Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

To use technology effectively, you need to discover its capabilities, its strengths, and its weaknesses. We urge you to practice graphing functions of various types using your own computer package or calculator. Let \(f(x)=\left(x^{3}+3 x-5\right) /\left(x^{2}+4\right)\). (a) Evaluate \(f(1.38)\) and \(f(4.12)\). (b) Construct a table of values for this function corresponding to \(x=-4,-3, \ldots, 3,4\).

Step-by-Step Solution

Answer

Evaluate \( f(x) \) at the specific points using a calculator, then construct a table for integer values from \(-4\) to \(4\) by computing \( f(x) \) at each point.

Step 1: Understand the Function

The given function is \( f(x) = \frac{x^3 + 3x - 5}{x^2 + 4} \). This is a rational function, meaning it is a ratio of a polynomial where the denominator must not be zero. This function can be evaluated for various values of \( x \) to understand its behavior.

Step 2: Evaluate the Function at Specific Points

We are asked to evaluate \( f(x) \) at \( x = 1.38 \) and \( x = 4.12 \).- For \( x = 1.38 \): \[ f(1.38) = \frac{(1.38)^3 + 3(1.38) - 5}{(1.38)^2 + 4}\] Calculate this using a calculator to find the numerical value.- For \( x = 4.12 \): \[ f(4.12) = \frac{(4.12)^3 + 3(4.12) - 5}{(4.12)^2 + 4}\] Again, use a calculator to find the result.

Step 3: Create a Table of Values

Construct a table to determine \( f(x) \) for integer values from \( -4 \) to \( 4 \). Evaluate the function at each of these points:- \( f(-4) \), \( f(-3) \), \( f(-2) \), \( f(-1) \), \( f(0) \), \( f(1) \), \( f(2) \), \( f(3) \), \( f(4) \).Use a calculator to substitute each value into the function and compute the result. This will help in understanding how the function behaves for different inputs.

Step 4: Verify Calculations

Double-check each computed value to ensure correctness by substituting back into the function using a calculator or computation tool. This verifies that the function values are reliable and accurate for all specified inputs.

Key Concepts

Function Evaluation

Function evaluation is a crucial aspect of understanding how algebraic expressions transform into numerical values.To evaluate a rational function like \( f(x) = \frac{x^3 + 3x - 5}{x^2 + 4} \), you substitute the value of \( x \) into the expression and simplify the result. This process helps to determine what the function's output is for specific inputs.Let’s take a look at the calculation steps. For example, to find \( f(1.38) \), plug in \( x = 1.38 \):

• First, compute the numerator: \( (1.38)^3 + 3(1.38) - 5 \)
• Then, compute the denominator: \( (1.38)^2 + 4 \)
• Simplify by dividing the numerator by the denominator to get a numerical output.

Similarly, evaluate \( f(4.12) \) by following the same steps: substitute, calculate, and simplify.These evaluations are particularly useful because they provide specific data points that enhance understanding of the function's behavior, showing how even slight changes in \( x \) can affect \( f(x) \). By computing these values, you develop a clearer insight into how the function operates at distinct points. This step-by-step function evaluation solidifies your grasp of rational functions and their properties.

Graphing Techniques

Graphing is an essential method to visually interpret the behavior of a function.When graphing a rational function like \( f(x) = \frac{x^3 + 3x - 5}{x^2 + 4} \), it involves plotting the output values or \( f(x) \) against the corresponding \( x \) values on a graph. This way, we can observe trends, patterns, or any peculiar behavior of the function.Constructing a table of values is the foundation of graphing. For example, to graph our function effectively, you would:

• Select a range for \( x \), like \( -4 \) to \( 4 \).
• Calculate \( f(x) \) for each \( x \) in this range. These values are plotted as points on the coordinate plane.
• Connect the points to reveal the curve or shape of the function.

The graph helps in identifying key features of the function like intercepts, asymptotes, and intervals of increase or decrease. By observing these features, students can infer the behavior of the function at more than just isolated points, giving a comprehensive view of how the function behaves globally across its domain.

Using Technology in Calculus

Utilizing technology in calculus, such as graphing calculators or computer software, greatly enhances the learning and application of mathematical concepts.These tools are invaluable for calculating complex functions or for graphing intricate expressions like \( f(x) = \frac{x^3 + 3x - 5}{x^2 + 4} \). They automate the cumbersome calculations and plot graphs with precision, aiding in more accurate conclusions than manual methods might allow.For effective use of technology:

• Start by inputting the function into a graphing tool.
• Use the tool's evaluation features to calculate specific numeric values.
• Leverage the graphing capability to visualize the function's complete curve within any desired domain.

By practicing with these technologies, students develop a deeper understanding of the function across multiple dimensions and can easily identify behaviors like asymptotic tendencies or sudden shifts in direction. Additionally, such technology saves time and mitigates errors when dealing with complex arithmetic. Learning to harness the power of technology in calculus not only enhances efficiency but also opens up new realms for exploration and analysis.