Use Newton's method to find the coordinates, correct to six decimal places, of the point on the parabola y =(x − 5)^2 that is closest to the origin.
To find the point on the parabola y = (x - 5)^2 that is closest to the origin, we can use Newton's method to minimize the distance between the origin and a point on the parabola.
Let's set up the problem and solve it step by step:
Step 1: Define the distance function
The distance between the origin (0, 0) and a point on the parabola can be calculated using the distance formula:
d = sqrt(x^2 + y^2)
Step 2: Minimize the distance function
We want to minimize the distance function d with respect to x. So, we need to find the value of x that minimizes d. To do this, we can minimize the square of the distance function, which is equivalent and easier to work with:
f(x) = x^2 + y^2 = x^2 + (x - 5)^2
Step 3: Find the derivative of f(x)
To use Newton's method, we need to find the derivative of f(x) with respect to x:
f'(x) = 2x + 2(x - 5)
Simplifying the equation, we get:
f'(x) = 4x - 10
Step 4: Set up the Newton's method iteration formula
Newton's method iteration formula is given by:
x[n+1] = x[n] - f(x[n]) / f'(x[n])
where x[n] is the nth approximation of x and x[n+1] is the (n+1)th approximation.
Step 5: Apply Newton's method
Let's start with an initial guess for x, denoted as x[0]. We'll use x[0] = 0 as a starting point:
x[0] = 0
Now, we can apply Newton's method to find subsequent approximations of x:
x[n+1] = x[n] - f(x[n]) / f'(x[n])
Applying this iteration process, we can find the approximations for x until we achieve the desired accuracy.
Step 6: Find the y-coordinate
Using the x-coordinate obtained from the previous step, we can find the corresponding y-coordinate using the equation of the parabola:
y = (x - 5)^2
Step 7: Round the coordinates to six decimal places
Finally, round the x and y coordinates to six decimal places to match the desired accuracy.
By following these steps, you will be able to find the coordinates of the point on the parabola nearest to the origin using Newton's method.