absolute minimum and maximums by: hannah ahluwalia and anita vellaichamy
TRANSCRIPT
ABSOLUTE MINIMUM AND MAXIMUMS
By: Hannah Ahluwalia and Anita Vellaichamy
LESSON OBJECTIVES
In this lesson, you will learn how to find
critical numbers analytically and
graphically, as well as using them to
determine the absolute minimum and
maximum values of a function
INTRODUCTION
In order to continue with the lesson,
you must know how to find the
derivative of a function
EXAMPLE
Find the derivative of a
function through the
power rule
INTRODUCTION PART 2
You must also know how to solve for
the zeroes of a function. This is done
simply by setting an equation equal to
zero and solving for x.
EXAMPLE 2
CRITICAL NUMBERS
Now that you know these basics, we
may move into the first part of the
lesson: finding critical numbers
CRITICAL NUMBERS CONT.
A critical point of a function is where
or
CRITICAL POINTS CONT.
To find the critical points of a function
analytically, you must take the
derivative of the function. Once you
have the derivative, set it equal to zero
and solve for X
CRITICAL POINTS EX. 1
Points A and B are
critical points because A
B
CRITICAL POINT EX. 2
These are your critical numbers
NOW TRY ONE FOR YOURSELF!
Find the critical points
of the function.
CRITICAL POINT QUIZ
A.
B. ,
C.
D.
LESSON CONT.
Now that you know how to find
critical points let’s learn how to you
them! If you want to review critical
points, feel free to go back at any time. Review Continue
LESSON CONT.
You must plug each x-value into the
function. This includes critical points
that you solved for, and the endpoints
of the function.
Continue
Back
ABS. MAXIMUM AND MINIMUM EX.1
Function:
1. Get the derivative (you can use the power rule here):
2. Factor the derivative you found:
3. Set each equal to zero to find the critical points:
This function has two critical points.
ContinueBack
http://tutorial.math.lamar.edu/Classes/CalcI/AbsExtrema.aspx
EXAMPLE CONT.
Next, lay out what you need to solve, and evaluate.
critical points
endpoints
Abs. minimumAbs. maximum
Continue
Back
FINISHING THE EXAMPLE
Now state your conclusion, depending on what the
question asks you, you may need to say it one of two
ways:
1. The abs. maximum occurs at and the abs. minimum
occurs at
2. The abs. maximum value is , and the abs. minimum
value is Back Continue
NOW YOU TRY ONE!
Start by finding the critical points
of:
Review how to find these Continue
to answer
https://mathway.com/examples/Calculus/Applications-of-Differentiation/Finding-the-Absolute-Maximum-and-Minimum-on-the-Given-Interval?id=825
DID YOU GET IT?
𝑥=0,2If no, try againIf yes, continue
EXAMPLE CONT.
Now plug the x- values into the
equation and find the absolute
maximum and minimum.
Back Continue
EXAMPLE CONT.
Did you get the abs. maximum at
with a value of and an abs.
minimum at and a value of ?
If no, try again
If yes, continue
LESSON CONT.
If you are finding the abs. maximum
and minimum by using your calculator,
you will need to graph the function to
find the critical points. Back to example problem
Continue with lesson
CALCULATOR EX 1.
Function: ,
Start again by finding the derivative:
Next, graph the derivative function, to see where it crosses
the x- axis ( these x- values are the critical points). Make
sure to use the interval they gave you.
BackContinue to example
http://tutorial.math.lamar.edu/Classes/CalcI/AbsExtrema.aspx
EXAMPLE CONT.
To help identify critical points on the graph, you can
use the following calculator directions: 2nd ,
calculate, zero, click left bound, right bound, and
one more time for the guess.
You should find these as the critical values:
Back
Continue
EXAMPLE CONT.
Next, test the values into the equation, include critical points and
endpoints.
𝑓(0)=100.0
𝑓(0.604)=102.4756
𝑓(0.9661)=102.2368
𝑓(2.1755)=107.1880
𝑓(2.5369)=106.9492
𝑓( 3.7463)=111.9004
𝑓(4)=111.7121
Abs. minimum
Abs. maximum
Back Finish the example
FINISH THE EXAMPLE
To finish identify the Abs. minimum and Abs.
maximum.
