Problem 22 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=-3 e^{-4 x} $$

Step-by-Step Solution

Answer

The general antiderivative is \( \frac{3}{4} e^{-4x} + C \).

Step 1: Identify the Function Form

The given function is \( f(x) = -3 e^{-4x} \). Notice that this function is an exponential function where the exponent is a linear expression in terms of \( x \).

Step 2: Apply the Antiderivative Formula

The antiderivative of \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \), where \( C \) is the constant of integration and \( a \) is a constant. In this problem, \( a = -4 \).

Step 3: Find the Antiderivative

To find the antiderivative of \( -3 e^{-4x} \), use the formula: \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \) and multiply by the constant coefficient \(-3\): \[ \int -3 e^{-4x} \, dx = -3 \left( \frac{1}{-4} e^{-4x} \right) + C = \frac{3}{4} e^{-4x} + C \]

Step 4: Final Step: Write the General Antiderivative

Thus, the general antiderivative of the given function is \( \frac{3}{4} e^{-4x} + C \).

Key Concepts

Exponential Functions

Exponential functions are an important class of mathematical functions where the variable, often denoted as \( x \), appears in the exponent. They have the general form \( f(x) = a e^{bx} \), where \( a \) and \( b \) are constants, and \( e \) is the base of the natural logarithms, approximately equal to 2.718. Exponential functions are used frequently in a variety of fields, such as:

• Population growth models, where the population increases exponentially.
• Radioactive decay processes, describing how substances decay exponentially over time.
• Financial calculations, including compound interest.

Using exponential functions helps describe real-world situations that grow or decay proportionally to their current value, which is why they are so prevalent in modeling various phenomena.

Integration Techniques

Integration is a fundamental concept in calculus used to find areas under curves, among other applications. Different techniques can be employed to find integrals, one of which is applying the antiderivative formula for exponential functions. When dealing with exponential functions of the form \( e^{ax} \), the antiderivative can be determined using a specific technique:

• Identify the core exponent \( ax \) and apply the formula: \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where \( C \) is a constant of integration.
• Multiply the result by any constant outside the exponential, to account for scaling effects.

These integration techniques provide a systematic method to solve integrals involving exponential functions, making the process straightforward once the formula is understood.

Constant of Integration

The constant of integration, represented by \( C \), is crucial when finding an antiderivative because it accounts for all possible vertical shifts of the function. In indefinite integrals (those without specified limits), the constant \( C \) appears because:

• Differentiation, the inverse process of integration, removes any constant shift of a function since the derivative of a constant is zero.
• Including \( C \) ensures all possible solutions to the antiderivative are covered, reflecting any initial values or conditions relevant to the problem.

For example, when finding the antiderivative of a function like \( -3 e^{-4x} \), the solution will be \( \frac{3}{4} e^{-4x} + C \). This allows the general solution to adapt to additional conditions, proving the importance of this constant in calculus.