Problem 17 - A Graphical Approach to College Algebra

Question

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=x^{5}-2 x^{3}+1 ;-1.6 \text { and }-1.5$$

Step-by-Step Solution

Answer

The function has a real zero between -1.6 and -1.5, approximately at -1.54.

Step 1: Verify Function Values at Given Points

First, substitute the endpoints \(-1.6\) and \(-1.5\) into the function \(P(x) = x^5 - 2x^3 + 1\) to find the function values at these points. Calculate:\[ P(-1.6) = (-1.6)^5 - 2(-1.6)^3 + 1 \]\[P(-1.6) = -10.48576 + 8.192 + 1 = -1.29376\]Calculate:\[P(-1.5) = (-1.5)^5 - 2(-1.5)^3 + 1\]\[P(-1.5) = -7.59375 + 6.75 + 1 = 0.15625\]

Step 2: Apply the Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function has values of opposite sign at two points, there must be at least one real zero between them. Since \(P(-1.6) = -1.29376 < 0\) and \(P(-1.5) = 0.15625 > 0\), and \(P(x) = x^5 - 2x^3 + 1\) is continuous (being a polynomial function), the theorem guarantees a real zero exists between \(-1.6\) and \(-1.5\).

Step 3: Approximate the Zero Using a Calculator

Use a graphing calculator or numerical solver to find the approximate value of the zero. Set the calculator to find the root of the function \(P(x)\) between \(-1.6\) and \(-1.5\). The approximate zero of the function within this interval is:\[x \approx -1.54\]

Key Concepts

Real Zeros

Real zeros are the values of \(x\) for which the function equals zero. In simple terms, they are the points where the graph of the function crosses the x-axis. Detecting real zeros is crucial in understanding the behavior of a function and solving equations.

In the context of the Intermediate Value Theorem, real zeros are identified when the function changes signs over an interval. This signifies the presence of at least one real zero between the two points.
When applying the theorem, you first compute the value of the function at the given endpoints. In our example, the function \(P(x) = x^5 - 2x^3 + 1\) changes signs between \(-1.6\) and \(-1.5\). This change in sign is a significant indicator that there is a real zero within this interval.

To approximate the real zero to a desired precision, numerical methods or graphing tools are often used. In our exercise, a calculator provided an approximation of the zero at \(x \approx -1.54\). Understanding and approximating real zeros helps us solve polynomial equations effectively.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers, with constants as coefficients. These functions form the foundation for a vast number of mathematical models and are characterized by smooth and continuous curves.

Polynomials can have various degrees, dictating the number of turns or changes in direction on their graphs.

• The degree of a polynomial is determined by the highest power of the variable.
• They might have several real or complex zeros.
• They exhibit continuous behavior, without breaks or jumps, which is crucial when applying the Intermediate Value Theorem.

In the exercise example of \(P(x) = x^5 - 2x^3 + 1\), this is a fifth-degree polynomial and hence can have up to five zeros in total.

Polynomial functions have simple derivatives, making them easy to analyze for slope and curvature. Their continuity and differentiability are what make them extremely important to study, especially when discussing real-world phenomena or computational models.

Continuous Functions

Continuous functions are those that have no gaps, holes, or jumps. This means you can draw their graph without lifting your pencil from the paper. A continuous function is essential when applying the Intermediate Value Theorem, as the theorem only holds true for these types of functions.

The polynomial function shown in the exercise, \(P(x) = x^5 - 2x^3 + 1\), exemplifies a continuous function because polynomials are inherently continuous by nature.

• This is due to their composition from basic operations (addition, subtraction, multiplication, and taking powers).
• Continuous functions ensure that for any two points, there is always a defined path in-between.
• This feature makes it feasible to identify zeros within specific intervals through sign changes.

Understanding continuity is crucial because it allows mathematicians to employ powerful theorems like the Intermediate Value Theorem.

In practical applications, more sophisticated mathematical models heavily rely on continuous functions to predict behavior accurately without unexpected interruptions or anomalies in data.