Find the point (x,y) on the curve y=sqrt(x+4) closest to the origin.
The distance from a point on the line from the origin is
((y-0)^2 + (x-0)^2)^0.5 =
(x^2+y^2)^.5 = (x^2 +(x+4)^0.5)
evaluate d/dx((x^2 +(x+4)^0.5)) = 0
This is the x value of the point closest to the origin, plug x into y = sqrt(x+4) to find the y value of this point
To find the point (x, y) on the curve y = √(x + 4) that is closest to the origin (0, 0), we need to minimize the distance between the points (x, y) and (0, 0).
The distance between two points (x₁, y₁) and (x₂, y₂) can be calculated using the distance formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, we want to minimize the distance d between the point (x, y) on the curve and the origin (0, 0). Therefore, we want to minimize the expression:
d = √(x² + y²)
Substituting the equation of the curve, we have:
d = √(x² + (√(x+4))²)
= √(x² + (x + 4))
= √(2x² + 4x + 4)
To find the minimum distance, we need to find the minimum value of the expression inside the square root:
f(x) = 2x² + 4x + 4
To find the minimum of f(x), we can take the derivative of the function and set it to zero:
f'(x) = 4x + 4
Setting f'(x) = 0:
4x + 4 = 0
4x = -4
x = -1
So, x = -1 is a critical point.
To determine if this critical point is a minimum, we can take the second derivative of f(x) and evaluate it at x = -1.
f''(x) = 4
Since f''(x) = 4 > 0, the critical point x = -1 is a minimum point.
To find the corresponding y-coordinate, we substitute x = -1 into the equation of the curve:
y = √((-1) + 4)
= √3
Therefore, the point (x, y) on the curve y = √(x + 4) closest to the origin is (-1, √3).
To find the point (x, y) on the curve y = sqrt(x+4) that is closest to the origin, we need to minimize the distance between the origin (0, 0) and any point on the curve.
Let's denote the distance between the origin (0, 0) and the point (x, y) on the curve as d.
The distance formula between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2-x1)^2 + (y2-y1)^2)
In this case, x1 = 0, y1 = 0 (origin), x2 = x, and y2 = sqrt(x+4).
Therefore, the distance formula becomes:
d = sqrt((x - 0)^2 + (sqrt(x+4) - 0)^2)
Simplifying the equation inside the square root:
d = sqrt(x^2 + (x+4))
Now, we need to minimize this distance d. To do so, we can find the minimum value by taking the derivative of d with respect to x and setting it equal to zero:
d/dx (d) = d/dx (sqrt(x^2 + x + 4)) = 0
To simplify the derivative, we can rewrite the equation as:
d^2/dx^2 (x^2 + x + 4) = 0
Expanding and simplifying:
2x + 1 = 0
Solving for x:
x = -1/2
Now, substitute x = -1/2 back into the equation y = sqrt(x+4) to find the corresponding y-coordinate:
y = sqrt(-1/2 + 4)
y = sqrt(7/2)
Therefore, the point (x, y) on the curve y = sqrt(x+4) closest to the origin is approximately (-1/2, sqrt(7/2)).