mean value theorem

1. Mean Value
Theorem
Here is where the lesson begins

2. What is the mean
value theorem?
The mean value theorem, often abbreviated as
MVT, is a fundamental concept in calculus. It
serves as a bridge between the average rate of
change of a function and its instantaneous rate of
change. In essence, the MVT tells us that if a
function is continuous on a closed interval [a, b]
and differentiable on the open interval (a, b), then
at some point 'c' within the open interval, the
instantaneous rate of change (derivative) of the
function is equal to the average rate of change
over the closed interval [a, b]

3. Applying the mean value theorem
Example problem 1
Scenario: Imagine a car is traveling
along a straight road, and you want
to calculate its average and
instantaneous velocity during a
specific time interval
Velocity of a moving car
Given:
■ Initial time, t1 = 2 seconds
■ Final time, t2 = 6 seconds
■ Initial position, s(t1) = 10 meters
■ Final position, s(t2) = 50 meters

4. Step-by-step solution
What is the average and instantaneous velocity of the car between t1 and t2?
■ Average velocity = (change in position) / (change in time)
■ Average velocity = [s(t2) - s(t1)] / [t2 - t1]
■ Average velocity = [50 m - 10 m] / [6 s - 2 s] = 40 m/s / 4 s = 10 m/s
Step 1: calculate average velocity
1

5. Step-by-step solution
What is the average and instantaneous velocity of the car between t1 and t2?
■ Let v(t) represent the velocity of the car
■ According to the MVT, there exists a time 'c' between t1 and t2 where v'(c) =
[v(t2) - v(t1)] / [t2 - t1]
■ We have the average velocity (10 m/s) as calculated above
■ Therefore, v'(c) = 10 m/s
The average velocity of the car between 2 seconds and 6 seconds is 10 meters per
second. By the mean value theorem, there exists a time 'c' within this interval
where the instantaneous velocity is also 10 meters per second
Step 2: apply the MVT to find instantaneous velocity
2

6. Applying the mean value theorem
Example problem 2
Scenario: You have a function f(x) = x^3 - 3x^2 + 2 on the interval [0, 3]. Prove the
existence of at least one point 'c' within this interval where the instantaneous rate of
change (derivative) equals the average rate of change
Proving the existence of a turning point

7. Step-by-step solution
Example problem 2
■ The function f(x) is continuous on the closed interval [0, 3]
■ The function f(x) is differentiable on the open interval (0, 3)
Step 1: check MVT conditions
1

8. Step-by-step solution
Example problem 2
■ Average rate of change = [f(3) - f(0)] / (3 - 0)
■ Average rate of change = [(3^3 - 3(3^2) + 2) - (0^3 - 3(0^2) + 2)] / 3
■ Average rate of change = [27 - 27 + 2] / 3 = ⅔
Step 2: calculate average rate of change
2

9. Step-by-step solution
Example problem 2
■ According to the MVT, there exists at least one point 'c' in the interval
(0, 3) where f'(c) = average rate of change
■ We need to find 'c' such that f'(c) = 2/3
Step 3: apply the MVT
3

10. Final conclusion
Example problem 2
By the mean value theorem, there exists at least one point 'c' within the interval (0,
3) where the instantaneous rate of change (derivative) of f(x) is equal to the
average rate of change, which is 2/3
Final conclusion

11. CREDITS: This presentation template was created by
Slidesgo, and includes icons by Flaticon, infographics &
images by Freepik and content by Swetha Tandri
Thanks!
Do you have any questions?
youremail@freepik.com
+91 620 421 838
yourwebsite.com
Please keep this slide for attribution

12. Alternative resources
Here’s an assortment of alternative resources whose style fits that of this template:
■ Gradient abstract geometric cover template

13. Resources
Did you like the resources used in this template? Get them on these websites:
Vectors
■ Gradient abstract geometric cover template
■ Gradient grainy colorful texture

14. Instructions for use
If you have a free account, in order to use this template, you must credit Slidesgo by keeping the Thanks slide. Please
refer to the next slide to read the instructions for premium users.
As a Free user, you are allowed to:
● Modify this template.
● Use it for both personal and commercial projects.
You are not allowed to:
● Sublicense, sell or rent any of Slidesgo Content (or a modified version of Slidesgo Content).
● Distribute Slidesgo Content unless it has been expressly authorized by Slidesgo.
● Include Slidesgo Content in an online or offline database or file.
● Offer Slidesgo templates (or modified versions of Slidesgo templates) for download.
● Acquire the copyright of Slidesgo Content.
For more information about editing slides, please read our FAQs or visit our blog:
https://slidesgo.com/faqs and https://slidesgo.com/slidesgo-school

15. As a Premium user, you can use this template without attributing Slidesgo or keeping the Thanks slide.
You are allowed to:
● Modify this template.
● Use it for both personal and commercial purposes.
● Hide or delete the “Thanks” slide and the mention to Slidesgo in the credits.
● Share this template in an editable format with people who are not part of your team.
You are not allowed to:
● Sublicense, sell or rent this Slidesgo Template (or a modified version of this Slidesgo Template).
● Distribute this Slidesgo Template (or a modified version of this Slidesgo Template) or include it in a database or in
any other product or service that offers downloadable images, icons or presentations that may be subject to
distribution or resale.
● Use any of the elements that are part of this Slidesgo Template in an isolated and separated way from this
Template.
● Register any of the elements that are part of this template as a trademark or logo, or register it as a work in an
intellectual property registry or similar.
For more information about editing slides, please read our FAQs or visit our blog:
https://slidesgo.com/faqs and https://slidesgo.com/slidesgo-school
Instructions for use (premium users)

16. This presentation has been made using the following fonts:
Epilogue
(https://fonts.google.com/specimen/Epilogue)
Albert Sans
(https://fonts.google.com/specimen/Albert+Sans)
#000000 #efefef #f8b546 #f68d56
#415ab2 #dd9fe7 #f578ae #666666
#71dafd
#ffffff
Fonts & colors used

17. Create your Story with our illustrated concepts. Choose the style you like the most, edit its colors, pick
the background and layers you want to show and bring them to life with the animator panel! It will
boost your presentation. Check out how it works.
Storyset
Pana Amico Bro Rafiki Cuate

18. You can easily resize these resources without losing quality. To change the color, just ungroup the resource and click
on the object you want to change. Then, click on the paint bucket and select the color you want. Group the resource again
when you’re done. You can also look for more infographics on Slidesgo.
Use our editable graphic resources...

21. JANUARY FEBRUARY MARCH APRIL MAY JUNE
PHASE 1
PHASE 2
Task 1
Task 2
Task 1
Task 2
JANUARY FEBRUARY MARCH APRIL
PHASE 1
Task 1
Task 2

24. You can resize these icons without losing quality.
You can change the stroke and fill color; just select the icon and click on the paint bucket/pen.
In Google Slides, you can also use Flaticon’s extension, allowing you to customize and add even more icons.
...and our sets of editable icons

25. Educational Icons Medical Icons

26. Business Icons Teamwork Icons

27. Help & Support Icons Avatar Icons

28. Creative Process Icons Performing Arts Icons

29. Nature Icons

30. SEO & Marketing Icons