Question

[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).
27. Converging to different zeros Use Newton's method to find the zeros of f(x)=4x^4-4x^2 using the given starting values.
c. x_0 = 0.8 and x_0 = 2, lying in (√2/2, ∞)

Accepted Answer

Step 1: Recall Newton's method formula: x_{n+1} = x_n - f(x_n) / f'(x_n). This iterative method is used to approximate the zeros of a function by starting with an initial guess $$x_0. $$Step 2: Compute the derivative of the given function f(x) = $$4x^4$$ - $$4x^2. $$Using the power rule, f'(x) = $$16x^3 - 8x. $$Step 3: Substitute the given initial values $$x_0 = 0.8 $$and $$x_0 = 2 $$into the Newton's method formula. For each starting value, calculate f($$x_0) $$and f'($$x_0$$), then compute the next iteration $$x_1$$ using the formula x_{n+1} = x_n - f(x_n) / f'(x_n). Step 4: Repeat the iterative process for each starting value ($$x_0 = 0.8 $$and $$x_0 = 2) $$until the difference between successive approximations (|x_{n+1} - x_n|) is smaller than a chosen tolerance (e.g., 10^-6). This ensures convergence to a zero of the function. Step 5: Analyze the results. For $$x_0 = 0.8$$, the method should converge to a zero in the interval (√2/2, ∞). Similarly, for $$x_0 = 2$$, the method should converge to another zero in the same interval. Use technology (e.g., a calculator or computer) to perform the iterations efficiently.

Verified video answer for a similar problem:

Key Concepts

Newton's Method

Zeros of a Function

Convergence