Using Derivatives to Sketch the Graph of a Function Lesson 4.3

How It Was Done BC (Before Calculators) How can knowledge of a function and it's derivative help graph the function? How much can you tell about the graph of a function without using your calculator's graphing? Regis might be calling for this information!

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

Monotone Function Theorem 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

Monotone Function Theorem 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

Monotone Function Theorem 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

Concavity Concave UP Concave DOWN Inflection point: Where concavity changes 

Inflection Point Consider the slope as curve changes through concave up to concave down  Slope starts negative Becomes less negative Slope becomes (horizontal) zero Slope becomes positive, then more positive At inflection point slope reaches maximum positive value After inflection point, slope becomes less positive Graph of the slope

Inflection Point What could you say about the slope function when the original function has an inflection point Graph of the slope Slope function has a maximum (or minimum Thus second derivative = 0 Slope function has a maximum (or minimum Thus second derivative = 0

Second Derivative This is really the rate of change of the slope When the original function has a relative minimum –Slope is increasing (left to right) and goes through zero –Second derivative is positive –Original function is concave up

Second Derivative When the original function has a relative maximum –The slope is decreasing (left to right) and goes through zero –The second derivative is negative –The original function is concave down

Second Derivative If the second derivative f ’’(x) = 0 –The slope is neither increasing nor decreasing If f ’’(x) = 0 at the same place f ’(x) = 0 –The 2 nd derivative test fails –You cannot tell what the function is doing Not an inflection point

Example Consider Determine f ‘(x) and f ’’(x) and when they are zero

Example f(x) f ‘(x) f ‘’(x)  f ‘’(x) = 0 this is an inflection point f ‘(x) = 0, f ‘’(x) < 0 this is concave down, a maximum f ’(x) = 0, f’’(x) > 0, this is concave up, a relative minimum

Example Try f ’(x) = ? f ’’(x) = ? Where are relative max, min, inflection point?

Algorithm for Curve Sketching Determine critical points –Places where f ‘(x) = 0 Plot these points on f(x) Use second derivative f’’(x) = 0 –Determine concavity, inflection points Use x = 0 (y intercept) Find f(x) = 0 (x intercepts) Sketch

Assignment Lesson 4.3 Page 214 Exercises 9 – 43 odd, 47, 49, 53