Problem 35 - Precalculus with Limits

Question

Evaluating a Natural Exponential Function In Exercises \(35-38\) , evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. \(\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ {f(x)=e^{x}} & {x=3.2}\end{array}\)

Step-by-Step Solution

Answer

The value of the function \(f(x) = e^{x}\) at \(x = 3.2\) rounded to three decimal places is approximately \(24.532\).

Step 1: Understand the function

The first step is to understand the function \(f(x) = e^{x}\). In this function, \(e\) is the base of the natural logarithm and \(x\) is the power to which \(e\) is raised.

Step 2: Substitute the given value

Substitute the given value \(x=3.2\) into the function \(f(x) = e^{x}\). When we do this substitution, the function becomes \(f(3.2) = e^{3.2}\).

Step 3: Evaluate the function

Next, evaluate the function \(f(3.2) = e^{3.2}\) using a calculator to get an approximation.

Step 4: Round off the result

Finally, round off the result to three decimal places as requested by the problem statement.

Key Concepts

Understanding Euler's Number

Euler's number, denoted as \(e\), is a mathematical constant approximated to be 2.71828. It forms the backbone of the natural exponential function \(f(x) = e^x\). This number is pivotal in mathematics, particularly in calculus, where it explains growth and compounded rates.

Here’s why Euler's number is so special:

• \(e\) is the limit of \((1 + 1/n)^n\) as \(n\) approaches infinity.
• It is the base of the natural logarithm, often appearing in problems involving exponential growth, such as populations and interest calculations.
• It provides a natural way to describe growth where the rate of change is proportional to the current value.

Understanding how \(e\) behaves in equations like \(f(x) = e^x\) allows for a deeper appreciation of natural growth patterns and the beauty of mathematics.

Function Evaluation Made Easy

Evaluating a function involves substituting a given value into the function and calculating the result. For the function \(f(x) = e^x\), evaluating means replacing \(x\) with an actual number. In our case, this number is 3.2.

Here's the simplified process:

• Start with the function: \(f(x) = e^x\).
• Substitute \(x = 3.2\) into the function to get \(f(3.2) = e^{3.2}\).
• Use a calculator capable of exponential calculations to determine \(e^{3.2}\).

This helps students practice substituting values and using tools like calculators for computations, which are crucial skills in applied mathematics.

Function evaluation transforms abstract mathematical concepts into tangible results, making it a valuable skill for solving real-world problems.

Mastering Rounding Decimals

Rounding decimals is the process of trimming a number to a fixed number of decimal places for simplicity. In mathematics, we often round to make results easier to work with or to meet specific problem requirements, such as having solutions accurate to three decimal places.

Here's how to round decimals effectively:

• Consider the third decimal place and the digit immediately after it.
• If this subsequent digit is 5 or greater, round the third decimal place up by one.
• If it is less than 5, retain the third decimal digit as it is.

For example, if your calculator gives you \(e^{3.2} = 24.5325\), you'll round it to 24.533.

Rounding is not just about simplicity, but also about reasoning. It ensures that solutions are precise enough while remaining comprehensible and manageable.