Problem 16 - Applied Calculus

Question

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(P=200 e^{0.12 t}\)

Step-by-Step Solution

Answer

The derivative of the function is \(24 e^{0.12t}\).

Step 1: Understand the Function

The function given is an exponential function of the form \(P = 200 e^{0.12 t}\). Here, \(e\) is the base of the natural logarithm, and the exponent is \(0.12t\).

Step 2: Use the Exponential Function Derivative Rule

Recall that the derivative of \(e^{kt}\), where \(k\) is a constant, with respect to \(t\) is \(ke^{kt}\). We'll use this rule to differentiate the given function.

Step 3: Differentiate the Function

Apply the derivative rule to the function. The derivative of \(P\) with respect to \(t\) is \(\frac{dP}{dt} = 200 \cdot 0.12 e^{0.12t}\).

Step 4: Simplify the Result

Now simplify \(\frac{dP}{dt} = 200 \times 0.12 e^{0.12t} = 24 e^{0.12t}\). So, the derivative of the function is \(24 e^{0.12t}\).

Key Concepts

Exponential Functions

Exponential functions are an essential concept in mathematics, particularly useful in modeling growth and decay processes. An exponential function typically has the form \( f(x) = a \cdot e^{kx} \), where:

• \(a\) is a constant called the initial value or coefficient.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
• \(k\) is a constant that can represent either growth (if \(k > 0\)) or decay (if \(k < 0\)).

The function represents how a quantity changes over time or another continuous measure. For example, the function \(P = 200 e^{0.12 t}\) shows exponential growth, with an initial quantity of 200, as \(t\) increases along with the factor 0.12 dictating the rate of growth. This is important in many real-life applications, including population growth, radioactive decay, and interest calculations. Whenever dealing with exponential functions, remember they have a unique property: the rate of change of the function is proportional to its current value, which is what makes them so useful in natural settings.

Derivative Rules

When differentiating exponential functions, specific derivative rules can simplify the process.

• The derivative of an exponential function \(e^{kx}\) with respect to \(x\) is \(k \cdot e^{kx}\). This rule stems from the unique property of the exponential function where its derivative is proportional to the function itself, a feature that no other function has without modification.
• If you have a product of a constant and an exponential function, such as \(c \cdot e^{kx}\), the derivative is simply \(c \cdot (k \cdot e^{kx}) = c \cdot k \cdot e^{kx}\).

For the given exercise, using these rules made the differentiation straightforward. Start with knowing the rule, and apply it while keeping the constants separate to simplify later. Once understood and applied, you can simplify the expression further, as seen when deriving \(24 e^{0.12t}\) from \(200 \cdot 0.12 e^{0.12t}\). Familiarize yourself with these derivative rules to solve related calculus problems efficiently.

Calculus Problem-Solving

Calculus problems, like differentiating exponential functions, often follow a set process of steps to solve effectively. Here's a simple approach to tackle these problems:

• Understand the function: Break down what's given and identify any constants, the variable, and what kind of function you're dealing with. Familiarize yourself with notations such as \(e^{x}\).
• Apply the right rules: For exponential functions, identify the right derivative rule. Always start by writing down the rules relevant to your problem.
• Calculate and simplify: Apply the formulas accurately and simplify your results by doing operations like multiplying constants and clearing fractions.
• Verify your results: Double-check each step of your solution process, ensuring that all arithmetic operations and applications of derivative rules are correct.

This problem-solving mindset involves a systematic approach; it ensures accurate results and reduces errors in calculations. By following these steps, it becomes easier for students to handle calculus problems not just accurately but also with confidence. The more one practices, the better one understands the underlying concepts of calculus.