Problem 1 - Calculus: Early Transcendental Functions

Question

In exercises \(1-4,\) sketch several members of the family of functions defined by the antiderivative. $$\int x^{3} d x$$

Step-by-Step Solution

Answer

The antiderivative of \(x^3\) is \(\frac{1}{4}x^4 + C\) which represents a family of quartic functions. It is a collection of upwards opening, parabola-like curves, which are displaced vertically depending on the value of C

Step 1: Finding the Antiderivative

The antiderivative of \(x^{3}\) can be found using the power rule for integration, which states that the integral of \(x^n\) with respect to x is \(\frac{1}{n+1}x^{n+1}\). So, by applying the power rule, the antiderivative of \(x^{3}\) is \(\frac{1}{4}x^{4}\). But the general antiderivative should also have the constant 'C'. So, the antiderivative should be \(\frac{1}{4}x^{4} + C\).

Step 2: Sketching the Function

The function \(\frac{1}{4}x^{4} + C\) is a quartic function. It represents a family of curves, where 'C' is any real number. For different values of C, you can generate different members of the family. This function will always be a upwards opening parabola-like curve. Their differences will be on the vertical position, based on the value of 'C'. For example, if C=0, the graph touches the origin. If C is a positive value, the graph shifts up, and if C is negative, the graph shifts down.

Step 3: Drawing the Graphs

So, to sketch several members of the family of functions, different values for C will be chosen, and the function will be plotted for each of those values. Each curve represents a different member of the family of functions

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental technique used to find antiderivatives of functions. It's particularly straightforward when dealing with polynomials, where integration essentially "undoes" the process of differentiation. By this rule, the integral of a function of the form \(x^n\) is computed as \[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \] where \( n eq -1 \). This method helps in finding the antiderivative by increasing the power of \(x\) by one and then dividing by this new power. For instance, for the problem \( \int x^3 \, dx \), applying the power rule gives us \[ \frac{1}{4} x^4 + C \] ensuring the process is simple and systematic. This rule highlights the relationship between integration and differentiation, further showing how each operation reverses the other.

Quartic Function

A quartic function is a polynomial of degree four, represented generally as \(ax^4 + bx^3 + cx^2 + dx + e\). In our exercise, the function \[ \frac{1}{4}x^4 + C \] is a specific type of quartic polynomial because it includes the term \(x^4\). The graph of a quartic function typically has a shape similar to a parabola but can have more complex behavior, like multiple turning points. However, in the case of \( \frac{1}{4}x^4 + C \), the function is simpler and resembles a wider U-shape, owing to the positive leading coefficient. This indicates that the graph opens upward, being symmetric about the y-axis when \( C = 0 \). Changes in the values of \( C \) shift the graph vertically up or down, but do not affect its quartic nature.

Family of Functions

The concept of a family of functions arises when considering functions that vary by a specific parameter, resulting in multiple related curves. The antiderivative \( \frac{1}{4}x^4 + C \) represents a family of quartic functions, where \( C \) is the flexible parameter. Each distinct value of \( C \) creates a different member of this family. Therefore, changing \( C \) translates the graph of the function vertically, leading to various curves all sharing the same shape.

• If \( C = 0 \), the graph passes through the origin.
• Positive \( C \) values move the graph up.
• Negative \( C \) values shift it down.

Thus, sketching several members by varying \( C \) offers a visual context to understand the influence \( C \) has, maintaining the structural integrity of their quartic nature.

Constant of Integration

In the integration process, the constant of integration, \( C \), holds important significance. It arises because antiderivatives are not unique; thus, the same derivative can correspond to infinitely many original functions differing only by a constant. While calculating indefinite integrals, this arbitrary constant \( C \) is added to represent all possible vertical shifts of the antiderivative's graph. In our problem, once we find \( \frac{1}{4}x^4 \) as the antiderivative of \(x^3\), we introduce \( C \) to reflect these infinite possibilities. This characteristic implies that there are endless versions of the function, all shifted by different amounts along the y-axis. Therefore, this constant is crucial for accurately depicting the complete set of potential functions derived through integration.