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Increasing & Decreasing Functions and 1 st Derivative Test

Increasing & Decreasing Functions and 1 st Derivative Test. Lesson 4.3. f(x). a. Increasing/Decreasing Functions. Consider the following function For all x < a we note that x 1 <x 2 guarantees that f(x 1 ) < f(x 2 ). The function is said to be strictly increasing. f(x). a.

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Increasing & Decreasing Functions and 1 st Derivative Test

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  1. Increasing & Decreasing Functions and 1st Derivative Test Lesson 4.3

  2. f(x) a Increasing/Decreasing Functions • Consider the following function • For all x < a we note that x1<x2 guarantees that f(x1) < f(x2) The function is said to be strictly increasing

  3. f(x) a Increasing/Decreasing Functions • Similarly -- For all x > a we note that x1<x2 guarantees that f(x1) > f(x2) • If a function is either strictly decreasing or strictly increasing on an interval, it is said to be monotonic The function is said to be strictly decreasing

  4. 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

  5. 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

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

  7. f ‘(x) < 0on left f ‘(x) > 0on right 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 

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

  9. First Derivative Test • What if they are positive on both sides of the point in question? • This is called aninflection point 

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

  11. Assignment A • Lesson 4.3A • Page 226 • Exercises 1 – 57 EOO

  12. Application Problems • Consider the concentrationof a medication in thebloodstream t hours afteringesting • Use different methods to determine when the concentration is greatest • Table • Graph • Calculus

  13. 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?

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

  15. Assignment B • Lesson 4.3 B • Page 227 • Exercises 95 – 101 all

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