Problem 27 - 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.) $$f(x)=x^{3}+x-1, \quad[0,1]$$

Step-by-Step Solution

Answer

Approximately, the zero of the function \(f(x) = x^{3} + x - 1\) in the interval \([0,1]\) is 0.6.

Step 1: Evaluating the Function at Interval Ends

Evaluate the function at \(a = 0\) and \(b = 1\) to get \(f(0)\) and \(f(1)\). This means substitute \(0\) and \(1\) into the given function \(f(x)\). Calculate \(f(0)\) by setting \(x = 0\) in \(f(x)\) to get \(0^{3} + 0 - 1 = -1\). Similarly, find \(f(1)\) by setting \(x = 1\) in \(f(x)\) to get \(1^{3} + 1 - 1 = 1\). Since \(f(a)\) and \(f(b)\) encompass \(0\) (as \(f(a) < 0\) and \(f(b) > 0\) or vice versa), we can confirm that the function must cross the \(x\)-axis in the interval \([a, b] = [0, 1]\)

Step 2: Using the Intermediate Value Theorem

Now the zero of \(f\) on the interval \([a, b]\) must be approximated. Check the midpoint of the interval, which is \((0+1)/2 = 0.5 \) and evaluate \(f\) at this new value. Substituting \(x = 0.5\) into the function \(f(x)\), we get \(f(0.5) = (0.5)^{3} + 0.5 - 1 = -0.375\). This is less than \(0\), which indicates that the zero of \(f\) still lies somewhere between \(0.5\) and \(1\). Narrow down the interval to \([0.5, 1]\) and repeat the above process till the required decimal accuracy is achieved.

Step 3: Repeating the Process to Approximate Zero

Now, evaluate the function at the new midpoint, which is \(0.75\) (average of \(0.5\) and \(1\)). When substituting \(x = 0.75\) into \(f(x)\), you obtain \(f(0.75) = (0.75)^{3} + 0.75 - 1 = 0.171875\), which is greater than zero. Thus, now you know the zero lies between \(0.5\) and \(0.75\). Next, take the midpoint between \(0.5\) and \(0.75\) which is \(0.625\) and evaluate \(f\) at \(0.625\) to determine the exact interval where the zero lies. Repeat this process until finding an approximation of the zero to the nearest tenth is achieved.

Key Concepts

Zero of a Function

A zero of a function, also known as a root, is a value of the variable that makes the function equal to zero. In other words, when you substitute this value into the function, the result is zero. Finding zeros is crucial in understanding the behavior of functions, especially when dealing with their graphs. Zeros tell us where the graph of the function will intersect the x-axis.

For polynomial functions like the one in this exercise, zeros can also give insight into factors of the polynomial. Remember, a zero of a function tells us something important about where the output changes sign. It's where the function is neither positive nor negative.

• Zeros are often found using solving techniques like factoring, graphing, or calculus-based methods like the Newton-Raphson method.
• The Intermediate Value Theorem is particularly valuable for finding zeros in continuous functions over a specific interval.

Polynomial Functions

Polynomial functions are equations that involve powers of a variable, called "polynomials." These equations form the backbone of algebra and are characterized by the sum of several terms that have the same variable raised to different powers. Typical polynomial functions could be linear, quadratic, cubic, or even higher degrees.

Here's what you need to know about polynomial functions:

• The degree of the polynomial, indicated by the highest power of the variable, dictates the number of roots or zeros and the general shape of its graph.
• Cubic polynomials, such as the given function in this exercise, can have up to three real roots.
• The coefficients in front of each term help determine the polynomial's behavior as the variable approaches positive or negative infinity.

In the exercise, the polynomial function is of degree three, hence why it's called a cubic function. These functions can appear wavy on a graph due to their bending nature around their zeros.

Approximating Zeros

Approximating zeros involves finding a value where the function gets close to zero but might not be precisely zero, due to the constraints like lack of exact factors. In context, the Intermediate Value Theorem is employed to offer a systematic approach to zero approximation by checking function values at specific points.

Here's a simplified way to approximate zeros:

• Select an interval where the function changes sign, meaning at one end of the interval the function value is positive and at the other end it's negative.
• Calculate the function at the midpoint of this interval. This is your first approximation step.
• If the function value at the midpoint is close enough to zero (within a desired accuracy), you've found your approximate zero.
• If not, adjust your interval based on the sign of the function at the midpoint, and repeat the process, halving the interval each time.

The repeated halving of the interval, as shown in the solution, hones in on the zero. By gradually narrowing down the interval and checking at each midpoint, an approximation to any decimal precision can be achieved.