Find the point on the curve y=x^(1/2) that is a minimum distance from the point (4,0).
My book says you use the distance formula.
Then you let L = D^2 because the minimum value of D^2 will occur at the same value of x as the minimum value of D.
What is L, though.
L is the distance^2. You don't have to do that way, as I will demonstrate.
D^2=(4- x)^2+(0-y)^2 that comes from the distance formula.
Doing it the way the L=D^2 did:
L= ..
dL/dx=0=2(4-x)+2(y)dy/dx
but dy/dx = d(sqrt x)/dx= 1/2sqrtx
so 0=-2x+2sqrtx/2sqrtx or
2x=2
x= 1/2, y= 1/sqrt2
Now, lets do it without the L substitution:
D^2=(4- x)^2+(0-y)^2 that comes from the distance formula.
2D dD/dx=0=2(4-x)+2(y)dy/dx
again, dy/dx= d(sqrtx)/dx= 1/(2sqrtx)
so 0=-2x+2sqrtx/2sqrtx
and again x=1/2, y= 1/sqrt2
They are saying, let D^2 = L
so when later on you differentiate
the result for L is simpler than that for D^2
They are using the property that if a > b
then a^2 > b^2.
let the closest point be P(x,y)
then
L = D^2 = (x-4)^2 + (y-0)^2
= (x-4)^2 + (x^(1/2))^2
= (x-4)^2 + x
dL/dx = 2(x-4) + 1 = 0 for a min distance
2x - 8 + 1 = 0
x = 7/2
if x=7/2 , then y = √(7/2) = √7/√2 = √14/2
the closest point is ((7/2 , √14/2)
In this problem, we are asked to find the point on the curve y = x^(1/2) that is at the minimum distance from the point (4,0). One way to approach this is by using the distance formula, which calculates the distance between two points in a coordinate plane.
The distance formula is given by:
D = √[(x2 - x1)^2 + (y2 - y1)^2]
For our problem, let's consider the point on the curve as (x, x^(1/2)), where x is the x-coordinate. Using the distance formula, we have:
D = √[(x - 4)^2 + (x^(1/2) - 0)^2]
To simplify the problem, we can square both sides of the equation to eliminate the square root:
D^2 = (x - 4)^2 + (x^(1/2))^2
Now, we can define a new variable L = D^2. This transformation helps us find the minimum value of D^2, which corresponds to the same value of x as the minimum value of D.
Therefore, L = (x - 4)^2 + x.
Now, the problem reduces to finding the value of x that minimizes L.