A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • 5 years ago

Find the critical points for f(t) = t/(1 + t^2)

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

    Can you do the derivative of the equation? Set that equal to zero.

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

    Also identify points that will make the derivative undefined.

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

    thanks

  4. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Get your complete answer?

  5. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I am stuck on finding the crtitical points from (1-2t^2(1+t^2)^-1). I know that the factored out piece (1 + t^2)^-1 is where the derivative is undefined...

  6. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Let me check something. That isn't the derivative that I ended up with.

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

    okay

  8. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    You should end up with (1-t^2)/[(1+t^2)^2\

  9. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Then your critical points should be pretty easy to find from there.

  10. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Was your f'(t) = (1+t^2)^-1 - 2t^2(1+t^2)^2

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

    ?

  12. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I am not sure how you came to your derivative.

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

    \[f(t) = t/(1+t ^{2})\] define \[g(t) = t\] \[h(t) = 1 + t ^{2}\] Quotient Rule: \[f'(x) = [h(x)g'(x) - g(x)h'(x)]/[h(x)]^{2}\]

  14. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    You can also use the product rule, of course. This is just an easy way to annotate online.

  15. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    If you use the product rule, remember the chain rule with the negative exponent.

  16. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I got \[f'(x) = ((1+t^2)-2t^2)/(1+t^2)^2\]

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

    Correct. Simplify the numerator.

  18. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So now I have \[-2t^2/(1+t^2)\]

  19. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Not quite

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

    Oh! I know what I did-- the actual answer is \[(1-t^2)/(1+t^2)^2\]

  21. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    There ya go

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

    So, what values of t make this 0 or undefined?

  23. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    No value can make this undefined because t is squared in the denominator...

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

    No real value, right. i will, if you are allowed complex solutions. I'm guessing not, however.

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

    I can't multiply the denominator on each side to get rid of it, right?

  26. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I can graph it...

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

    You can, but it isn't needed. You have done all the steps you need.

  28. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    But the critical points are when I set f' equal to zero...

  29. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    A number line would be unrealistic...

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

    The derivative is 0 when \[1 - t ^{2} = 0\]

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

    Oh---we only set the numerator equal to zero? Does that apply for all fraction derivatives...because I know that if I set the denominator equal to zero, it would show the values where the function is undefined.

  32. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Right. If the numerator is 0, the fraction is 0. If the denominator is 0, the fraction is undefined. There are some interesting things that happen if both are 0, but you'll learn those later :D

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

    So I have my critical points as +/- 1. This means that one of them can be a global max/min when I test the critical points.

  34. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    They can be global, local, or neither.

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

    Thanks so much for helping me and staying online this entire time! I really appreciate it!

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

    My pleasure

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

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

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.