Problem 45 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

\(y=x^{3} e^{2-x}\) where \(x=2\)

Step-by-Step Solution

Answer

8

Step 1 - Substitute the Value

First, substitute the given value of x into the function. Here, x = 2.

Step 2 - Evaluate the Exponent Term

Calculate the value of the exponent term: \[ e^{2-2} = e^{0} \]

Step 3 - Simplify the Exponent

Simplify the exponent value: \[ e^{0} = 1 \]

Step 4 - Calculate the Remaining Term

Now, evaluate the remaining term involving x to the power of 3: \[ 2^{3} = 8 \]

Step 5 - Multiply the Results

Finally, multiply the results from Step 3 and Step 4 together: \[ y = 8 \times 1 = 8 \]

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation. It involves raising a number (called the base) to a power (exponent). For instance, in the expression \(x^{3}\), x is the base and 3 is the exponent. This means you multiply x by itself three times: \(x \times x \times x\). In our example, raising 2 to the power of 3 (or \(2^{3}\)) gives us 8.
Please remember that exponentiation follows certain rules:

• Power of Zero: Any number raised to the power of zero is 1, for example, \(2^{0} = 1\).
• Negative Exponents: A negative exponent indicates the reciprocal, \(x^{-n} = \frac{1}{x^n}\).
• Multiplication of Similar Bases: \(x^a \times x^b = x^{a+b}\).

Understanding these concepts is crucial in simplifying and evaluating functions involving exponents.

Simplification

Simplification in mathematics involves reducing expressions to their simplest form. This makes them easier to work with and understand. Let's look at our problem step by step:
First, substitute x = 2 into the expression \(y = x^{3} e^{2-x}\). This gives:\[\begin{equation} y = 2^{3} e^{2-2} ewline y = 2^{3} e^{0} ewline y = 8 \times 1 ewline y = 8ewline \end{equation}\]
Notice how breaking down each part of the expression simplifies our problem. We calculated \(e^{0} = 1\) and \(2^{3} = 8\).
Simplifying exponent terms is crucial. It ensures correct function evaluation. To handle these simplifications:

• Break down complex terms into basic operations.
• Apply arithmetic rules step by step.
• Double-check each step for accuracy.

This approach not only clarifies the problem but also avoids errors.

Calculus

Calculus is a branch of mathematics dealing with continuous change. It consists of two main parts: differentiation and integration. Although the given problem doesn't directly involve differentiation or integration, understanding these concepts is essential.

• Differentiation: This involves finding the rate of change or the slope of a function. For example, if we wanted to find how y changes as x changes in \(y = x^{3} e^{2-x}\), we'd use differentiation.
• Integration: This is about finding the total accumulation of quantities. For instance, integrating a rate of change gives the total quantity.

While our primary focus here was simplifying the function via algebraic operations, calculus comes into play in more complex scenarios involving rates of change and accumulated values.
In future problems involving functions like this, understanding calculus will help you analyze how functions behave and interpret real-world phenomena accurately.