Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.8.36a

Chapter 4, Problem 4.8.36a

{Use of Tech} Investment problem A one-time investment of $2500 is deposited in a 5-year savings account paying a fixed annual interest rate r, with monthly compounding. The amount of money in the account after 5 years is a(r) = 2500(1 + r/12)⁶⁰.

a. Use Newton’s method to find the value of r if the goal is to have $3200 in the account after 5 years.

Verified step by step guidance

Identify the function for which we need to find the root. We want to find the interest rate r such that the amount in the account after 5 years is \$3200. Set up the equation: a(r) = 2500(1 + r/12)^60 = 3200.

Rearrange the equation to form a function f(r) whose root we need to find: f(r) = 2500(1 + r/12)^60 - 3200 = 0.

Apply Newton's method, which is an iterative process to find successively better approximations to the roots of a real-valued function. The formula is: r_{n+1} = r_n - f(r_n) / f'(r_n).

Calculate the derivative f'(r) of the function f(r). Use the chain rule and power rule to differentiate: f'(r) = 2500 * 60 * (1 + r/12)^59 * (1/12).

Choose an initial guess for r, say r_0, and use the Newton's method formula iteratively: r_{n+1} = r_n - (2500(1 + r_n/12)^60 - 3200) / (2500 * 60 * (1 + r_n/12)^59 * (1/12)). Continue the iterations until the change in r is sufficiently small.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Compound Interest Formula

The compound interest formula calculates the amount of money in an account after a certain period, considering the principal, interest rate, and compounding frequency. In this problem, the formula a(r) = 2500(1 + r/12)⁶⁰ represents the future value of a $2500 investment compounded monthly over 5 years. Understanding this formula is crucial for setting up the equation to solve for the interest rate r.

Newton's Method

Newton's Method is an iterative numerical technique used to find approximate solutions to equations. It involves starting with an initial guess and refining it using the formula xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ). In this context, Newton's Method helps find the interest rate r by iteratively solving the equation 2500(1 + r/12)⁶⁰ = 3200, where f(r) is the difference between the desired amount and the formula's output.

Derivative Calculation

Calculating the derivative is essential for applying Newton's Method, as it requires the derivative of the function involved. For the function a(r) = 2500(1 + r/12)⁶⁰, the derivative with respect to r helps determine the slope of the tangent line at any point r. This derivative is used in Newton's Method to adjust the guess for r, moving closer to the solution with each iteration.

Recommended video:

05:44

Derivatives

Related Practice

Textbook Question

Folded boxes

a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.

Textbook Question

Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.

a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?

Textbook Question

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for x < 3

Textbook Question

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for x < -1

Textbook Question

Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.

a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.

Textbook Question

Two poles of heights m and n are separated by a horizontal distance d. A rope is stretched from the top of one pole to the ground and then to the top of the other pole. Show that the configuration that requires the least amount of rope occurs when Θ₁ = Θ₂ (see figure). <IMAGE>