Calculus

Find the point on the graph of y = x^2 + 1 that’s closest to the point 8, 1.5. Hint: Remember the distance formula.

  1. 👍
  2. 👎
  3. 👁
  1. Distance to point, squared, is:
    R^2 = (x-8)^2 + (y-1.5)^2
    = (x-8)^2 + (x^2 -0.5)^2

    Solve for the x value when d(R^2)/dx = 0

    d/dx [x^2 -16x +64 + x^4 -x^2 + 1/4] = 0
    2x -16 +3x^3 -2x = 0
    3x^3 = 16
    x = 1.747
    y = 4.053

    1. 👍
    2. 👎
  2. Note that d/dx x^4 = 4x^3 not 3x^3
    From there on, we have

    2x - 16 + 4x^3 - 2x = 0
    4x^3 = 16
    x = cbrt(4)
    y = cbrt(16)+1

    Or, looking at things in another way, the normal line from point (p,q) on the curve will have the shortest distance to (8,1.5)

    At any point (p,q) on the curve, the slope is 2p, so the normal line has slope -1/2p

    Now we have a point and a slope:

    (y-q)/(x-p) = -1/2p
    y - p^2 - 1 = (p-x)/2p
    1.5 = (p-8)/2p + p^2 + 1
    3p = p - 8 + 2p^3 + 2p
    2p^3 = 8
    p^3 = 4
    p = cbrt(4)
    q = cbrt(16)+1

    1. 👍
    2. 👎
  3. Thanks Steve for noticing my error

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. linear algebra

    Let T be the plane 2x−3y = −2. Find the shortest distance d from the point P0=(−1, −2, 1) to T, and the point Q in T that is closest to P0. Use the square root symbol '√' where needed to give an exact value for your

  2. Linear algebra

    Let L be the line passing through the point P(−2, −2, 2) with direction vector d=[1, −2, 3]T. Find the shortest distance d from the point P0(−4, −4, −5) to L, and the point Q on L that is closest to P0. Use the square

  3. Calc

    Find the point on the line –2x+2y–3=0 which is closest to the point (–4–4). Please provide solution. ( , ) Thanks guys!

  4. calc

    let f be function given by f(x)= Ln(x)/x for all x> 0. the dervative of f is given by f'(x)= (1 - Ln(x))/x squared. a) write equation for the line tangent to the graph of f at x=e squared b) Find the x-coordinate of the critical

  1. Calculus

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

  2. Linear Algebra!

    Let T be the plane 3x−2z = 14. Find the shortest distance d from the point P0=(5, 5, −3) to T, and the point Q in T that is closest to P0. Use the square root symbol '√' where needed to give an exact value for your answer.

  3. Calculus

    Find the point on the line 4x+3y-4=0 which is closest to the point (5,2)

  4. math

    Find the point P on the graph of the function y=sqrt{x} closest to the point (10,0)

  1. college algebra, Please help!!

    Answer the following function. f(x)=2x^2-x-1 A. Is the point (-2,9) on the graph of f? B. If x equals 2, what is fx? What point(S) are on the graph of f? c. if f(x)= -1, what is x? what point(s) are on the graph of f? d. what is

  2. Math-Calculus

    Find the point on the line -2x + 4y + 3 =0 which is closest to the point (-2,1). Note: I have been struggling on this for hours!

  3. calc

    Find the point on the graph of y = x2+1 that is closest to the point (3,1). d = √[(x-3)^2+(y-1)^2] d = √[(x-3)^2+(x^2+1-1)^2] d = √[(x-3)^2+(x^2)^2] d = [(x-3)^2+x^4]^1/2 d' = 1/2 [(x-3)^2+x^4)^1/2 [2(x-3)+4x^3] 0 =

  4. math

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

You can view more similar questions or ask a new question.