Calculus

posted by .

Find the point on the graph of function that is closest to the point.
f(x)=x^2 (2,1/2)

  • Calculus -

    From the given point P0(x0,y0)=(2,1/2), we draw a line to join a point on the curve
    f(x)=x^2 at P(x,y).

    The distance P-P0 will be shortest when P-P0 is perpendicular to the tangent of f(x) at P.

    The slope of P-P0 is given by:
    m1=(y0-y)/(x0-x)
    =(y0-x^2)/(x0-x) since P is on curve f(x)=x^2.

    The tangent of the curve f(x) at P is given by
    m2=f'(x)=2x

    Perpendicularity requires that m1*m2=-1, or
    [(y0-x^2)/(x0-x)]*(2x)=-1
    which simplifies to a cubic equation
    x³-1=0 which factorizes easily to
    (x-1)(x²+x+1)=0
    The first factor gives x=1, and the second factor results in two complex roots which do not concern us.
    Therefore the required point is P(1,1), or P(1,1) is the point on f(x) that is closest to the given point (1, 1/2).

  • Calculus -

    Thanks!

  • Calculus :) -

    You're welcome!

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. calculus

    let f be the function f(x) = x^3 + 3x^2 - x + 2 a. the tangent to the graph of f at the point P = (-2,8) intersects the graph of f again at the point Q. Find the coordinates of point Q. b. Find the coordinates of point R, the inflection …
  2. calculus

    Find the point on the graph of f(x)=√x that is closest to the value (4,0). (differentiate the square of the distance from a point (x,√x) on the graph of f to the point (4,0).)
  3. math

    Find the point on the graph of the function closest to the given point. Function f(x) = xsquared Point (2, 1/2)
  4. Math

    Find the point on the graph of the function that is closest to the given point. Function Point f(x) = x^2 (16, 1/2) (x, y) = ( ) ?
  5. Calculus

    find the point on the graph of the function that is closest to the given point: f(x)=(x+1)^2 (5,3)
  6. calculus

    optimization find the point on the graph of the function that is closest to the given point f(X)= square root of x point:(8,0)
  7. Calculus

    Find the point P on the graph of the function y=\sqrt{x} closest to the point (2,0)
  8. calculus

    1. Which of the following expressions is the definition of the derivative of f(x) = cos(x) at the point (2, cos(2))?
  9. calculus

    8. Find the derivative of the function f(x) = cos^2(x) + tan^2(x) 9. Find f '(x) for f(x) = xln(x) - 5ex. 10. Find f '(x) for f (x) = e(^lnx) + e^(x)ln(x) 11. Find the x-coordinate where the graph of the function f(x) = e^(-cosx) has …
  10. graphing help?

    Graph ​ g(t)=4sin(3t)+2 ​. Use 3.14 for π . Use the sine tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point. How would …

More Similar Questions