{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 = 1/x and y = 4 - x²

Verified step by step guidance

1. Step 1: Set the equations equal to each other to find the intersection points. This means solving 1/x = 4 - x².
2. Step 2: Rearrange the equation to form a single function f(x) = 4 - x² - 1/x. We need to find the roots of this function, which represent the intersection points.
3. Step 3: Use preliminary graphing or analysis to estimate initial guesses for the intersection points. Graph both y = 1/x and y = 4 - x² to visually identify where they might intersect.
4. Step 4: Apply Newton's method, which uses the formula x_{n+1} = x_n - f(x_n)/f'(x_n). Calculate the derivative f'(x) = -2x + 1/x².
5. Step 5: Iterate using Newton's method starting from your initial guesses until the values converge to a satisfactory approximation of the intersection points.

Key Concepts

Newton's Method

Newton's Method is an iterative numerical technique used to find approximate solutions to equations. It starts with an initial guess and refines it using the function's derivative, converging to a root. The formula involves evaluating the function and its derivative at the current approximation, allowing for successive approximations that ideally get closer to the actual solution.

Intersection Points

Intersection points occur where two curves meet, meaning their y-values are equal for the same x-value. To find these points, one typically sets the equations of the curves equal to each other and solves for x. The solutions can then be substituted back into either original equation to find the corresponding y-values, yielding the coordinates of the intersection points.

Graphical Analysis

Graphical analysis involves plotting functions to visually identify features such as intersection points, behavior, and trends. By sketching the curves, one can gain insights into where they might intersect, which aids in selecting effective initial guesses for numerical methods like Newton's. This visual approach can simplify complex problems and enhance understanding of the relationships between functions.