Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. (Value) $$x=11$$ $$x=18.31$$ $$x=\frac{1}{2}$$ $$x=\sqrt{0.65}$$ (Function) $$f(x)=-\ln x$$

Step-by-Step Solution

Answer

The function values for the given 'x' values are: \[f(11)\], \[f(18.31)\], \[f(1/2)\], \[f(\sqrt{0.65})\]. The exact numerical values can be computed using a scientific calculator, making sure to round to three decimal places.

Step 1: Evaluate at x = 11

To find the value of the function at x = 11, substitute 11 into the function for x: \[f(11)=-\ln 11\]. This operation can be done using a scientific calculator. Make sure to round the final answer to three decimal places.

Step 2: Evaluate at x = 18.31

To find the value of the function at x = 18.31, substitute 18.31 into the function for x: \[f(18.31)=-\ln 18.31\]. This operation can be done using a scientific calculator. Ensure to round the final answer to three decimal places.

Step 3: Evaluate at x = 1/2

To find the value of the function at x = 1/2, substitute 1/2 into the function for x: \[f(1/2)=-\ln (1/2)\]. This operation can be done using a scientific calculator. Remember to round the final answer to three decimal places.

Step 4: Evaluate at \(x = \sqrt{0.65}\)

To find the value of the function at x = \(\sqrt{0.65}\), substitute \(\sqrt{0.65}\) into the function for x: \[f(\sqrt{0.65})=-\ln \sqrt{0.65}\]. This operation can be done using a scientific calculator. Round the final answer to three decimal places.

Key Concepts

Understanding Natural Logarithms

The natural logarithm, denoted by \( \ln \), is a special type of logarithm with the base \( e \). Here, \( e \) is a mathematical constant approximately equal to 2.71828. The natural logarithm \( \ln(x) \) represents the power to which the base \( e \) must be raised to produce the given number \( x \). For example, if \( e^y = x \), then \( y = \ln(x) \). This logarithmic function is frequently used in various mathematical fields, such as calculus, physics, and exponential growth or decay models.
In this exercise, we use the natural logarithm as part of the function \( f(x) = -\ln(x) \). Notice the minus sign in the function \(-\ln(x)\), which means that we essentially take the natural logarithm of \( x \) and then flip its sign. This subtraction can significantly change the results we get compared to the plain natural logarithm.

The Art of Rounding Decimals

Rounding decimals is a crucial aspect of mathematics that helps simplify numbers to a manageable precision. In our exercise, we are required to round the results to three decimal places. This means adjusting the number such that only three digits remain after the decimal point.
To round correctly, look at the digit in the fourth decimal place:

• If it's 5 or greater, increase the third decimal place by 1.
• If it's less than 5, leave the third decimal place as it is.

For example, if you have a result of 3.4567:

• The fourth decimal is 7, which is greater than 5.
• Thus, you would round up, leaving you with 3.457.

Being consistent with rounding ensures that our solutions are both precise and concise. It's particularly useful in ensuring compatibility across results, especially when we are working with irrational numbers or logarithms.

How to Use a Scientific Calculator

A scientific calculator is an essential tool for evaluating complex mathematical expressions, and it's equipped to handle natural logarithms directly. When using a scientific calculator to evaluate expressions like \(- \ln(x)\), follow these steps:

• Input the value of \( x \) using the numeric keypad.
• Press the \( \ln \) button to compute the natural logarithm.
• Multiply the result by -1 to account for the negative sign in \(- \ln(x)\).
• Apply any necessary rounding to achieve the required decimal places.

Most scientific calculators have a direct button for \( \ln \), and they typically allow you to enter complex expressions directly, streamlining calculations. Practicing with your calculator can make you faster and more accurate, especially when tackling a series of calculations or handling multiple expressions.

Introduction to Function Evaluation

Function evaluation is the process of finding the output of a function given an input value. In mathematical notation, we express this as \( f(x) \), where \( x \) is the input value, and \( f(x) \) is the output.
To evaluate the function \( f(x) = -\ln(x) \) for different \( x \) values:

• Substitute \( x \) with specific values like 11, 18.31, 1/2, or \( \sqrt{0.65} \).
• Use your calculator to find \( \ln(x) \).
• Apply the negative sign to the logarithm to get \( -\ln(x) \).
• Round the results to three decimal places as instructed.

Function evaluation is a foundational concept in mathematics, as it allows you to understand how functions transform inputs into outputs. Every step in function evaluation requires careful calculation to ensure accuracy, whether it's substituting values, performing operations, or rounding results.