Textbook Question
Optimal soda can
a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm³ that minimize the surface area.
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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 guidance1
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.
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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.
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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.
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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.
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Related Practice
Textbook Question
Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.
a. Evaluate g(2), h(2), g'(2), and h'(2).
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Textbook Question
Sketch a graph of a function f with the following properties.
f' < 0 and f" < 0, for x < 3
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Textbook Question
Do dogs know calculus? A mathematician stands on a beach with his dog at point A. He throws a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve his ball. Assume C is the point on the edge of the beach closest to the tennis ball (see figure). <IMAGE>
a. Assume the dog runs at speed r and swims at speed s, where r > s and both are measured in meters per second. Also assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes for the dog to get to the ball.
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Textbook Question
{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.
a. Plot a graph of the curve when a = 3.
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Textbook Question
{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).
a. Without using a calculator, find the values of a, with 0 ≤ a ≤ 4, such that f has a fixed point. Give the fixed point in terms of a.
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