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Increasing & Decreasing Functions and 1 st Derivative Test Lesson 4.3

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Increasing/Decreasing Functions Consider the following function For all x < a we note that x 1

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Increasing/Decreasing Functions Similarly -- For all x > a we note that x 1 f(x 2 ) If a function is either strictly decreasing or strictly increasing on an interval, it is said to be monotonic f(x) a The function is said to be strictly decreasing

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Test for Increasing and Decreasing Functions If a function is differentiable and f ’(x) > 0 for all x on an interval, then it is strictly increasing If a function is differentiable and f ’(x) < 0 for all x on an interval, then it is strictly decreasing Consider how to find the intervals where the derivative is either negative or positive

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Test for Increasing and Decreasing Functions Finding intervals where the derivative is negative or positive Find f ’(x) Determine where Try for Where is f(x) strictly increasing / decreasing f ‘(x) = 0 f ‘(x) > 0 f ‘(x) < 0 f ‘(x) does not exist Critical numbers

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Test for Increasing and Decreasing Functions Determine f ‘(x) Note graph of f’(x) Where is it pos, neg What does this tell us about f(x) f ‘(x) > 0 => f(x) increasing f ‘(x) f(x) decreasing

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First Derivative Test Given that f ‘(x) = 0 at x = 3, x = -2, and x = 5.25 How could we find whether these points are relative max or min? Check f ‘(x) close to (left and right) the point in question Thus, relative min f ‘(x) < 0 on left f ‘(x) > 0 on right

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First Derivative Test Similarly, if f ‘(x) > 0 on left, f ‘(x) < 0 on right, We have a relative maximum

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First Derivative Test What if they are positive on both sides of the point in question? This is called an inflection point

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Examples Consider the following function Determine f ‘(x) Set f ‘(x) = 0, solve Find intervals

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Assignment A Lesson 4.3A Page 226 Exercises 1 – 57 EOO

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Application Problems Consider the concentration of a medication in the bloodstream t hours after ingesting Use different methods to determine when the concentration is greatest Table Graph Calculus

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Application Problems A particle is moving along a line and its position is given by What is the velocity of the particle at t = 1.5? When is the particle moving in positive/negative direction? When does the particle change direction?

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Application Problems Consider bankruptcies (in 1000's) since 1988 Use calculator regression for a 4 th degree polynomial Plot the data, plot the model Compare the maximum of the model, the maximum of the data 1988198919901991199219931994 594.6643.0725.5880.4845.31042.1835.2

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Assignment B Lesson 4.3 B Page 227 Exercises 95 – 101 all

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