Problem 10 - Applied Calculus

Question

Decide whether the expression is a number or a family of functions. (Assume \(f(x)\) is a function.) $$\int f(5) d x$$

Step-by-Step Solution

Answer

The expression is a family of functions.

Step 1: Identify the Components

We have an integral, which is usually used to find the area under a curve, represented as \(\int f(5) \, dx\). In this expression, \(f(5)\) is a constant because \(f\) is a function applied to the constant number 5, which results in a constant value.

Step 2: Determine the Nature of the Integral

Integrating a constant with respect to \(x\) results in a linear function. Based on integral calculus, \(\int c \, dx = cx + C\), where \(c\) is a constant and \(C\) is the constant of integration.

Step 3: Solve the Integral

Applying the integration rule for a constant, we get \(\int f(5) \, dx = f(5)x + C\). This result represents a family of functions, where each function is of the form \(f(5)x\) plus some constant \(C\).

Step 4: Conclusion

The expression represents a family of functions, as varying the constant \(C\) results in different functions respecting the linear form.

Key Concepts

Family of Functions

When we talk about a "family of functions," we're referring to a set of functions that share a similar structure but can differ by certain parameters. In the context of integral calculus, an integral can represent numerous functions, differing only by a constant term. This is an important concept because when you integrate, you're not just getting one "answer," but rather a family of potential solutions.
For example, in the solution to the exercise, \[ \int f(5) \, dx = f(5)x + C \]each function in this family has the same form of \( f(5)x \), with \( C \) being the variable element.

• The variable \( C \), known as the constant of integration, allows the functions to differ slightly.
• Even if two functions look identical apart from the value of \( C \), they are considered different.

So, each unique function from the integration process is part of the broader family of functions.

Constant of Integration

In integral calculus, the constant of integration, denoted as \( C \), is an essential part of solving indefinite integrals. The reason this constant is crucial is because integration is the reverse of differentiation, and when you differentiate a constant, it disappears. So, to cover all possibilities, we add this constant back when finding an antiderivative.
Imagine you're given a slope, and you need to find the original line; many lines can have the same slope but different y-intercepts.

• The constant \( C \) represents this unknown starting point.
• It ensures that all possible antiderivatives are accounted for.

In our example from the exercise, adding \( C \) turns the expression \( f(5)x \) into a general solution, showing its dependence on this constant to encompass a family of lines, each slightly varied with \( C \).

Definite and Indefinite Integrals

Integrals in calculus come in two main types: definite and indefinite, each serving different purposes. Indefinite integrals, like the one in our problem \[ \int f(5) \, dx = f(5)x + C \]do not include limits of integration and result in a general form or "family" of solutions. They provide a function whose derivative gives the integrand, plus the constant of integration \( C \).

• Indefinite integrals tell us about a whole set of functions.
• They are essential when the upper and lower bounds of integration are not specified.

On the other hand, definite integrals do involve limits, providing a numerical value representing the area under a curve between those limits. While definite integrals focus on precise answers, indefinite integrals encompass broader possibilities by accounting for every potential y-intercept through the constant \( C \). This makes indefinite integrals crucial when you're asked to find a family of functions rather than a single, fixed value.