First and Second Derivatives

About First and Second Derivatives

The graph of a function \(f(x)\) is closely related to the graphs of its first and second derivatives:

When the graph of the function \(f(x)\) is increasing, the value of \(f'(x)\) is positive, so the graph of \(f'(x)\) will lie above the \(x\)-axis.
When the graph of the function \(f(x)\) is decreasing, the value of \(f'(x)\) is negative, so the graph of \(f'(x)\) will lie below the \(x\)-axis.
When the slope of the line tangent to \(f(x)\) is 0, the value of \(f'(x)\) is 0, so the graph of \(f'(x)\) will have an \(x\)-intercept.
When the graph of the function \(f(x)\) is concave up, the value of \(f''(x)\) is positive, so the graph of \(f''(x)\) will lie above the \(x\)-axis.
When the graph of the function \(f(x)\) is concave down, the value of \(f''(x)\) is negative, so the graph of \(f''(x)\) will lie below the \(x\)-axis.
When the graph of \(f(x)\) has an inflection point, the value of \(f''(x)\) is 0, so the graph of \(f''(x)\) will have an \(x\)-intercept.
If \(f(x)\) is a polynomial with degree \(n\), then \(f'(x)\) will be a polynomial of degree \(n-1\) and \(f''(x)\) will be a polynomial of degree \(n-2\).

Using these clues, it is possible to determine which of the graphs in this applet is the original function, which is its first derivative, and which is its second derivative.

Using the Applet

The applet will generate three graphs, representing the functions \(f(x)\), \(f'(x)\), and \(f''(x)\). The function \(f(x)\) will be a polynomial (that is randomly generated using a selection of points within the graphing window and Lagrange interpolation). By studying the three graphs and the properties above, decide which of the three graphs is the actual function \(f(x)\), and click on it. You'll then be asked to decide which of the two remaining graphs is \(f'(x)\).

About First and Second Derivatives

Using the Applet

About this Applet