{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.


y = 4√x and y = x² + 1

Verified step by step guidance

1. First, set the equations equal to each other to find the intersection points: \( 4\sqrt{x} = x^2 + 1 \). This will give us the equation to solve: \( 4\sqrt{x} - x^2 - 1 = 0 \).
2. Define the function \( f(x) = 4\sqrt{x} - x^2 - 1 \). We need to find the roots of this function, which correspond to the intersection points of the curves.
3. Calculate the derivative of the function, \( f'(x) \), which is needed for Newton's method. The derivative is \( f'(x) = \frac{2}{\sqrt{x}} - 2x \).
4. Choose initial approximations for \( x \) based on a preliminary graph or analysis. For example, you might choose \( x_0 = 1 \) and \( x_0 = 4 \) as starting points.
5. Apply Newton's method iteratively using the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \) until the values converge to a satisfactory level of accuracy. Repeat this process for each initial approximation to find all intersection points.