Critical Points and Inc & Dec Intervals Miyo and Fareeha

Pre Req Knowledge  How to find the derivative of a function  Product Rule:  Quotient Rule  Chain Rule  Trig Rules:

Pre Req Review Try this derivative (click for answer) Now try a chain rule derivative (Answer) work

Lesson Objective  Now that we’ve done some review, you’re ready to begin the main lesson  We will teach you how to apply the derivative to find critical points of a function, and find on which intervals the function is increasing/decreasing

Critical points…  What is a critical point?  F has a critical point where .  is undefined f’(x)=0 here, at x=-1, so the critical point is at x=-1

Increasing/Decreasing Intervals…  When is increasing? When is it decreasing?  is increasing when is positive  is decreasing when is negative f’(x) is negative here, and as you can see, f(x) is decreasing here also f’(x)Is positive here, and as you can see, f(x) is increasing here also

Now we’ll show you how to find all of this analytically [without the graph]

Steps to find critical points analytically 1)Derive f(x) 2)Factor (if needed) 3)Find where f’(x)=0 or where f’(x) is undefined 1)Simply set f’(x) equal to 0 2)If it is a fraction, you must set the numerator and denominator equal to 0, so that you can find where f’(x) is undefined 4)Solve for x

Example 1 Lets use the equation from the graph before: You were right! The critical point is at x=-1!  Derive  Factor  Set  Solve for x

How to find if f(x) is increasing or decreasing on an interval 1) Find the critical points of f(x) 2) Place all critical points on a number line 3) Test values in between the critical points in f’(x) to determine if the interval is increasing or decreasing 1) if f’(x) > 0, then f is increasing on this interval 2) If f’(x) < 0, then f is decreasing on this interval

Example 1 (cont.) Now let’s apply those steps to our example:  Critical points  Place critical points on a number line  Test a value on either side of the critical point (plug it in to f’(x)  Determine your answer Let’s test the points x=-5 and x=0, since they are on either side of -1 Since f’(-5) is negative, f is decreasing on this interval (-∞,-1) Since f’(0) is positive, f is increasing on this interval (-1,∞) - +

Example 1 Conclusion  F(x) has a critical point at x=-1  F(x) is decreasing on (-∞,-1) increasing on (-1,∞)

Example 2  Derive f(x)  Find the critical points  Find where f’(x) is equal to zero OR where it is undefined Let’s try another problem!

Example 2 (cont.)  Choose points between the critical points to test—let’s use 01

Example 2 Conclusion  F(x) has critical points at x= -1, 0, 1  F(x) is increasing on (- ∞, -1) and (1, ∞)  F(x) is decreasing on (-1,1)

Lesson Review  How to find critical points: 1) Find f’(x) 2) Find where f’(x)=0 or where f’(x) is undefined  How to find if f(x) is increasing or decreasing between critical points 1) Place critical points on a number line 2) Test points between the critical points by plugging them into f’(x) 1) if f’(x) > 0, then f is increasing on this interval 2) If f’(x) < 0, then f is decreasing on this interval

Quiz!! Now we’re going to test your knowledge. Grab a pencil and a piece of paper!

Question 1  Determine the intervals where f(x) is increasing and decreasing for F(x) is increasing on (- ∞,1) and (-1/3, ∞), and decreasing on (-1, -1/3) F(x) is increasing on (-1, -1/3), and decreasing on (- ∞,1) and (-1/3, ∞) F(x) is increasing on (-3, -1), and decreasing on (∞, -3) and (-1,∞,)

Oops!! Try again! Click for Hint Click to go back to quiz Have you found the critical points yet? Try putting your critical points on a number line! 3 Did you get the right derivative? 12

Yay!! Correct!!! Next Question!

Question 2  Determine the intervals where g(x) is increasing and decreasing for g(x) is increasing on (- ∞,-1), and decreasing on (-1, ∞) g(x) is increasing on (- ∞, ∞) g(x) is increasing on (-∞,3), and decreasing on (3,∞)

Oops!! Try again! Remember, to find the critical points of fractions, you need to set both the numerator and the denominator of f’(x) equal to zero! Try plugging in -1 to find if g is inc or dec on (-∞.0). Then try that for the other intervals. Click for Hint 12 Click to go back to quiz

Yay!! Correct!!!

Question 3  Determine the intervals on [-6,6] that f(x) is increasing and decreasing for F(x) is increasing on (- 6,0) and decreasing on (0,6) F(x) is increasing on (2, 6), and decreasing on (- 6,2) F(x) is increasing on (-6, 4), and decreasing on (4,6)

Oops!! Try again! Remember the product rule and chain rule! Remember! The number line must be restricted on the domain [-6,6] Click for Hint 12 Click to go back to quiz

Yay!! Correct!!!

Question 4  Determine the intervals on [0, π ] where f(x) is increasing and decreasing for F(x) is increasing on (0, ), and decreasing on (, π ) F(x) is increasing on (0, π ) F(x) is increasing on (, π ), and decreasing on (0, )

Oops!! Try again! Click for Hint 12 Derivative of sinx is cosx! Unit Circle Labeled In 30° Increments With Values | ClipArt ETC etc.usf.edu1024 × 1024Search by image Unit Circle Labeled In 30° Increments With Values Use the unit circle to find the critical points! Click to go back to quiz

Yay!! Correct!!!