Quantcast

A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

mlddmlnog

  • 2 years ago

verify that the hypothesis of the Mean value theorem are satisfied on the given interval and findall values of c that satisfy theconclusion of the theorem. f(x)=x^2+x [-4,6]

  • This Question is Closed
  1. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @zepdrix

  2. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Do you understand how the Meal-Value Theorem works? :O

  3. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    hm.. i'll try to do this one myself and ask for help if i still can't get the answer :)

  4. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1358281444723:dw|Ok but here's a brief explanation of the concept c: just in case it helps.

  5. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1358281503651:dw|

  6. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Remember the formula for finding the slope of a line between 2 points? (a secant line) If we use the notation from the graph,\[\large m=\frac{f(b)-f(a)}{b-a}\] In order for the function to be continuous, this slope must be equal to the slope of a TANGENT line somewhere in the middle.\[\large \frac{f(b)-f(a)}{b-a}=f'(c)\] A derivative at a particular point represents the slope of a tangent line.

  7. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok, i need help... i dont know how to do this.. haha i thought i did but i dont. so, please continueee :)

  8. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So in this problem we're starting with a secant line, passing through x=-4 and x=6. Let's start by finding the slope \(m\) of that line.\[\large m=\frac{f(6)-f(-4)}{6-(-4)}\]

  9. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[\large m=\frac{\color{orangered}{f(6)}-\color{cornflowerblue}{f(-4)}}{6-(-4)}\] \[\large m=\frac{(\color{orangered}{6^2+6})-(\color{cornflowerblue}{(-4)^2-4})}{6+4}\]

  10. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Understand how those got plugged in?

  11. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    YES :)

  12. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Simplifying that down, I think we end up withhhhhhhhh 3ish.

  13. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes. i got 3.

  14. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So the theorem says - In order for this function to be continuous from -4 to 6, there must be a TANGENT line somewhere between those points that has this slope of 3.

  15. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Let's first find f'(x).

  16. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    What'd u get? :O

  17. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    2x+1 :)

  18. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    k cool :D

  19. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So then,\[\large f'(c)=2c+1\]And we're claiming, that by the Mean-Value Theorem this tangent line at x=c has a slope of 3. It has to happen somewhere between -4 and 6 in order for the function to be continuous. So let's set our f'(c) equal to 3 and solve for c. \[\large 2c+1=3\]

  20. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    c=1

  21. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    When you get an answer, verify whether or not it is BETWEEN -4 and 6. Because that will help to verify that we didn't make a mistake somewhere.

  22. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    since it's 1, it is between -4 and 6 :)

  23. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yay team c:

  24. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes. but is that the final answer? i thought we were supposed to use the mean value theorem.. which says: if F is continuous on [z,b], and if F is an anti-derivative of F on [a,b], then \[\int\limits_{a}^{b}f(x)dx=f(b)-f(a)\].

  25. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oops, i meant [a,b]..

  26. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Oh crap maybe i confused it with the intermediate value theorem... sec.. thinking :3

  27. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ahh darn. haha ok ;p

  28. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    hmm no it doesn't appear i did... hmmmm

  29. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    The thing you posted is the FTC (Fundamental Theorem of Calculus, Part 2).\[\large \int\limits_a^b f(x)dx=F(b)-F(a)\] I'm not quite sure what that has to do with the mean value theorem D: hmmm

  30. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh sorryyy yea thta IS the fundamental theorem of calculus.. ha but the mean value theorem of integrals says : if f is conb], then there exists at least one number x* in [a,b] such that \[\int\limits_{a}^{b}f(x)dx=f(x*)(b-a).\]

  31. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    if f is continuous on [a,b] i meant.

  32. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Hmm crap I dunno D: I'm familiar with seeing it that way.

  33. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    aw darn it.. is there anyone you could ask help for? because that formula is what my teacher uses... :'(

  34. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @hartnn @hba @campbell_st Maybe one of these fellas know what you're talking about c:

  35. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ahh thankss :)

  36. mlddmlnog
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh nvermind! we got the correct answer. so i';lljust do it the way @zepdrix did it! :D thank you for comming anyway! :)

  37. hartnn
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ohh..okay :)

  38. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.