To find the point on the graph of a function that is closest to a specific point, we can use the concept of minimizing the distance between the function’s point and that specific point in a coordinate system. For the function f(x), we’ll denote the point we want to minimize the distance to as P(2, 12).
Here are the steps to find that nearest point:
- Define the distance formula: The distance D between any point on the function (x, f(x)) and the point P(2, 12) is given by the formula:
- Square the distance: To make calculations easier, we square the distance to avoid dealing with the square root:
- Differentiate: Take the derivative of the squared distance D² with respect to x. Set this derivative to zero and solve for x:
- Find critical points: The solution from the derivative will give you the x-coordinate of the point on the graph that minimizes the distance between (x, f(x)) and P(2, 12).
- Evaluate the function: Once you have the value of x, substitute it back into the original function to find the corresponding y-coordinate, which gives you the coordinates of the closest point.
D = √[(x – 2)² + (f(x) – 12)²]
D² = (x – 2)² + (f(x) – 12)²
d(D²)/dx = 0
For example, if our function is f(x) = x², we would follow the steps outlined. First, calculate f(x)
In conclusion, this approach allows us to efficiently find the point on the graph of the function that is closest to the specified point (2, 12) using calculus techniques!