Problem 31 - Thomas Calculus in SI Units

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. \(y = \frac{1}{2}x^2 + C\); b. \(y = \frac{1}{3}x^3 + C\); c. \(y = \frac{1}{4}x^4 + C\).

Step 1: Understanding Antiderivatives

The problem involves finding all functions whose derivatives match the given expressions. This involves finding the antiderivative, also known as the indefinite integral, of the given expression. The antiderivative may include a constant, as the derivative of a constant is zero.

Step 2: Solving part a

For part a, we are looking for a function whose derivative is \(y' = x\). The antiderivative of \(x\) is found by increasing the power of \(x\) by one and dividing by the new power:\[ y = \int x \, dx = \frac{1}{2}x^2 + C \]where \(C\) is the constant of integration.

Step 3: Solving part b

For part b, find the function whose derivative is \(y' = x^2\). The antiderivative of \(x^2\) involves increasing the power by one to get \(x^3\) and dividing by the new power:\[ y = \int x^2 \, dx = \frac{1}{3}x^3 + C \]where \(C\) is the constant of integration.

Step 4: Solving part c

In part c, we seek a function whose derivative is \(y' = x^3\). Increase the power of \(x^3\) by one and divide by the new power to get the antiderivative:\[ y = \int x^3 \, dx = \frac{1}{4}x^4 + C \]where \(C\) is the constant of integration.

Key Concepts

Indefinite Integral

An indefinite integral, also known as an antiderivative, is essentially the reverse process of differentiation. In this sense, while differentiation gives us the rate of change of a function, the indefinite integral helps us find the original function when we know the derivative. The process aims to determine all possible functions whose derivative is the given function. These functions are unique up to an additive constant.

• For example, if we know the derivative of a function is \(x\), then the indefinite integral will be \(\int x \, dx\),
• The result includes an integration constant, denoted as \(C\), making the solution: \(\frac{1}{2}x^2 + C\).

This general formula is possible because the derivative of a constant is zero, meaning any constant added to our function would not change its derivative.

Constant of Integration

The constant of integration, represented by \(C\), arises from the fact that integration is essentially finding a family of functions. All functions in this family have the same derivative, as the derivative of a constant is zero.

• For instance, for \(\int x^2 \, dx\), which solves to \(\frac{1}{3}x^3 + C\),
• Every possible value of \(C\) represents a different antiderivative of the same derivative.

This inclusion of \(C\) ensures all potential initial conditions and scenarios are covered, providing the most general solution possible for the antiderivative. Without including \(C\), we would miss the broader family of solutions.

Power Rule for Integration

The Power Rule for Integration is a fundamental tool that helps us find antiderivatives quickly and easily. It is applied when integrating functions of the form \(x^n\). The rule states:

• When integrating \(x^n\), add 1 to the exponent and divide by the new exponent: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\).

This is effective because this operation reverses the Power Rule for differentiation.

• For example, applying it to \(y' = x\) yields \(\frac{1}{2}x^2 + C\),
• Similarly, applying it to \(y' = x^3\) results in \(\frac{1}{4}x^4 + C\).

The Power Rule is simple, yet powerful, and offers an rapid way to tackle polynomials. It's essential to practice this rule to become comfortable applying it to various problems.