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Math Help - Finding k of a Tangent Line

  1. #1
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    Finding k of a Tangent Line



    Maybe we can walk through this? At first glance, the first thing I would do is find the derivative of y=5x+45, but that's not right.
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  2. #2
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    Quote Originally Posted by BeSweeet View Post
    At first glance, the first thing I would do is find the derivative of y=5x+45, but that's not right.
    first of all set the derivative - (k\sqrt{x})' = 5 and solve for x.
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    Quote Originally Posted by pickslides View Post
    first of all set the derivative - (k\sqrt{x})' = 5 and solve for x.
    I'm still pretty lost :\.
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    Quote Originally Posted by BeSweeet View Post
    I'm still pretty lost :\.
    y = k\sqrt{x} find \frac{dy}{dx}

    Then make \frac{dy}{dx}=5 as that is the gradient of the tangent.

    Now solve for x
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  5. #5
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    Quote Originally Posted by pickslides View Post
    y = k\sqrt{x} find \frac{dy}{dx}

    Then make \frac{dy}{dx}=5 as that is the gradient of the tangent.

    Now solve for x
    That's one of the things... We never learned the whole \frac{dy}{dx} thing. We just used:
    (f(x+h)-f(x))/h

    ...
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  6. #6
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    If you haven't learned "the whole dy/dx thing", why did you refer to finding the derivative in your first post? The formula you give is for the slope of a straight line. If f(x) is not linear, it is the slope of a "secant" line, a line that crosses the graph at x and x+y, not the tangent line. Since you don't yet know the derivative (I suspect this problem is preliminary to introducing the derivative), here's a method that predates the calculus:

    If y= 5x+ 45 and y= k\sqrt{x} meet at all, we must have y= 5x+ 45= k\sqrt{x} or 5x- k\sqrt{x}+ 45= 0. Let u= \sqrt{x} so that x= u^2. The equation becomes 5u^2- ku+ 45= 0. If x= a at the point of intersection then x- a= u- \sqrt{a} must be a factor of that polynomia. If fact, to be tangent that must be a double factor- we must have 5u^2- ku+ 45= 5(u- \sqrt{a})^2 Multiplying the right side, we get 5u^2- ku+ 45= 5u^2- 10\sqrt{a} u+ 5a for all u. Comparing coefficients, -k= -10\sqrt{a} and 5a= 45. Solve the second equation for a and use that to solve the first equation for k.
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