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Math Help - [SOLVED] Racetrack Problems

  1. #1
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    Question [SOLVED] Racetrack Problems

    PLEASE HELP ASAP!!!!!!!!!!
    1). Consider the functions f(x)=lnx and g(x)=x−1. These are continuous and differentiable for x>0. In this problem we use the Racetrack Principle to show that one of these functions is greater than the other.

    (a) Find a point c such that f(c)=g(c). c=???

    (b) Find the equation of the tangent line to f(x)=lnx at x=c for the value of c that you found in (a).
    For part a, I thought the answer was c=1, but that's not correct.



    2). Suppose that f(t) is continuous and twice-differentiable for t>=0. Further suppose f''(t)<=9 for all t>=0 and f(0)=f'(0)=0.

    Using the Racetrack Principle, what linear function g(t) can we prove is greater than than f(t) (for t>=0)?
    g(t)= ???

    Then, also using the Racetrack Principle, what quadratic function h(t) can we prove is greater than than f(t) (for t>=0)?
    h(t)= ???

    I have NO clue!
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  2. #2
    ux0
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    Quote Originally Posted by iheartthemusic29 View Post
    PLEASE HELP ASAP!!!!!!!!!!
    1). Consider the functions f(x)=lnx and g(x)=x−1. These are continuous and differentiable for x>0. In this problem we use the Racetrack Principle to show that one of these functions is greater than the other.

    (a) Find a point c such that f(c)=g(c). c=???

    (b) Find the equation of the tangent line to f(x)=lnx at x=c for the value of c that you found in (a).
    For part a, I thought the answer was c=1, but that's not correct.

    Why is c=1 not correct?

    a)
    f(1)=ln(1)=0
    g(1)=1-1=0

    b) the slop of the tangent line is just f'(x)= \frac{1}{x} ... use that to figure out the equation of the line..
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  3. #3
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    I think there's an error in the answer key, because, as you said, c should equal 1.

    Can someone help me with question 2??? I don't even know where to begin on this one. Please! And thanks!
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  4. #4
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    Never mind! I figured it out.
    g(t)=9t and h(t)=(9/2)t^2
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