Problem 32 - Applied Calculus

Question

Find an antiderivative. $$f(z)=e^{z}+3$$

Step-by-Step Solution

Answer

The antiderivative of \( f(z) = e^z + 3 \) is \( e^z + 3z + C \).

Step 1: Recognize the Form of the Function

The given function is \( f(z) = e^z + 3 \). This function is composed of two terms: \( e^z \) and a constant \( 3 \). We know from calculus that the antiderivative of a sum of functions is the sum of their antiderivatives.

Step 2: Find the Antiderivative of \( e^z \)

The function \( e^z \) is known to have itself as an antiderivative, because the derivative of \( e^z \) with respect to \( z \) is \( e^z \). Therefore, the antiderivative of \( e^z \) is \( e^z \).

Step 3: Find the Antiderivative of the Constant Term

The antiderivative of a constant \( c \) is \( cz \), where \( z \) is the variable of integration. Here, the constant is \( 3 \), so its antiderivative is \( 3z \).

Step 4: Combine Antiderivatives

Adding the antiderivatives from the previous steps, we get: the antiderivative of \( f(z) = e^z + 3 \) is \( e^z + 3z \).

Step 5: Add the Constant of Integration

Since antiderivatives can differ by a constant, we add a constant \( C \) to the complete antiderivative. Therefore, the general antiderivative is \( e^z + 3z + C \).

Key Concepts

Calculus

Calculus is a branch of mathematics that helps us understand changes. It is divided primarily into two areas: differentiation and integration. Differentiation focuses on rates of change, similar to figuring out how the speed of a car changes over time. On the other hand, integration is about accumulating quantities, much like how much distance a car has covered. In this exercise, we focus on integration, specifically finding an antiderivative. An antiderivative is the reverse process of differentiation. Instead of finding how something changes, we try to determine the original function before the change. Calculus provides us with methods to find these antiderivatives, helping us solve problems in physics, engineering, and everyday life.

Integration

Integration is often described as the process of finding the whole from its parts. When we integrate a function, we sum up infinitely small parts to get the total. In our exercise, the goal is to find the antiderivative, which is essentially a type of integration.There are specific techniques and rules for integration. Here, we used the rule that the antiderivative of a sum is the sum of their antiderivatives:

• The antiderivative of an exponential function, such as \( e^z \).
• The antiderivative of a constant, such as 3.

These are basic integration rules, and they pave the way for finding more complex antiderivatives. Integrating exponential functions requires knowledge of these inherent characteristics, making it a fascinating topic in calculus.

Exponential Function

Exponential functions are functions where the variable is in the exponent, such as \( e^z \). In calculus, \( e^x \) is unique because it is its own derivative and antiderivative. This property is a defining feature of exponential functions and simplifies integration significantly.Understanding exponential functions is key in many real-world applications, such as:

• Growth models, like population or finance.
• Decay processes, such as radioactive decay.

In the exercise, the term \( e^z \) was a straightforward part to integrate, knowing its antiderivative is the same. Such knowledge emphasizes the beauty and simplicity inherent within exponential functions.

Constant of Integration

The constant of integration, typically represented by \( C \), is added when finding antiderivatives. This constant accounts for any constant that might have been lost when differentiating.In the exercise, when calculating the antiderivative, we ended with an expression \( e^z + 3z + C \). Here, \( C \) represents an unknown constant because differentiation of a constant is zero. Thus, adding \( C \) ensures all possible antiderivatives are covered.Including the constant of integration is crucial:

• It allows for general solutions for differential equations.
• Ensures completeness of the integration process.

Remembering \( C \) is vital in calculus, as it acknowledges the full scope of potential solutions.