Problem 12 - 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. $$P(x)=3 x^{3}+7 x^{2}-4 ; \quad \frac{1}{2} \text { and } 1$$

Step-by-Step Solution

Answer

The function has a real zero at approximately \( x = 0.71 \).

Step 1: Evaluate Function at Given Points

First, evaluate the function \( P(x) = 3x^3 + 7x^2 - 4 \) at the given points \( x = \frac{1}{2} \) and \( x = 1 \).\\( P\left(\frac{1}{2}\right) = 3\left(\frac{1}{2}\right)^3 + 7\left(\frac{1}{2}\right)^2 - 4 = \frac{3}{8} + \frac{7}{4} - 4 = -\frac{11}{8} \). \\( P(1) = 3(1)^3 + 7(1)^2 - 4 = 3 + 7 - 4 = 6 \).

Step 2: Apply Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function \( f(x) \) has values of opposite signs at two points, \( a \) and \( b \), there is at least one real zero between \( a \) and \( b \). Here, since \( P\left(\frac{1}{2}\right) = -\frac{11}{8} \) (negative) and \( P(1) = 6 \) (positive), there exists at least one real zero between \( x = \frac{1}{2} \) and \( x = 1 \).

Step 3: Use Calculator to Approximate Zero

Use a graphing calculator or numerical solver to find the zero of \( P(x) \) between \( x = \frac{1}{2} \) and \( x = 1 \). After inputting the function and checking the interval, the calculator gives a zero at approximately \( x = 0.71 \).

Key Concepts

Continuous Functions

A function is considered continuous if you can draw it on a graph without lifting your pencil. This means that there are no holes, jumps, or gaps in the graph of the function.
In mathematical terms, a function \( f(x) \) is continuous at a point \( c \) if the limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \). In other words, you can find the value of the function at that point simply by looking at its nearby values.
Continuous functions are crucial in calculus, particularly when applying the Intermediate Value Theorem, which assures us that if a function is continuous over an interval \([a, b]\), it takes on every value between \( f(a) \) and \( f(b) \). This property is very handy when seeking real zeros.

Real Zeros

When we talk about real zeros of a function, we are discussing the points where the function's graph intersects the x-axis. These intersections represent the values of \( x \) for which the function \( f(x) = 0 \).
Real zeros are incredibly important in understanding the behavior of functions, especially polynomials. They can tell us where the function changes sign and help in sketching the graph accurately.

• The Intermediate Value Theorem uses real zeros to establish that, if a continuous function changes sign over an interval, it must cross the x-axis somewhere in that interval.
• This is why the theorem is a key tool for confirming the existence of real zeros within specified bounds.

Consequently, finding real zeros often involves checking intervals for sign changes in the function values.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, grouped together using only addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial function is \( P(x) = 3x^3 + 7x^2 - 4 \).

• These functions have powers (exponents) that are whole numbers, and they form smooth, continuous graphs with no breaks.
• The degree of the polynomial (the highest power of \( x \)) tells us about the general shape of its graph and the number of real zeros it may have.

Understanding polynomial functions is vital since they are used in various areas of math and applied sciences.
By employing the Intermediate Value Theorem, we can explore their behavior across intervals. This particular theorem allows us to verify the existence of real zeros by checking the signs of the polynomial at its endpoints.