Problem 18 - Applied Calculus

Question

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(P(t)=12.41(0.94)^{t}\)

Step-by-Step Solution

Answer

The derivative of \(P(t) = 12.41 (0.94)^t\) is \(12.41 \cdot (0.94)^t \ln(0.94)\).

Step 1: Identify the function type

The function given is in the form of an exponential function where the base of the exponent is a constant (0.94) and the variable is in the exponent (\(t\)). This function is expressed as \(P(t) = 12.41 \cdot (0.94)^t\).

Step 2: Apply the differentiation formula for an exponential function

The derivative of an exponential function \(a^x\) is \(a^x \ln(a)\). In this case, the base \(a = 0.94\). Therefore, the differentiation of the function \((0.94)^t\) with respect to \(t\) is \((0.94)^t \ln(0.94)\).

Step 3: Multiply by the constant factor

The entire function is multiplied by the constant 12.41. So, the derivative of the function \(P(t) = 12.41 \cdot (0.94)^t\) is obtained by multiplying the constant 12.41 with the derivative of \((0.94)^t\). Thus, the derivative is \(12.41 \cdot (0.94)^t \ln(0.94)\).

Step 4: Simplify the expression

The expression for the derivative of the function is \(12.41 \cdot (0.94)^t \ln(0.94)\). This is already a simplified expression, representing the rate of change or the slope of \(P(t)\) at any point \(t\).

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In this exercise, the base is 0.94, which is a fixed number, and the exponent is the variable \(t\). This forms the function \(P(t) = 12.41 \cdot (0.94)^t\).

There are a few key characteristics of exponential functions:

• The base of the exponent is constant, while the exponent itself is a variable.
• They model scenarios where growth or decay happens at a consistent rate.
• They are continuous and differentiable across their entire domain.

Exponential functions are widely used in various fields like finance, science, and population dynamics where processes change rapidly.

Derivative Formula

Differentiation allows us to find the rate at which one quantity changes with respect to another. For an exponential function like \(a^x\), the derivative can be found using the formula \(a^x \ln(a)\). Here, \(a\) is the base of the exponential term. This formula is derived from the general rule of differentiating exponential functions.

In our exercise, we have \((0.94)^t\), so the derivative with respect to \(t\) becomes \((0.94)^t \ln(0.94)\).

This principle is useful for understanding how exponential functions change and helps in determining the slope at any given point. Remember that the natural logarithm \(\ln(a)\) signifies how many times a base 'e' must be multiplied by itself to reach another number.

Constant Multiplication

When a constant is multiplied by a function, its effect on differentiation is simple yet powerful. The constant multiplication rule in calculus states that if you have a function \(f(x)\) multiplied by a constant \(c\), the derivative is \(c \cdot f'(x)\).

In our example, \(P(t) = 12.41 \cdot (0.94)^t\), the factor 12.41 is constant. Thus, the derivative becomes \(12.41 \cdot (0.94)^t \ln(0.94)\).

This rule emphasizes that only the scale of the derivative changes and not its functional nature. This process allows us to handle differentiations of complex functions where constants are involved, making calculus computations more manageable.

Simplification in Calculus

Simplification is a crucial step in calculus to ensure expressions are in the neatest and most usable form. In our problem, the derivative \(12.41 \cdot (0.94)^t \ln(0.94)\) is already simplified.

Simplification involves:

• Reducing expressions to fewer terms without changing their value.
• Making the expression easier to interpret and use in further calculations.

Simplified derivatives allow us to easily understand the rate of change of a function at any moment, aiding in interpretations and predictions in real-world processes.