5/2/2015Calculus - Santowski6

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(B) Extreme Value Theorem A function, f(x), that is CONTINUOUS on a CLOSED interval [a,b] will have BOTH an absolute maximum value and an absolute minimum value on the closed interval Now, sketch an example of what has just been described. 5/2/2015Calculus - Santowski7

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(C) Extrema and Open Intervals We now consider the importance of an open vs closed interval using the functions f(x) = x 3 and g(x) = ln(x) 5/2/2015Calculus - Santowski8

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(C) Extrema and Open Intervals To illustrate the point about intervals, consider the functions, f(x) = x 3 and g(x) = ln(x) If we have an open interval (- ,+ ) for f(x) and (0,+ ) for g(x), then we should consider the end behaviours of the two functions: For f(x) = x 3, lim x , f(x) and lim x - , f(x) - For g(x) = ln(x), lim x 0+, g(x) - and lim x , g(x) So, on OPEN intervals, absolute extrema MAY NOT exist 5/2/2015Calculus - Santowski9

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(D) Terms – Relative or Local We say that f(x) has a relative (or local) maximum at x = c if f(x) < f(c) for every x in some open interval around x = c. Now, sketch an example of what has just been described. We say that f(x) has a relative (or local) minimum at x = c if f(x) > f(c) for every x in some open interval around x = c. Now, sketch an example of what has just been described. 5/2/2015Calculus - Santowski10

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(E) Terms - Diagram To understand the terms, a visualization will help: 5/2/2015Calculus - Santowski11

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(F) Example Problem solving approach for finding absolute extrema of f(x) on [a,b]. 1. Verify that the function is continuous on [a,b]. 2. Find all critical points of f(x) that are in the interval [a,b]. This makes sense if you think about it. Since we are only interested in what the function is doing in this interval we don’t care about critical points that fall outside the interval. 3. Evaluate the function at the critical points found in step 1 and the end points. 4. Identify the absolute extrema. 5/2/2015Calculus - Santowski12

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(F) Example #1 Determine the absolute extrema for the following function and interval: g(x) = 2x 3 + 3x 2 – 12x + 4 on [-4,2] 5/2/2015Calculus - Santowski13

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(F) – Example #2 Suppose that the population (in thousands) of a certain kind of insect after t months is given by P(t) = 3t + sin(4t) + 100 Determine the minimum and maximum population in the first 4 months. 5/2/2015Calculus - Santowski14

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(F) Example #3 Suppose that the amount of money in a bank account after t years is given by, Determine the minimum and maximum amount of money in the account during the first 10 years that it is open. 5/2/2015Calculus - Santowski15

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(G) Example Find the absolute extrema of g(x) = sin(x)cos(2x) on the interval [-3 /4, /3] Step 1 => determine d/dx g(x) in order to work toward the critical numbers Step 2 => determine max/min values Step 3 => evaluate function values at end points Step 4 => the largest value found in steps 2&3 is the absolute max and the smallest value is the absolute min 5/2/2015Calculus - Santowski16

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(G) Example Find the absolute extrema of g(x) = sin(x)cos(2x) on the interval [-3 /4, /3] To find d/dx g(x): 5/2/2015Calculus - Santowski17

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(G) Example Find the absolute extrema of g(x) = sin(x)cos(2x) on the interval [-3 /4, /3] ==> find critical numbers 5/2/2015Calculus - Santowski18

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(G) Example Find the absolute extrema of g(x) = sin(x)cos(2x) on the interval [-3 /4, /3] ==> evaluate 5/2/2015Calculus - Santowski19

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(G) Example So our 5 function values are: And the absolute max is and the absolute min is at 5/2/2015Calculus - Santowski20

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(H) Practice Find the absolute extrema of the following functions: 5/2/2015Calculus - Santowski21

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(I) Internet Links Max Min Values from Paul Hawkins at Lamar U Extreme Values from Visual Calculus Extreme Value Theorem from Pink Monkey 5/2/2015Calculus - Santowski22

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(J) Homework Textbook, S6.1, p331 (1) Graphs, Q1-8 (2) Algebra, Q10-32 as needed + variety 5/2/2015Calculus - Santowski23

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(D) Extrema and Continuity We now consider the importance of continuity using the functions y = tan(x) and y = 1/x 2 5/2/2015Calculus - Santowski24

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(D) Extrema and Continuity If we have discontinuity in y = tan(x) at x = π/2 and at x = 0 for g(x) = 1/x 2, then we should consider the behaviours of the two functions at the discontinuity: For f(x) = tan(x), lim x π/2-, f(x) - and lim x π/2+, f(x) + For g(x) = 1/x2, lim x 0+, g(x) + and lim x 0-, g(x) + Since the function values increase (or decrease) without bound (+ ), there clearly is NO max or min value for the function 5/2/2015Calculus - Santowski25

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