# math

posted by .

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

• math -

using distance from (2,1/2) to y=x^2,

d^2 = (x-2)^2 + (y-1/2)^2
= (x-2)^2 + (x^2 - 1/2)^2
= x^2 - 4x + 4 + x^4 - x^2 + 1/4
= x^4 - 4x + 17/4

d = sqrt(x^4 - 4x + 17/4)
d' = 2(x^3 - 1)/sqrt(x^4 - 4x + 17/4)
d' = 0 when x = 1

so, (2,1/2) is closest to (1,1)
______________________

using normal line, we want the line from (2,1/2) normal to the curve. The distance from P to the curve will be minimum along the line normal to the curve.

At (x,y) the slope of x^2 = 2x
so, the normal has slope -1/2x

So, the equation of the normal line from (2,1/2) is

(y - 1/2)/(x-2) = -1/2x

2x(x^2 - 1/2) = -(x-2)
2x^3 - x = -x + 2
2x^3 = 2
x = 1

So, the normal line from (2,1/2) is the line to (1,1)

## Similar Questions

1. ### 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 …
2. ### 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 …
3. ### 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) = ( ) ?
4. ### Calculus

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

If the point P is on the graph of a function f, find the corresponding point on the graph of the given function. P(4,9) y=1/3f(1/2x)-1
9. ### calculus

1. Which of the following expressions is the definition of the derivative of f(x) = cos(x) at the point (2, cos(2))?
10. ### 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 …

More Similar Questions