Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Use the Intermediate Value Theorem to approximate the zero of \(f\) in the interval \([a, b]\). Give your approximation to the nearest tenth. (If you have a graphing utility, use it to help you approximate the zero.) $$f(x)=x^{5}+x+1, \quad[-1,0]$$

Step-by-Step Solution

Answer

The approximate zero of the function is ~-0.8.

Step 1: Evaluate the function at the endpoints

First calculate the value of the function at the endpoints of the interval, -1 and 0. So you find \(f(-1)\) and \(f(0)\). If \(f(-1) = (-1)^{5}+(-1)+1 = -1\) and \(f(0) = (0)^{5}+(0)+1 = 1\).

Step 2: Determine if 0 falls between the function's values

In step 1, it was found that the function takes the values -1 and 1 at the endpoints of the interval. Zero does indeed fall between these two values, therefore, according to the Intermediate Value Theorem, the function must take a value of 0 at some point within the interval [-1,0].

Step 3: Approximate the zero of the function

The zero of the function lies somewhere between -1 and 0. Given this is a fifth degree polynomial, the exact root might be complex to find algebraically. Therefore, the exercise suggests using a graphing utility or a method of approximation such as bisection to find the root to the nearest tenth, which is approximately -0.8.

Key Concepts

Function Evaluation

Grasping the idea of function evaluation is like learning to peek into a function's behavior at specific points. Imagine a function as a magical box that does mathematical operations on any number you put in. Every function consists of an equation with variables, like our given function, which is a polynomial:

• The function is defined as \( f(x) = x^5 + x + 1 \).
• The remarkable part of evaluating is simply replacing the \( x \) with chosen values to calculate what comes out.

When solving an exercise, evaluating at endpoints of an interval helps determine the direction the function is heading. Here, we've evaluated \( f(x) \) at both \( x = -1 \) and \( x = 0 \):

• At \( x = -1 \), the function gives \( -1 \).
• At \( x = 0 \), the output is \( 1 \).

These evaluations help illustrate where the function crosses the x-axis and therefore, hint towards roots in a particular interval.

Polynomials

Polynomials are a special type of mathematical expression consisting of variables and coefficients, and they exhibit fascinating behavior. When examining polynomials, especially those of a higher degree like fifth-degree ones, you're often delving into some intriguing patterns and results.

• The function we are discussing, \( f(x) = x^5 + x + 1 \), is a polynomial.
• It combines terms of varying powers of \( x \), which is typical of polynomial functions.

Understanding polynomials is pivotal because their shape isn't always straightforward; they can have many twists and turns. Recognize that each polynomial has:

• A degree, which is the highest power of \( x \). Here, it is 5.
• Various roots, which are values where the polynomial equals zero.

These properties are the key to uncovering where the polynomial crosses the x-axis—giving rise to its zeros or roots in a solution.

Root Approximation

Finding the root of a polynomial symbolizes finding an \( x \)-value where the function touches or crosses the x-axis, and this problem can sometimes be complex. The Intermediate Value Theorem assists in figuring out where this happens, suggesting that if a function thermometer changes signs across an interval, a "zero" lies somewhere in between.

• According to the theorem, there is a change between negative \( f(-1) \) to positive \( f(0) \).
• Therefore, there must be a root in between these values.

Sometimes it's hard to pinpoint the exact root mathematically, especially with higher degree polynomials. In our case:

• The problem recommends graphing or using numerical means like bisection to close in on this zero.
• Through such techniques, it's been approximated to be around \( -0.8 \), lending an insightful estimation without needing precise algebraic methods.

Root approximation is a crucial tool in understanding the veins of complex functions, hinting at certain values without deriving an exact solution every time.