Problem 19 - University Calculus: Early Transcendentals

Question

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(e^{3 x}\) b. \(e^{-x}\) c. \(e^{x / 2}\)

Step-by-Step Solution

Answer

Antiderivatives are: a. \(\frac{1}{3}e^{3x} + C\), b. \(-e^{-x} + C\), c. \(2e^{x/2} + C\).

Step 1: Understand the Exercise

The task is to find the antiderivatives of different exponential functions. For each function, we will integrate and find the function whose derivative gives us the original function. The functions we have are in the form of exponential functions which can generally be integrated using direct formulas.

Step 2: Find the Antiderivative of \( e^{3x} \)

To find the antiderivative of \( e^{3x} \), we apply the formula for the antiderivative of exponential functions. The integral of \( e^{ax} \) with respect to \( x \) is \( \frac{1}{a}e^{ax} + C \), where \( a \) is a constant and \( C \) is the constant of integration. Here \( a = 3 \), so the antiderivative is \( \frac{1}{3}e^{3x} + C \).

Step 3: Find the Antiderivative of \( e^{-x} \)

For \( e^{-x} \), we use the same rule for exponential integrals. Here, \( a = -1 \). Therefore, the antiderivative is \( \frac{1}{-1}e^{-x} + C = -e^{-x} + C \).

Step 4: Find the Antiderivative of \( e^{x/2} \)

For \( e^{x/2} \), we have \( a = \frac{1}{2} \). Applying the formula, the antiderivative is \( \frac{1}{\frac{1}{2}}e^{x/2} + C = 2e^{x/2} + C \).

Step 5: Verify by Differentiation

To ensure the solutions are correct, differentiate each antiderivative.- Differentiating \( \frac{1}{3}e^{3x} + C \) gives \( e^{3x} \), which matches the original function.- Differentiating \( -e^{-x} + C \) results in \( e^{-x} \), confirming its correctness.- Differentiating \( 2e^{x/2} + C \) produces \( e^{x/2} \), verifying the solution.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function that have the form \( e^{ax} \), where \( e \) is Euler's number (approximately 2.71828) and \( a \) is a constant. These functions are extremely important in mathematics and various applied fields due to their unique property of constant growth or decay rates.

The base \( e \) is special because the derivative of \( e^x \) is simply \( e^x \), making it an excellent candidate for modeling natural growth processes like population increase and radioactive decay. In the context of calculus, understanding how to work with exponential functions, especially in integration and differentiation, is key for tackling more complex mathematical problems.

For the functions in the exercise, each includes the exponential base \( e \) with different constants \( a \) affixed to \( x \). These constants affect the rate of growth or decay. Integrating these exponential functions helps us find antiderivatives, which is a necessary skill for solving differential equations and analyzing changes over time.

Integration Formulas

Integration is the reverse operation of differentiation. When integrating exponential functions of the form \( e^{ax} \), we use specific formulas designed for simplicity and precision. The general antiderivative of an exponential function \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \), where \( C \) is the constant of integration.

This formula reflects that while the act of integration is fairly straightforward, remembering to multiply by \( \frac{1}{a} \) is crucial. The constant \( C \) appears because integration always accounts for an additional constant, since differentiating a constant yields zero. Thus, without \( C \), we cannot represent all possible antiderivatives.

In the exercise provided:

• For \( e^{3x} \), the antiderivative is \( \frac{1}{3}e^{3x} + C \).
• For \( e^{-x} \), it becomes \( -e^{-x} + C \).
• And for \( e^{x/2} \), the solution is \( 2e^{x/2} + C \).

These calculations demonstrate the use of the integration formula effectively and show how important constants in exponential functions play a role when integrating.

Differentiation Verification

To check the correctness of an antiderivative, differentiation verification is a key step. By differentiating the antiderivative, you should recover the original function.

Differentiation undoes integration, so a correct antiderivative, when differentiated, will return the original function. This verification process ensures accuracy, especially when constants affect the exponential rate. Using differentiation helps confirm our understanding and assures that no steps were missed during integration.

Let's verify:

• The derivative of \( \frac{1}{3}e^{3x} + C \) yields \( e^{3x} \), as expected.
• The derivative of \( -e^{-x} + C \) results in \( e^{-x} \), confirming it correctly undoes the integration.
• The derivative of \( 2e^{x/2} + C \) yields \( e^{x/2} \), returning the initial function accurately.

Differentiation verification is a practical tool, pushing students to apply their knowledge in both directions—integrating functions and then verifying via differentiation—ensuring that comprehension is complete and robust.