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Higher Derivatives Concavity 2 nd Derivative Test

Higher Derivatives Concavity 2 nd Derivative Test. Lesson 5.3. Think About It. Just because the price of a stock is increasing … does that make it a good buy? When might it be a good buy? When might it be a bad buy? What might that have to do with derivatives?. Think About It.

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Higher Derivatives Concavity 2 nd Derivative Test

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  1. Higher DerivativesConcavity2nd Derivative Test Lesson 5.3

  2. Think About It • Just because the price of a stock is increasing … does that make it a good buy? • When might it be a good buy? • When might it be a bad buy? • What might that have to do with derivatives?

  3. Think About It • It is important to knowthe rate of the rate of increase! • The faster the rate of increase, the better. • Suppose a stock price is modeled by • What is the rate of increase for several months in the future?

  4. Think About It • Plot the derivative for 36 months • The stock is increasing at a decreasing rate • Is that a good deal? • What happens really long term? Consider the derivative of this function … it can tell us things about the original function

  5. Higher Derivatives • The derivative of the first derivative is called the second derivative • Other notations • Third derivative f '''(x), etc. • Fourth derivative f (4)(x), etc.

  6. Find Some Derivatives • Find the second and third derivatives of the following functions

  7. Velocity and Acceleration • Consider a function which gives a car's distance from a starting point as a function of time • The first derivative is the velocity function • The rate of change of distance • The second derivative is the acceleration • The rate of change of velocity

  8. Point of Inflection where function changes from concave down to concave up Concavity of a Graph • Concave down • Opens down • Concave up • Opens up

  9. Concavity of a Graph • Concave down • Decreasing slope • Second derivativeis negative • Concave up • Increasing slope • Second derivative is positive

  10. Test for Concavity • Let f be function with derivatives f ' and f '' • Derivatives exist for all points in (a, b) • If f ''(x) > 0 for allx in (a, b) • Then f(x) concave up • If f ''(x) < 0 for all x in (a, b) • Then f(x) concave down

  11. Test for Concavity Strategy • Find c where f ''(c) = 0 • This is the test point • Check left and right of test point, c • Where f ''(x) < 0, f(x) concave down • Where f ''(x) > 0, f(x) concave up • Try it

  12. Determining Max or Min • Use second derivative test at critical points • When f '(c) = 0 … • If f ''(c) > 0 • This is a minimum • If f ''(c) < 0 • This is a maximum • If f ''(c) = 0 • You cannot tell one way or the other!

  13. Assignment • Lesson 5.3 • Page 345 • Exercises 1 – 85 EOO

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