Increasing & Decreasing Functions and 1 st Derivative Test Lesson 4.3

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 ) f(x) a The function is said to be strictly increasing

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

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)  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

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

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

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 4.3A 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 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