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Chapter 0: Problem 105

Use the ZERO feature or the INTERSECT feature to approximate the zeros of each function to three decimal places. $$f(x)=2 x^{3}-x^{2}-14 x-10$$

Short Answer

Expert verified
The zeros of the function \(f(x) = 2x^3 - x^2 - 14x - 10\) are found by using the ZERO feature on a graphing calculator.

Step by step solution

01

- Understand the Problem

We need to find the zeros of the function \(f(x) = 2x^3 - x^2 - 14x - 10\) to three decimal places. A zero of a function is a value of \(x\) for which \(f(x) = 0\).
02

- Use a Graphing Calculator

To find the zeros, use a graphing calculator. Enter the function \(f(x) = 2x^3 - x^2 - 14x - 10\) into the calculator and graph it.
03

- Identify Approximate Zeros

Look for the points where the graph crosses the x-axis. These points are the zeros of the function.
04

- Use the ZERO Feature

Use the ZERO feature on the calculator to find the x-values where the function crosses the x-axis. On most calculators, this involves selecting 'Calc', then 'Zero', and moving the cursor near each point where the graph crosses the x-axis.
05

- Interpret Results

Once you use the ZERO feature, the calculator provides the x-values where the function equals zero. Ensure these values are accurate to three decimal places.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
Polynomial functions are expressions involving sums of powers of variables with constant coefficients, such as \(f(x) = 2x^3 - x^2 - 14x - 10\). Polynomial functions can have several features, such as roots, which are the values of the variable that make the function equal to zero. These can be straightforward when dealing with lower-order polynomials like linear or quadratic functions but can become more complex with higher-order polynomials like the cubic function in this exercise.
Graphing Calculators
Graphing calculators are powerful tools for visualizing functions and finding important features, such as zeros or intercepts. To solve our exercise, you鈥檒l need to enter the function \(f(x) = 2x^3 - x^2 - 14x - 10\) into the graphing calculator. Viewing the graph allows you to see where it crosses the x-axis, indicating potential zeros. Most graphing calculators have specific features like 'ZERO' or 'INTERSECT' to help find these precise points.Don't forget to zoom in and out or adjust the viewing window to find all the points where the function might cross the x-axis.
Approximation Techniques
When finding zeros of functions, especially higher-order ones, you often rely on approximation techniques. Graphing calculators provide tools like the ZERO feature to find zeros to a specified accuracy, often to several decimal places. These techniques involve settings on your calculator that aim to locate points where the function crosses the x-axis with high precision. An important practice is:
  • Ensuring the cursor stays close to where the function seems to cross the x-axis when using the ZERO feature.
  • Re-checking results by plotting the zeros onto the graph to see if they hold true visually.
Roots of Equations
Roots or zeros of equations are fundamental concepts in algebra and calculus. For polynomial functions like \(f(x) = 2x^3 - x^2 - 14x - 10\), they represent the points where the polynomial equals zero. Finding these zeros helps in understanding key features of the function and where it intersects the x-axis. These roots can be real or complex numbers. In our exercise, we are interested in the real roots, which can be found using a graphing calculator's ZERO or INTERSECT features and verifying them to three decimal places.Accurate roots allow us to solve polynomial equations effectively, factorize polynomials, and understand the graph's behavior.

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Most popular questions from this chapter

Find the domain of each function given below. $$f(x)=\frac{x-2}{6 x-12}$$

Find the equilibrium point for the following demand and supply functions, where \(q\) is the quantity, in thousands of units, and \(x\) is the price per unit, in dollars. Demand: \(q=83-x\) Supply: \(\quad q=\frac{x^{2}}{576}-1.9\)

It is theorized that the price per share of a stock is inversely proportional to the prime (interest) rate. In January 2010 , the price per share 5 of Apple Inc. stock was \(\$ 205.93\) and the prime rate \(R\) was \(3.25 \%\). The prime rate rose to \(4.75 \%\) in March 2010. (Source: finance, yahoo.com and Federal Reserve Board.) What was the price per share in March 2010 if the assumption of inverse proportionality is correct?

While driving a car, you see a child suddenly crossing the street. Your brain registers the emergency and sends a signal to your foot to hit the brake. The car travels a distance \(D\), in feet, during this time, where \(D\) is a function of the speed \(r,\) in miles per hour, that the car is traveling when you see the child. That reaction distance is a linear function given by \(D(r)=\frac{11 r+5}{10}\). a) Find \(D(5), D(10), D(20), D(50),\) and \(D(65).\) b) Graph \(D(r).\) c) What is the domain of the function? Explain.

Jimmy decides to mow lawns to earn money. The initial cost of his lawnmower is 250 dollars. Gasoline and maintenance costs are 4 dollars per lawn. a) Formulate a function \(C(x)\) for the total cost of mowing \(x\) lawns. b) Jimmy determines that the total-profit function for the lawnmowing business is given by \(P(x)=9 x-250\) Find a function for the total revenue from mowing \(x\) lawns. How much does Jimmy charge per lawn? c) How many lawns must Jimmy mow before he begins making a profit?
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