Problem 36 - 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. $$functions$$ $$f(x)=1.5 e^{x / 2}$$ $$values$$ $$x=240$$

Step-by-Step Solution

Answer

The final result, being a very large number in decimal, is calculated using a calculator. It is crucial to remember to round the final answer to three decimal places.

Step 1: Understand the function

The function given is \(f(x)=1.5 e^{x / 2}\). This is an exponential function with base e (Euler's Number) which is approximately equal to 2.71828.

Step 2: Substitution

Substitute the given value \(x=240\) into the function: \(f(240)=1.5 e^{240 / 2}\).

Step 3: Simplify exponent

First, alleviate the exponent in the function. It becomes \(f(240)=1.5 e^{120}\).

Step 4: Calculation

Calculate the exponential part using a calculator or software, to get the value in decimal. Multiply this value with 1.5 to get the final answer.

Key Concepts

Understanding Exponential Functions

Exponential functions are a fascinating part of mathematics. They describe situations where things grow or decay at a constant rate. In an exponential function, such as \( f(x) = a \, e^{x} \), \( a \) is a constant multiplier, and \( e^{x} \) is the exponential expression. Here, the base of the exponential expression is \( e \), also known as Euler's number.
Exponential functions can model all kinds of real-world phenomena. For instance, they are used to describe populations, interest in banking, and even the spread of diseases. The consistent growth rate is what makes them unique and useful.

• Key Aspect 1: **Base** - Constant in an exponential function, often Euler's number \( e \).
• Key Aspect 2: **Exponent** - The variable part affecting growth.
• Key Aspect 3: **Constant Multiplier** - The initial size or amount.

Understanding how to manipulate and evaluate these functions is crucial for solving problems in science and economics as well.

The Significance of Euler's Number

Euler's number, denoted as \( e \), is approximately 2.71828. It is a fundamental constant in mathematics, much like \( \pi \). It forms the base of the natural logarithm and is integral to exponential functions.
Regarded as the natural base due to its unique property, \( e^x \) is the only function that is the derivative of itself. This makes it highly valuable in calculus, helping solve differential equations and model growth processes efficiently.

• Importance in **Calculus**: The derivative of \( e^x \) simplifies to itself, making calculations and predictions straightforward.
• Usage in **Natural Phenomena**: Models processes like radioactive decay and compound interest.
• **Universal Nature**: Appears in various branches of mathematics and science, highlighting exponential growth and decay processes.

Understanding Euler's number helps in grasping concepts in advanced mathematics and its applications in daily life.

Mastering Function Evaluation

Function evaluation is an essential skill in mathematics. It involves substituting a given value into a function to find the outcome. Consider the function \( f(x) = 1.5 \, e^{x/2} \). To evaluate this function for a specific \( x \), follow these steps:
First, substitute the provided \( x \) value into the equation. If \( x = 240 \), then \( f(x) = 1.5 \, e^{240/2} \). This requires simplifying the exponent first, becoming \( f(240) = 1.5 \, e^{120} \).
To find the result, calculate the exponent using a calculator to handle the potentially large number. Then multiply by any constants, here being 1.5. This process requires understanding how functions operate and how to execute calculations precisely.

• Step 1: **Substitution**: Insert given \( x \) values.
• Step 2: **Simplification**: Work through arithmetic simplifications like dividing exponents.
• Step 3: **Calculation**: Use tools to calculate large numbers; finalize with constants.

Mastering this technique allows for better comprehension of complex mathematical models and predictions.