Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 55b

Chapter 4, Problem 55b

{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?

Verified step by step guidance

Step 1: Understand the problem. We want to approximate the reciprocal of a number 'a' using Newton's method. The function we will use is f(x) = 1/x - a, where 'a' is the number whose reciprocal we want to find.

Step 2: Set up Newton's method. Newton's method is an iterative process given by the formula: x_{n+1} = x_n - f(x_n)/f'(x_n). We need to find the derivative of f(x) = 1/x - a.

Step 3: Calculate the derivative f'(x). The derivative of f(x) = 1/x - a is f'(x) = -1/x^2. This will be used in the Newton's method formula.

Step 4: Choose a starting value x_0. Since we are approximating the reciprocal of 7, a reasonable starting value might be x_0 = 0.1, as it is close to the reciprocal of 7.

Step 5: Apply Newton's method iteratively. Use the formula x_{n+1} = x_n - (1/x_n - 7)/(-1/x_n^2) to update x_n. Continue iterating until the approximation reaches eight digits of accuracy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 formula x_{n+1} = x_n - f(x_n)/f'(x_n), where f is the function and f' is its derivative. This method is particularly useful for finding roots of functions, such as approximating reciprocals.

Function and Derivative

In calculus, a function represents a relationship between inputs and outputs, while its derivative measures the rate of change of the function at any given point. For the function f(x) = 1/x - a, the derivative f'(x) = -1/x^2 is essential for applying Newton's Method, as it helps determine the slope of the tangent line at the current approximation.

Convergence and Accuracy

Convergence refers to the process by which an iterative method approaches a final value or solution. In the context of Newton's Method, achieving eight digits of accuracy means that the approximation is refined until the result is stable to that level of precision. Understanding how quickly and accurately the method converges is crucial for effectively using it to approximate values.

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Derivatives

Related Practice

Textbook Question

Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest growing plants in the world.

b. Based on the Mean Value Theorem, what can you conclude about the instantaneous growth rate of bamboo measured in millimeters per second between 10:00 A.M. and 3:00 P.M.?

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Textbook Question

Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest growing plants in the world.

a. A bamboo shoot was 500 cm tall at 10:00 A.M. and 515 cm tall at 3:00 P.M. Compute the average growth rate of the bamboo shoot in cm/hr over the period of time from 10:00 A.M. to 3:00 P.M.

Textbook Question

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x - 2 tan⁻¹ x on [-√3,√3)

Textbook Question

{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.

Textbook Question

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

Textbook Question

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

c. Find the rate at which water flows from the tank and plot the flow rate function.