Find the point on the graph of function that is closest to the point.
f(x)=x^2 (2,1/2)
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).
You're welcome!
Well, I must say, finding that point on the graph must be quite a "square" task. But don't worry, I'm here to help you out and add some "function-al" humor along the way!
To find the point on the graph of the function f(x) = x^2 that is closest to the point (2, 1/2), we need to use a little bit of "distance magic". We're looking for the point on the graph that minimizes the "distance" between the two points.
So, let's get "crazy" with calculations! The distance between two points (x1, y1) and (x2, y2) can be found using the formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
In our case, (x1, y1) is (2, 1/2) and (x2, y2) is (x, x^2). So the distance becomes:
Distance = √((x - 2)^2 + (x^2 - 1/2)^2)
Now, to find the point on the graph that minimizes this distance, we need to minimize the above expression. It might sound a bit "complex," but hang in there!
To minimize this distance, we can take the derivative of the expression with respect to x, set it equal to zero, and solve for x. This will give us the x-coordinate of the point on the graph that is closest to (2, 1/2).
However, I hope you're patient enough to solve it on your own because this is where my "math skills" start to get a little "juggled"!
To find the point on the graph of the function f(x) = x^2 that is closest to the point (2, 1/2), you can follow these steps:
1. Start by writing the equation for the distance between two points. The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, (x1, y1) is the point on the graph of f(x) = x^2, and (x2, y2) is the given point (2, 1/2).
2. Since we're looking for the closest point, we want to minimize the distance. Thus, we'll minimize the distance formula d.
3. Substitute the y-coordinate of the given point into the equation for f(x):
f(x) = x^2
We have:
y = x^2
The y-coordinate of the given point is 1/2, so we set y equal to 1/2:
x^2 = 1/2
4. Solve the equation x^2 = 1/2 to find the x-coordinates of the points that are closest to (2, 1/2).
Taking the square root of both sides, we have:
x = ±sqrt(1/2)
Simplifying, we get:
x = ±sqrt(2)/2
Therefore, the x-coordinates of the points closest to (2, 1/2) on the graph of f(x) = x^2 are ±sqrt(2)/2.
5. To find the corresponding y-coordinates of these points, substitute the x-coordinates obtained in step 4 back into the equation for f(x):
f(x) = x^2
Plugging in x = sqrt(2)/2, we have:
y = (sqrt(2)/2)^2
= 2/4
= 1/2
Plugging in x = -sqrt(2)/2, we have:
y = (-sqrt(2)/2)^2
= 2/4
= 1/2
6. Therefore, the two points on the graph of f(x) = x^2 that are closest to the point (2, 1/2) are (sqrt(2)/2, 1/2) and (-sqrt(2)/2, 1/2).