Problem 12 - Calculus with Concepts in Calculus

Question

Find all antiderivatives of the given function. $$ e^{-x} $$

Step-by-Step Solution

Answer

The antiderivative is \[-e^{-x} + C\].

Step 1: Identify the Function

The function given is the exponential function with a negative exponent, specifically \[f(x) = e^{-x}\]We are tasked with finding the antiderivative of this function.

Step 2: Recall the Antiderivative Rule for Basic Exponentials

Recall from calculus that the antiderivative of the exponential function \(e^{ax}\) is given by \[\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\]where \(a\) is a constant and \(C\) is the constant of integration.

Step 3: Apply the Rule to the Function

Applying the above rule to our specific function \(f(x) = e^{-x}\), we see that \(a = -1\). Thus, the antiderivative is \[\int e^{-x} \, dx = \frac{1}{-1} e^{-x} + C = -e^{-x} + C\]

Step 4: Conclusion

The antiderivative of \(e^{-x}\) is \[-e^{-x} + C\]where \(C\) is any constant.

Key Concepts

Exponential Function

An exponential function is one of the most commonly encountered functions in mathematics. It is written in the form \(e^{x}\), where \(e\) is Euler's number, approximately equal to 2.71828. Exponential functions are unique because they grow at a constant percentage rate. This means that the rate at which \(e^{x}\) increases is proportional to its current value.
A key feature of exponential functions is their behavior with negative exponents. In the function \(e^{-x}\), the negative sign in the exponent indicates that the function represents exponential decay rather than growth. While \(e^{x}\) grows rapidly, \(e^{-x}\) decreases as \(x\) increases, approaching zero.
This kind of behavior makes it very useful in modeling real-world processes, such as radioactive decay or cooling of substances, where quantities decrease over time.

Calculus

Calculus is a branch of mathematics that deals with change and motion. It has two main branches: differential calculus and integral calculus. While differential calculus is concerned with the concept of a derivative and rates of change, integral calculus is all about accumulation and areas under curves.
When we find the antiderivative or the integral of a function, we're essentially reversing the process of differentiation. In this way, we're calculating the family of functions which, when differentiated, yield the given function. The act of finding antiderivatives or integrals encompasses the vast world of integration, a core part of calculus.
In the exercise provided, solving for the antiderivative involves applying the rules of integral calculus to an exponential function. Applying the specific antiderivative formula for exponentials allows us to find the general solution, showcasing how calculus provides the tools necessary for both solving and understanding complex mathematical problems.

Constant of Integration

The constant of integration, denoted as \(C\), is critical in the context of finding antiderivatives. In calculus, when we determine an antiderivative, we often face an infinite number of possible solutions, differing by a constant. This happens due to the nature of differentiation: the derivative of any constant is zero.
Thus, when working backward to find an antiderivative, there are countless functions that have the same derivative as the function given. To account for this infinite set of solutions, we add a constant, \(C\), to the antiderivative expression.
This constant reflects that any function differing only by a constant from another will also be an antiderivative of the same original function. In practical terms, it helps adjust the function to fit specific conditions, such as initial values or boundary conditions, that might be provided in a problem scenario.