Question

Graphical Reasoning In Exercises 45 and \(46,\) use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window where \(g(x)=\frac{f(x+0.01)-f(x)}{0.01}\) Label the graphs and describe the relationship between them. \(f(x)=3 \sqrt{x}\)

Short answer

Both \(f(x)\) and \(g(x)\) are graphed. The function \(g(x)\) follows the derivative of \(f(x)\), so it represents the tangent's slope of \(f(x)\) at any given point.

Step-by-step solution

Define the functions

First, define the functions. You are explicitly given \(f(x) = 3 \sqrt{x}\). The function \(g(x)\), on the other hand, is defined as \(g(x)=\frac{f(x+0.01)-f(x)}{0.01}\). Here, \(g(x)\) represents an approximation of the derivative of \(f(x)\) using the difference quotient.

Graph the functions

Next, use a graphing utility to plot both functions in the same window. Pay specific attention to the scales on each axis so that both graphs can be clearly seen. Remember to label each graph accordingly.

Analyze the graphs

Finally, analyze the graphs. Look closely for any changes in \(g(x)\) when \(f(x)\) changes, and vice versa. Also, see if there are any patterns between the graphs of \(f\) and \(g\). The graph \(g(x)\) is expected to be the slope of the curve \(f(x)\) at given points as it is built on the derivative.