Problem 60 - Precalculus with Limits

Question

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. \(g(x)=-\ln x \quad x=\frac{1}{2}\)

Step-by-Step Solution

Answer

The result is approximately 0.693, after rounding to three decimals.

Step 1: Identify the function

The given function is \(g(x) = -\ln x\). We need to find the value of this function when \(x=\frac{1}{2}\).

Step 2: Substitute the value into the function

Insert the value \(x = \frac{1}{2}\) into the function to get \(g\left(\frac{1}{2}\right) = -\ln\left(\frac{1}{2}\right)\).

Step 3: Calculate the result

Use the calculator to find the value of \(-\ln\left(\frac{1}{2}\right)\). Make sure your calculator is set to calculate natural logarithms.

Step 4: Round the result

After calculation, round off the result to three decimal places as per the instruction of the problem.

Key Concepts

Function Evaluation

Evaluating a function means finding the output when a specific value is substituted into the function. Consider the function given: \( g(x) = -\ln x \). Here, the task is to evaluate this function when \( x = \frac{1}{2} \). To do so, we substitute \( \frac{1}{2} \) in place of \( x \) in the function. This transforms the expression to \( g\left(\frac{1}{2}\right) = -\ln\left(\frac{1}{2}\right) \). Evaluating a function, especially with logarithms, can provide insights into its behavior and value at specific points. Understanding how to properly substitute and interpret function notation is crucial in solving mathematics problems involving functions.

Rounding

Rounding is a mathematical process used to reduce the number of digits in a number. This can make it easier to understand and use. When you're tasked with evaluating a function like \(g(x) = -\ln x\) at \(x = \frac{1}{2}\), rounding the result to three decimal places means you must show three digits after the decimal point.

• This process helps in reducing the precision of a number while maintaining a level of accuracy similar to the original value.
• Sometimes, decimals have many digits which complicates analysis or reporting in practical applications.

Thus, once the calculation is made with a calculator, check to ensure your answer is presented in three decimal place format. Remember that rounding is applied according to certain rules: - If the digit after your required decimal place is 5 or more, you round up. - If it's less than 5, you round down. Applying these rules will ensure your results conform to the exercise's requirements.

Calculator Usage

Using a calculator effectively is crucial, especially when dealing with more complex functions like natural logarithms. Make sure that your calculator is capable of computing natural logarithms and is set in the correct mode. Here’s a simplified guide to using calculators for natural logarithms:

• First, make sure your calculator is in the right mode. Some calculators may need to be switched to scientific mode to use advanced functions.
• Locate the 'ln' function key, as this is what you’ll use for computing natural logarithms.
• After inputting your number, in this case, \( \frac{1}{2} \), simply press the 'ln' function key to receive your result.

Double-check your input to avoid mistakes, especially with parentheses or negative signs. Calculators can often handle the arithmetic easily and quickly if used correctly, dramatically reducing errors and saving time. Understanding how to use your calculator competently thus becomes an essential skill.