Increasing Decreasing Functions and 1 st Derivative Test

Increasing & Decreasing Functions and 1 st Derivative Test Lesson 4. 3

Increasing/Decreasing Functions • Consider the following function f(x) a • 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

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

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

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

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

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 minf ‘(x) < 0 f ‘(x) > 0 on left on right

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

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

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

Assignment A • Lesson 4. 3 A • Page 226 • Exercises 1 – 57 EOO

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

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?

Application Problems • Consider bankruptcies (in 1000's) since 1988 1989 1990 1991 1992 1993 1994 594. 6 643. 0 725. 5 880. 4 845. 3 1042. 1 835. 2 • 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

Assignment B • Lesson 4. 3 B • Page 227 • Exercises 95 – 101 all