Abs. maximum occurs at , with a value of
Abs. minimum occurs at , with a value of
Try one on your own
Back
NOW YOU TRY ONE!
First find the derivative of
Review how to find derivative
Continue to answer
http://archives.math.utk.edu/visual.calculus/3/max.1/3.html
PROBLEM CONT.
Did you get this as the derivative?
If no, try again If yes, continue
PROBLEM CONT.
Did you get these as the critical points?
If no, try again
If yes, continue
PROBLEM CONT.
Were these your final answers?
Abs. maximum at with a value of
Abs. minimum at with a value of
If yes, continue
If no, try again
GOOD JOB!
If you want to review, go back, if
you would like to move forward to
the quiz, click continue. Review from beginning of lesson
Continue to quiz
QUESTION 1
What is the critical points of 𝑓 (𝑥 )=𝑥2−5 𝑥+7 , [−1,3 ]
2.5 8.4 4.2
Review topic http://archives.math.utk.edu/visual.calculus/3/max.1/1.html
QUESTION 2
Find the Abs. maximum value of
**Use calculator for this question http://archives.math.utk.edu/visual.calculus/3/max.1/4.html
Review
QUESTION 3
Find the Abs. minimum value of
http://archives.math.utk.edu/visual.calculus/3/max.1/2.html
Review
QUESTION 4
Now find both extrema of
**Use calculator for this problemContinue to first part of question
http://archives.math.utk.edu/visual.calculus/3/max.1/7.html
QUESTION 4 CONT.
Abs. maximum is at
14
𝑥=?
Review
QUESTION 4 CONT.
Abs. minimum is at
Review
GREAT JOB!
Now you can find both absolute
extrema, but feel free to review if you
need to.
Back to beginning Finis
h
INCORRECT!
Try this one again!
INCORRECT!
Try this one again!
Back to quiz question 1
CORRECT!
After taking the derivative of the function, you
should get
Then, set this equal to zero.
And solve!
You should get ,
CORRECT!
Next questionBack to question 1
CRITICAL NUMBERS CONT.
A critical point of a function is where
or
Back to quiz question 1
CRITICAL NUMBERS CONT.
A critical point of a function is where
or
Back to lesson
EXAMPLE
Find the derivative of a
function through the
power rule
Back to example
CRITICAL POINT EX. 2
These are your critical numbers
Back to example
ABS. MAXIMUM AND MINIMUM EX.1
Function:
1. Get the derivative (you can use the power rule here):
2. Factor the derivative you found:
3. Set each equal to zero to find the critical points:
This function has two critical points.
Back
INCORRECT!
Try this one again!
Back to quiz question 2
CORRECT!
Abs. Maximum
Continue to question 3Back to question 2
EXAMPLE CONT.
Next, lay out what you need to solve, and evaluate.
critical points
endpoints
Abs. minimumAbs. maximum
Back to question 2
EXAMPLE CONT.
Next, lay out what you need to solve, and evaluate.
critical points
endpoints
Abs. minimumAbs. maximum
Back to question 3
INCORRECT!
Try this one again!
Back to quiz question 3
CORRECT!
Abs. minimum value
Continue to question 4
Back to question 3
EXAMPLE CONT.
Next, lay out what you need to solve, and evaluate.
critical points
endpoints
Abs. minimumAbs. maximum
Back to question 4 part 1
INCORRECT!
Try this one again!
Back to quiz question 4 part 1
CORRECT!
Abs. Maximum
Continue to next part of question 4
Back to part 1
INCORRECT!
Try this one again!
Back to quiz question 4 part 2
CORRECT!
Abs. Minimum
Finish quiz
Back to part 2
EXAMPLE CONT.
Next, lay out what you need to solve, and evaluate.
critical points
endpoints
Abs. minimumAbs. maximum
Back to question 4 part 2
FINISH!