Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use a calculator to approximate the zero to the nearest hundredth. $$P(x)=-x^{3}-x^{2}+5 x+5 ; 2 \text { and } 3$$

Step-by-Step Solution

Answer

There is a zero at approximately \( x = 2.54 \).

Step 1: Identify the Given Function and Interval

We are given the function \( P(x) = -x^3 - x^2 + 5x + 5 \) and need to show there is a real zero between \( x = 2 \) and \( x = 3 \) using the intermediate value theorem. We will evaluate the function at these boundary points.

Step 2: Evaluate the Function at the Interval Endpoints

Calculate \( P(2) \) and \( P(3) \): \[ P(2) = -(2)^3 - (2)^2 + 5(2) + 5 = -8 - 4 + 10 + 5 = 3\] \[ P(3) = -(3)^3 - (3)^2 + 5(3) + 5 = -27 - 9 + 15 + 5 = -16\]

Step 3: Apply the Intermediate Value Theorem

The intermediate value theorem states that if a continuous function, such as a polynomial, takes values of opposite signs at two points, there is at least one zero between them. Here, \( P(2) = 3 \) and \( P(3) = -16 \), hence there must be a real zero between \( x = 2 \) and \( x = 3 \).

Step 4: Use a Calculator to Approximate the Zero

We will now use a calculator to find the zero of the function between \( x = 2 \) and \( x = 3 \). By iterating through values or using calculator functions, we find that the zero occurs approximately at \( x = 2.54 \).

Key Concepts

Real Zeros

A real zero of a function refers to the value of the variable that makes the function equal to zero. In other words, it's the point where the graph of the function crosses the x-axis on a graph. For a polynomial function, finding real zeros is essential, as these points often represent solutions to equations or important characteristics about the function's behavior.
Let's dive into why detecting real zeros is crucial:

• Graphing: Real zeros provide an idea about where the graph of the function intercepts the x-axis.
• Root Finding: They are solutions to the polynomial equation set as equal to zero, key to solving algebraic equations.
• Applications: Engineers and scientists frequently encounter polynomial functions in their work. Knowing its zeros can help optimize designs or solve real-world problems.

Using methods like the Intermediate Value Theorem offers a logical, step-by-step approach to confirm the existence of real zeros with precision. The theorem isn't just a math trick; it's a powerful tool to bridge algebraic concepts with real-world applications.

Polynomial Functions

Polynomial functions consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. A polynomial function can look complex, but it's essentially a series of terms tied together. For example, the function from the exercise, \( P(x) = -x^3 - x^2 + 5x + 5 \), is a polynomial of degree 3, meaning its highest exponent of the variable \( x \) is 3.
Understanding polynomial functions is beneficial for several reasons:

• Versatile Modeling: They can represent a wide array of phenomena, from simple measurements to advanced physical behaviors.
• Smoothness: Polynomials are remarkably smooth, continuous functions, which makes them easier to analyze than other functions might be.
• Predictable Behavior: Their graphs have distinctive, predictable shapes depending on the degree of the polynomial. For instance, a cubic polynomial like \( P(x) \) usually has a unique curve shape with one or two turning points.

Despite their potentially complex appearance, polynomial functions embody balance and order that are instrumental in mathematical analysis.

Continuous Functions

A continuous function is one where small changes in the input result in small changes in the output. Imagine drawing the graph without picking up your pencil — that's continuity! Polynomial functions, such as the one in our exercise, are perfect examples of continuous functions.
Why is continuity significant?

• Robust Analysis: Continuous functions allow comprehensive analysis using calculus and other analytical tools.
• Function Predictability: Being continuous means that there are no sudden jumps or breaks, offering a reliable view of the function's behavior.
• Applications: Since many real-world processes are smooth and unbroken, modeling these with continuous functions makes sense in disciplines like engineering and physics.

Continuity is crucial when applying the Intermediate Value Theorem—which we used in the given exercise. This theorem rests heavily on the fact that the function in question is continuous over the interval analyzed. It's this property that assures there's a place where the function's graph meets the x-axis — our real zero!