View
217
Download
1
Category
Preview:
Citation preview
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
63)( 24 xxxf
xxxf 64)(' 3
Try this derivative (click for answer)
23 )4()( xxxf Now try a chain rule derivative
(Answer)
(Answer)
work)43()4(2)(' 3 xxxxf
)43)(4(2)(' 3 xxxxf
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
0)(' xf)(' xf
f’(x)=0 here, at x=-1, so the critical point is at x=-1 963)( 2 xxxf
Increasing/Decreasing Intervals… When is increasing? When is it
decreasing? is increasing when is positive is decreasing when is negative
)(xf
)(xf )(' xf)(xf )(' xf
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
963)( 2 xxxf
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
undefined1)Simply set f’(x) equal to 02)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
963)( 2 xxxf
66)(' xxf
1
)1(60
)1(6)('
x
x
xxf
0)(' xf
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 interval2) 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
1x
66)(' xxf
-1
Let’s test the points x=-5 and x=0, since they are on either side of -1
24
6)5(6)5('
f
-1
Since f’(-5) is negative, f is decreasing on this interval (-∞,-1)
6
6)0(6)0('
f
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
0
0
1
1
10
1
12
)1)(1()2()('
1)(
2
2
2
2
2
2
22
2
2
2
x
x
x
x
x
x
x
x
xx
x
xxxxf
x
xxf
Let’s try another problem!
Example 2 (cont.)
Choose points between the critical points to test—let’s use
-1 0 1
102
12
1
10
x
x
x
x
100
99
100
1100)10('
3
41
141
)2
1('
3
41
141
)2
1('
100
99
100
1100)10('
f
f
f
f
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 points1) Place critical points on a number line2) Test points between the critical points by
plugging them into f’(x)1) if f’(x) > 0, then f is increasing on this interval2) 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 142)( 23 xxxf
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?
xxxf 86)(' 2
1 2
Question 2 Determine the intervals where g(x) is
increasing and decreasing for 3
2 1)(
x
xxg
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
1 2
Click to go back
to quiz
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)
xxxf 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
1 2
Click to go back
to quiz
Question 4 Determine the intervals on [0,π] where f(x) is
increasing and decreasing for 30sin5)( xxf
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,
)
2
2
2
2
Oops!! Try again!
Click for Hint
1 2
Derivative of sinx is cosx!
Use the unit circle to find the critical points!
Click to go back to quiz
Yay!! Correct!!!
You’re Done!
Now you know how tofind critical points and
inc/dec intervals!
Recommended