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Math Help - Intersection of tangent lines

  1. #1
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    Intersection of tangent lines

    1. f(x)= 2^x^2
    Find the point of the intersection of the tangent lines to f(x) at x=2 and -2. Can you explain why the function and these two tangent lines are symmetric?


    2. What is the approx. slope of the tangent to the curve  x^3+y^3=xy at x=1
    I got dy/dx for this problem...but i don't know how to get y when i plug in 1

    Thank you
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  2. #2
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    Quote Originally Posted by Linnus View Post
    1. f(x)= 2^x^2
    Find the point of the intersection of the tangent lines to f(x) at x=2 and -2. Can you explain why the function and these two tangent lines are symmetric?


    2. What is the approx. slope of the tangent to the curve  x^3+y^3=xy at x=1
    I got dy/dx for this problem...but i don't know how to get y when i plug in 1

    Thank you
    Hello,

    to 1.):

    To calculate the equation of the tangent lines you need a) the coordinates of the tangent point and b) the gradient of the graph of the function at the tangent point.

    f(x) = 2^{x^2}~\implies~f(2)=2^{2^2} = 16. Thus the tangent point is T_1(2, 16) and T_2(-2, 16)

    To calculate the gradient (=first derivative) I rearrange the term of the function to:

    f(x) = \left(e^{\ln(2)}\right)^{x^2}=e^{x^2 \cdot \ln(2)}~\implies~f'(x)=2x\cdot \ln(2) \cdot 2^{x^2}. Use chain rule!

    Therefore the slope of the tangent line is m_1=f'(2)=64 \cdot \ln(2) or m_2=f'(-2)=-64 \cdot \ln(2)

    Use the coordinates of the tangent point and the value of the slope to calculate the equation of the tangentline.

    t_1:y=64 \cdot \ln(2)\cdot x \underbrace{- 128 \cdot \ln(2) + 16}_{\text{-72.7228...}} and

    t_2:y=-64 \cdot \ln(2)\cdot x \underbrace{- 128 \cdot \ln(2) + 16}_{\text{-72.7228...}}

    Calculate the intersection between these 2 lines and you'll get C(0, - 128 \cdot \ln(2) + 16)

    Two graphs f and g are symmetric to the y-axis if their equations satisfy the condition
    f(x) = g(-x) for every x. You can easily show that the equation t_2 follows from t_1 if you change the x into (-x).

    to 2.:

    I can't help you to solve y - ył = 1

    Maybe you have to use Newton-Raphson method to get an approximate value for y.
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