Problem 58 - Precalculus with Limits

Question

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. \(f(x)=3 \ln x \quad x=0.74\)

Step-by-Step Solution

Answer

The short answer might vary slightly depending on the calculator used but should be approximately -0.301

Step 1: Substitute the value of \(x\)

The required value to substitute for \(x\) in the function \(f(x)=3 \ln x\) is 0.74. When done that we get \(f(0.74)=3 \ln 0.74\).

Step 2: Evaluate the function

Using a calculator, evaluate the value of \(3 \ln 0.74\). Make sure the calculator is set to use the natural logarithm (\(\ln\)).

Step 3: Round the result

Finally, round the result obtained to three decimal places, as required by the exercise.

Key Concepts

Function Evaluation

Function evaluation might sound complicated, but it's a straightforward process that involves finding the output of a function for a specific input. In our exercise, the function is given as \( f(x) = 3 \ln x \). Here, \( \ln \) represents the natural logarithm, which is a mathematical function that tells us how many times we must multiply \( e \) (approximately 2.718) to get another number. When you evaluate the function for \( x = 0.74 \), you substitute \( 0.74 \) for \( x \) in the function. This process is called function evaluation. So, you calculate \( f(0.74) = 3 \ln 0.74 \). By using a calculator, this gives you a numerical value that is the result of this evaluation.

Rounding

Rounding numbers is an essential skill in mathematics as it makes complex numbers easier to work with. In our particular exercise, we are asked to round our answer to three decimal places. Rounding helps in limiting the number of digits and thus simplifies the result for practical use without losing considerable accuracy. Consider the value you calculated earlier, for instance, if the result from your calculator reads 5.267489. To round this to three decimal places:

• Identify the third decimal place, which in our example is 7.
• Look at the fourth decimal number, which is 4.
• If this number is less than 5, retain the number in the third decimal place. In our example, it remains 7.

In this way, rounding gives us an answer that is tidy but still sufficiently accurate for most purposes.

Substitution

Substitution is a fundamental process in algebra and calculus, where you replace a variable with a given number. This allows you to simplify expressions and solve equations effectively. It's no different in the given exercise, where we need to substitute the value of \( x \) with \( 0.74 \).To perform substitution correctly in the function \( f(x) = 3 \ln x \):

• You must replace every occurrence of \( x \) with 0.74.
• This forms the expression \( f(0.74) = 3 \ln 0.74 \).
• After substitution, you are ready to evaluate the function with the specific number you have replaced in for \( x \).

Understanding substitution will allow you to engage more deeply with math problems as it is a foundational concept used widely across various types of mathematical problems.