Problem 3 - Calculus: Early Transcendental Functions

Question

Sketch several members of the family of functions defined by the antiderivative. $$\int\left(x^{3}-x\right) d x$$

Step-by-Step Solution

Answer

The antiderivative of the function \(x^{3} - x\) is \(\frac{1}{4}x^4 - \frac{1}{2}x^2 + C\), where C is the constant of integration. The family of functions defined by this antiderivative consists of an infinite number of functions, each corresponding to a different value of C. The sketch of the functions will have the same shape but will be translated vertically based on the value of C.

Step 1: Compute the Integration

To start with the solution, compute the integral of the function \(x^3 - x\). The integral of this function is obtained using the power rule of integration which states that the integral of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\). So to integrate \(x^3\), you get \(\frac{1}{4}x^4\), and to integrate \(-x\), you get \(-\frac{1}{2}x^2\). Therefore, the antiderivative of \(x^3 - x\) is \(\frac{1}{4}x^4 - \frac{1}{2}x^2 + C\), where C is the constant of integration.

Step 2: Identify the Family of Functions

Since the constant C can take any real value, there exists a family of functions expressed by the antiderivative \(\frac{1}{4}x^4 - \frac{1}{2}x^2 + C\). Each member of this family can be represented by a different value of C.

Step 3: Sketch the Function

To sketch the family of functions, several functions with different values of C should be drawn. You can start by sketching the function for C = 0, which will be \(\frac{1}{4}x^4 - \frac{1}{2}x^2\). This function has a graph that is symmetric with respect to y-axis and has a minimum at x = 0. For various other values of C, the graph will shift vertically but maintain the same shape. For example, for C = 1, the whole graph will shift one unit up.

Key Concepts

Integral Calculus

Integral calculus is a foundational part of higher mathematics, concerned with understanding and computing the area under a curve, known as the integral of a function. This field of study is essential in physics, engineering, economics, and beyond, as it allows us to calculate quantities like displacement, total accumulated wealth, or volumes of 3D objects.

Imagine a curve on a graph that represents a car's speed over time. Integral calculus helps us find out not just the speed at any given moment but the total distance traveled across a period. We achieve this by finding the antiderivative of the function representing speed, which gives us the accumulation of the car's movement over time.

Power Rule of Integration

One of the most important and commonly used rules in integral calculus is the power rule of integration. It is a shortcut method for integrating polynomials and states that for a term with the form \( x^n \), where n is a real number, the integral is \( \frac{1}{n+1}x^{n+1} \) plus the constant of integration, often denoted as C.

• For example, integrating \( x^3 \) involves increasing the exponent by 1 (to 4) and dividing by this new exponent, resulting in \( \frac{1}{4}x^4 \).
• Similarly, integrating \( -x \) (which is equivalent to \( x^1 \) with a negative sign), we get \( -\frac{1}{2}x^2 \).

This power rule doesn't apply if the exponent n is -1, as this leads to a natural logarithm function, not a power function.

Family of Functions

When we speak of a family of functions, we are referring to a set of functions that share a basic form but have slight variations due to different values of certain parameters. In the context of antiderivatives, these parameters are the constants of integration. Each different value of the constant results in a new member of the family.

The exercise provided illustrates this idea perfectly. The antiderivative \( \frac{1}{4}x^4 - \frac{1}{2}x^2 + C \) forms a big family with an infinite number of members, each corresponding to a unique value of C. When graphed, these functions look alike in shape—they are all parabolic curves—but are located at different positions along the y-axis.

Constant of Integration

Lastly, the constant of integration, often denoted as C, is an all-important concept in the world of integral calculus. Whenever we take an indefinite integral, we're finding a general antiderivative which includes the constant of integration. It accounts for the fact that there are infinitely many antiderivatives for a given function, each differing from the others by a constant amount.

In the exercise, after finding the antiderivative \( \frac{1}{4}x^4 - \frac{1}{2}x^2 \) of the given function \( x^3 - x \), we include the constant of integration C. This means we are not just looking at one function but an infinite set of functions that have the same basic curve but are translated vertically. The constant of integration is what makes the sketch of these functions look like a sequence of waves, each one the same shape but at a different level.