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 4.8.58a

Chapter 4, Problem 4.8.58a

{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.

Verified step by step guidance

Start by setting up the equation for a fixed point. A fixed point of the function f(x) = ax(1 - x) is a value of x such that f(x) = x. Therefore, set ax(1 - x) = x.

Rearrange the equation ax(1 - x) = x to form a quadratic equation. This can be done by expanding the left side to get ax - ax^2 = x, and then moving all terms to one side: ax - ax^2 - x = 0.

Factor the quadratic equation ax - ax^2 - x = 0. Start by factoring out x from the terms: x(a - ax - 1) = 0. This gives two potential solutions: x = 0 or a - ax - 1 = 0.

Solve the equation a - ax - 1 = 0 for x. Rearrange it to find x in terms of a: ax = a - 1, which simplifies to x = (a - 1)/a, provided a ≠ 0.

Determine the values of a for which the fixed points are valid. Since 0 ≤ a ≤ 4, check the conditions for x = 0 and x = (a - 1)/a to be valid fixed points. Note that x = 0 is always a fixed point, and x = (a - 1)/a is valid when a > 1.

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.

Fixed Points

A fixed point of a function f(x) is a value x such that f(x) = x. This means that when the function is applied to this value, it returns the same value. Finding fixed points often involves solving the equation f(x) - x = 0. In the context of the given function, identifying fixed points helps determine the values of a for which the function intersects the line y = x.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax^2 + bx + c. In this case, the function f(x) = ax(1 - x) can be expanded to a quadratic form, which allows for the analysis of its properties, such as its vertex, axis of symmetry, and roots. Understanding the behavior of quadratics is essential for finding fixed points and analyzing their stability.

Parameter Variation

Parameter variation involves changing the values of parameters in a function to observe how the function's behavior changes. In this problem, the parameter a affects the shape and position of the quadratic function f(x). By varying a within the specified range (0 ≤ a ≤ 4), we can determine how many fixed points exist and their corresponding values, which is crucial for solving the problem.

Recommended video:

03:52

Critical Points Example 2

Related Practice

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The function f(x) = √x has a local maximum on the interval [0,∞).

234

views

Textbook Question

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

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

347

views

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.

346

views

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.

997

views

Textbook Question

{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.

173

views

Textbook Question

{Use of Tech} Every second counts You must get from a point P on the straight shore of a lake to a stranded swimmer who is 50 from a point Q on the shore that is 50 m from you (see figure). Assuming that you can swim at a speed of 2 m/s and run at a speed of 4 m/s, the goal of this exercise is to determine the point along the shore, x meters from Q, where you should stop running and start swimming to reach the swimmer in the minimum time. <IMAGE>

a. Find the function T that gives the travel time as a function of x, where 0 ≤ x ≤ 50.

202

views