Problem 2 - 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 \(\frac{x^4}{4} - \frac{x^2}{2} + C\). Different functions within the family, associated to different values of \(C\), can be graphed by shifting the original function vertically. For \(C = 0, C = 1\), and \(C = -1\), the functions are \(\frac{x^4}{4} - \frac{x^2}{2}, \frac{x^4}{4} - \frac{x^2}{2} + 1\), and \(\frac{x^4}{4} - \frac{x^2}{2} - 1\), respectively.

Step 1: Calculation of the antiderivative

To calculate the antiderivative of \(x^3-x\), apply the basic power rule for integration and the property of the integral that the integral of a difference is the difference of the integrals. The antiderivative will be \(\frac{x^4}{4} - \frac{x^2}{2} + C\), where \(C\) is a constant of integration and represents the indefinite nature of the antiderivative.

Step 2: Choose different C values

To sketch different members of the family of functions, one can pick several values for the constant \(C\). For example, one can choose \(C = 0, C = 1\), and \(C = -1\). These values will yield three different functions: \(\frac{x^4}{4} - \frac{x^2}{2}, \frac{x^4}{4} - \frac{x^2}{2} + 1\), and \(\frac{x^4}{4} - \frac{x^2}{2} - 1\), respectively.

Step 3: Sketching the functions

Sketch the function \(\frac{x^4}{4} - \frac{x^2}{2}\) first. Then, to sketch the functions for different values of \(C\), one can shift the original function upward or downward by \(1\) unit to obtain the sketch for the functions \(\frac{x^4}{4} - \frac{x^2}{2} + 1\) (upward shift) and \(\frac{x^4}{4} - \frac{x^2}{2} - 1\) (downward shift). The shape of the function will not change; the different values of \(C\) only change the vertical position of the function in the graph.