Problem 33 - University Calculus: Early Transcendentals

Question

Find all possible functions with the given derivative. a. \(y^{\prime}=x\) b. \(y^{\prime}=x^{2}\) c. \(y^{\prime}=x^{3}\)

Step-by-Step Solution

Answer

a. \( y = \frac{x^2}{2} + C \); b. \( y = \frac{x^3}{3} + C \); c. \( y = \frac{x^4}{4} + C \)

Step 1: Understanding the Problem

The task is to find the original functions whose derivatives are given. Each part provides a derivative, and our goal is to find the antiderivative, or the integral, which reverses the differentiation.

Step 2: Finding the Antiderivative for Part a

For part a, we have the derivative \( y' = x \). The antiderivative of \( x \) is \( \frac{x^2}{2} + C \), where \( C \) is the constant of integration. This means the function \( y \) could be \( y = \frac{x^2}{2} + C \).

Step 3: Finding the Antiderivative for Part b

For part b, the derivative is \( y' = x^2 \). The antiderivative of \( x^2 \) is \( \frac{x^3}{3} + C \), where \( C \) is the constant of integration. Therefore, the function \( y \) is \( y = \frac{x^3}{3} + C \).

Step 4: Finding the Antiderivative for Part c

For part c, we have the derivative \( y' = x^3 \). The antiderivative of \( x^3 \) is \( \frac{x^4}{4} + C \), where \( C \) is the constant of integration. Thus, the function \( y \) is \( y = \frac{x^4}{4} + C \).

Key Concepts

Constant of Integration

When dealing with antiderivatives, there’s an important component called the "constant of integration". Whenever you find an indefinite integral, you need to remember this constant, denoted as \( C \).Here's why:

• An indefinite integral represents a family of functions.
• Each function in this family has the same derivative but differs by a constant value \( C \).
• This constant \( C \) accounts for all the potential vertical shifts of the function along the y-axis.

For instance, when you compute the antiderivative of \( x \), you get \( \frac{x^2}{2} + C \). The term \( C \) shows that there are infinitely many possible original functions, each differing by a constant amount. This is because differentiation wipes out constant terms, making the process of recovering the original function slightly uncertain without initial conditions.

Indefinite Integral

The concept of an indefinite integral is at the heart of finding antiderivatives. It's essentially the reverse process of taking a derivative.Here’s what you should know about indefinite integrals:

• An indefinite integral is written as \( \int f(x) \, dx \).
• It gives us a general function whose derivative is the integrand (the function being integrated).
• Unlike definite integrals, indefinite integrals do not have bounds, reflecting that they give us a family of potential functions and not a specific numerical answer.

For example, with \( y' = x \), the indefinite integral \( \int x \, dx = \frac{x^2}{2} + C \) provides the set of all functions whose derivative is \( x \). The inclusion of the constant \( C \) ensures the solution represents the entire family of functions that could have produced the derivative \( x \).

Differential Calculus

Differential calculus is a branch in mathematics focusing on how functions change. Key points about differential calculus include:

• It deals primarily with the concept of the derivative, which represents the rate of change of a function.
• The process of differentiation gives insight into the behavior of functions, such as slopes, tangents, and velocities.
• Finding derivatives is a central task in differential calculus.

When you're given a task like recovering a function from its derivative, you are essentially reverse-engineering a problem in differential calculus. By applying integration, you trace back from a function's rate of change to discover the original function set. This highlights how integral and differential calculus are complementary, with one tracing change and the other tracing the origin of that change.