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Integration given slope of tangent at specific point

  1. Nov 26, 2012 #1

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    1. The problem statement, all variables and given/known data
    Find f if f"(x)=12x2+2 for which the slope of the tangent line to its graph at (1,1) is 3.


    2. Relevant equations



    3. The attempt at a solution
    What I did first was found f(x)=x4+x2+cx+d (cx and d being constants of integration.) and from this point I attempted solving for cx and d but was stuck here. It may just be the wording of the problem but I'm not sure where else to go. How do I know what to set f(x) equal to in order to solve cx and d?

    Thank you for any suggestions/help.
     
  2. jcsd
  3. Nov 26, 2012 #2
    "The tangent to the graph of f through the point (α,β) is τ" means:

    (1) The graph of f passes through this point (α,β).
    (2) The value of f'(x) for x=α is τ.

    This gives you two equations, f(...)=... and f'(...)=..., and that's all you need to find the two unknowns, c and d in your case.
     
  4. Nov 26, 2012 #3

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    So once I find c and d I plug them into f(x). Which is my final answer. I managed to find c= -3 by solving f'(1)=3. When I go to solve d what am I setting f(x) equal to? I found f(x) to be x^4+x^2-3x+d. I set this equal to 3 and plugged in 1 for x and got d= 4 making my answer f(x)=x^4+x^2-3x+4 however that equal does not pass through the point (1,1)?
     
  5. Nov 26, 2012 #4

    Dick

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    If you want your curve to pass through (1,1) you want to set f(1)=1. Not f(1)=3.
     
  6. Nov 26, 2012 #5

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    Ok my final answer is f(x)=x4+x2-3x+2. Does that seem correct?
     
  7. Nov 26, 2012 #6
    You can check that quite easily by calculating f(1) and f'(1), which should be 1 and 3, respectively. (They are, so your solution is correct.)
     
  8. Nov 26, 2012 #7

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    Wonderful, thank you for the help guys.
     
  9. Nov 26, 2012 #8

    Mark44

    Staff: Mentor

    Make that f(x)=x4+x2-3x+2.

    To add to what Michael said, it's easy to check, and something you should get into the habit of doing. You've already done all the heavy lifting.

    There are basically three things you need to check:
    1. Is f''(x) = 12x2 + 2?
    2. Is f(1) = 1?
    3. Is f'(1) = 3?

    If you can answer yes to all three, then your solution is correct.
     
  10. Nov 29, 2012 #9

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    Thank you mark. I just had a hard time remember what statements meant what. If that makes sense. I did do the check and they all worked out thanks again.
     
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