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 4.8.58b

Chapter 4, Problem 4.8.58b

{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).

b. Consider the polynomial g(x) = f(f(x)). Write g in terms of a and powers of x. What is its degree?

Verified step by step guidance

First, understand that a fixed point of a function f(x) is a value x such that f(x) = x. For the function f(x) = ax(1 - x), we need to find f(f(x)) to form the polynomial g(x).

Start by substituting f(x) into itself: f(f(x)) = a(f(x))(1 - f(x)). This means we need to substitute f(x) = ax(1 - x) into the expression again.

Calculate f(x) = ax(1 - x) and substitute it into the expression: f(f(x)) = a(ax(1 - x))(1 - ax(1 - x)).

Simplify the expression: Expand the terms inside the parentheses and multiply them out. This will involve expanding (1 - ax(1 - x)) and multiplying it by ax(1 - x).

Determine the degree of the polynomial g(x): After simplifying, observe the highest power of x in the expression. The degree of g(x) will be the highest power of x present in the expanded form.

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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 is a value of x for which the function evaluates to the same value, meaning f(x) = x. This concept is crucial in understanding the behavior of functions, particularly in iterative processes and stability analysis. In the context of the given function f(x) = ax(1 - x), finding fixed points involves solving the equation ax(1 - x) = x.

Composition of Functions

The composition of functions involves applying one function to the result of another. In this case, g(x) = f(f(x)) means we first apply f to x, and then apply f again to the result. Understanding function composition is essential for manipulating and simplifying expressions, especially when dealing with polynomials and their degrees.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It provides insight into the polynomial's behavior, such as the number of roots and the end behavior of the graph. For the polynomial g(x) derived from f(f(x)), determining its degree involves analyzing the composition and identifying the maximum exponent of x in the resulting expression.

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Related Practice

Textbook Question

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

b. Find the position of the object for all relevant times.

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

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Textbook Question

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>

c. Find the time at which the object passes the rest position for the second time.

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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.

c. For arbitrary positive values of K and b, when does the maximum growth rate occur (in terms of K and b)?

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Textbook Question

c. Find the time when the object reaches its highest point. What is the height?

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

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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>

b. Find the critical point of T on (0, 50).

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Textbook Question

{Use of Tech} Let f(x) = ln((x+1)/(x-1)) and g(x) = ln ((x+1)/(x-1)).

b. Sketch graphs of f and g to show that these functions do not differ by a constant.

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