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04:16{Use of Tech} Approximating reciprocals To approximate the reciprocal of a number a without using division, we can apply Newton’s method to the function f(x) = 1/x - a.
b. Apply Newton’s method with a = 7 using a starting value of your choice. Compute an approximation with eight digits of accuracy. What number does Newton’s method approximate in this case?
{Use of Tech} Modified Newton’s method The function f has a root of multiplicity 2 at r if f(r) = f'(r) = 0 and f"(r) ≠ 0. In this case, a slight modification of Newton’s method, known as the modified (or accelerated) Newton’s method, is given by the formula xₙ + 1 = xₙ - (2f(xₙ)/(f'(xₙ), for n = 0, 1, 2, . . . . This modified form generally increases the rate of convergence.
b. Apply Newton’s method and the modified Newton’s method using x₀ = 0.1 to find the value of x₃ in each case. Compare the accuracy of these values of x₃.
Minimizing related functions Complete each of the following parts.
b. What value of x minimizes ƒ(x) = (x- a₁)² + (x - a₂)² + (x - a₃)² , for constants a₁, a₂, and a₃?
Verify that the following functions satisfy the conditions of Theorem 4.9 on their domains. Then find the location and value of the absolute extrema guaranteed by the theorem.
f(x) = 4x + 1/√x
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→1⁻ (x/(x-1) - x/(ln x)
Evaluate the following limits in two different ways: with and without l’Hôpital’s Rule.
lim_x→∞ (2x⁵ - x + 1) / (5x⁶ + x)
