Problem 27 - Thomas Calculus

Question

Find all possible functions with the given derivative. a. \(y^{\prime}=x \quad\) b. \(y^{\prime}=x^{2} \quad\) 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: Understand the Problem

We're asked to find functions whose derivatives are given: (a) For the derivative \(y' = x\), we need a function whose rate of change with respect to \(x\) is \(x\).(b) For \(y' = x^2\), the function's rate of change is \(x^2\).(c) For \(y' = x^3\), the function's rate of change is \(x^3\). The task is essentially to find the antiderivatives of these expressions.

Step 2: Recall the Antiderivative Concept

The antiderivative, or the indefinite integral, of a function \(f(x)\) is a function whose derivative is \(f(x)\). So, for this problem, we will integrate each given derivative to find the corresponding function \(y(x)\), plus the constant of integration \(C\).

Step 3: Integrate the First Derivative

For \(y' = x\): - Integrate \(x\) with respect to \(x\). - The integral is \(\int x \, dx = \frac{x^2}{2} + C\). - So, the function is \(y(x) = \frac{x^2}{2} + C\).

Step 4: Integrate the Second Derivative

For \(y' = x^2\): - Integrate \(x^2\) with respect to \(x\). - The integral is \(\int x^2 \, dx = \frac{x^3}{3} + C\). - So, the function is \(y(x) = \frac{x^3}{3} + C\).

Step 5: Integrate the Third Derivative

For \(y' = x^3\): - Integrate \(x^3\) with respect to \(x\). - The integral is \(\int x^3 \, dx = \frac{x^4}{4} + C\). - So, the function is \(y(x) = \frac{x^4}{4} + C\).

Step 6: Summarize Results

After completing the integration, we have the following functions:- (a) \(y(x) = \frac{x^2}{2} + C\).- (b) \(y(x) = \frac{x^3}{3} + C\).- (c) \(y(x) = \frac{x^4}{4} + C\). The constant \(C\) represents the family of solutions.

Key Concepts

Indefinite Integrals

Indefinite integrals play a key role in solving problems related to finding antiderivatives. An indefinite integral of a function is presented as \(\int f(x) \, dx\). It represents a family of functions whose derivatives yield \(f(x)\). These functions are also known as antiderivatives.
The main goal of indefinite integrals is to "reverse" the process of differentiation. For instance, given the derivative \(y' = x\), the indefinite integral \(\int x \, dx\) helps us find the original function \(y(x)\) that resulted in that derivative.

• For \(y' = x\), the indefinite integral is \(\frac{x^2}{2} + C\).
• For \(y' = x^2\), the indefinite integral is \(\frac{x^3}{3} + C\).
• For \(y' = x^3\), the indefinite integral is \(\frac{x^4}{4} + C\).

The symbol \(C\) is key here as it represents the constant of integration, which will be explained further. Indefinite integrals are essential because most functions have infinite possible antiderivatives, hence the "indefinite" nature.

Integration Techniques

Integration involves various techniques to find the antiderivative of a function. Here, we'll focus on the power rule, which is particularly useful for polynomials.
The power rule for integration is a fundamental technique and is expressed as: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \] where \(n eq -1\). This technique is applicable when integrating simple power functions like \(x\), \(x^2\), and \(x^3\). Let's break it down with examples:

• Example 1: \(y' = x\)
  Using the power rule, \(\int x \, dx = \frac{x^2}{2} + C\).
• Example 2: \(y' = x^2\)
  Apply the rule to get \(\int x^2 \, dx = \frac{x^3}{3} + C\).
• Example 3: \(y' = x^3\)
  Integrating gives \(\int x^3 \, dx = \frac{x^4}{4} + C\).

By mastering integration techniques like the power rule, you can tackle a wide range of calculus problems. Each integration step yields a family of functions due to the constant \(C\), which will be elaborated on next.

Constant of Integration

The constant of integration, denoted by \(C\), is a critical part of the solution to indefinite integrals. It accounts for the fact that differentiation of a constant results in zero, meaning any constant could potentially be the correct solution.
Whenever an indefinite integral is evaluated, such as \(\int x^2 \, dx = \frac{x^3}{3} + C\), \(C\) reflects infinitely many possible solutions because adding any constant to the antiderivative still satisfies the same derivative equation. This ensures that the set of all possible functions is captured.
The concept of the constant of integration can be visualized as having an infinite number of vertical shifts of the graph of the antiderivative function. For example:

• Adding \(C\) to \(\frac{x^2}{2}\) produces multiple unique functions, such as \(y = \frac{x^2}{2} + 2\), \(y = \frac{x^2}{2} - 3\), and so on, each corresponding to different initial conditions.

Understanding \(C\) is vital to grasp how integration results in a family of functions, giving you the flexibility to apply specific conditions when needed to find a particular solution.