Question

(a) Graph the function \(g\left( x \right) = {e^x} - 3{x^2}\)in the viewing rectangle \(\left( { - 1,4} \right)\)by \(\left( { - 8,8} \right)\).

(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of \(g'\). (See Example 2.8.1.)

(c) Calculate \(g'\left( x \right)\) and use this expression to graph \(g'\). Compare with your sketch in part (b).

Short answer

1. The graph of \(g\left( x \right)\) is represented below.

b. The slope of the function is 0 at \(x \approx 0.2,\,2.8\). Thus, it can be deduced that the slope is positive for \(\left( { - \infty ,\,0.2} \right)\) and negative for\(\left( {2.8,\,\infty } \right)\). The rough sketch is

c. The derivative \(g'\left( x \right)\) is \(g'\left( x \right) = {e^x} - 6x\). The graph of \(g'\left( x \right)\) is

The graph in part (b) is an approximation of this graph, which is actual one.

Step-by-step solution

Graph\(g\left( x \right)\)

The procedure to draw the graph the\(g\left( x \right) = {e^x} - 3{x^2}\)by using the graphing calculator is as follows:

Draw the graph of the function\(g\left( x \right) = {e^x} - 3{x^2}\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\({e^x} - 3{x^2}\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of graph of the function \(g\left( x \right) = {e^x} - 3{x^2}\) is shown below:

Estimate the slope

On reading the graph, it can be noted that the slope of the function is 0 at\(x \approx 0.2,\,2.8\). Thus, it can be deduced that the slope is positive for\(\left( { - \infty ,\,0.2} \right)\)and negative for\(\left( {2.8,\,\infty } \right)\).

Draw the rough graph of this function using above information as follows

Draw the graph of \(g'\left( x \right)\)

The derivative\(g'\left( x \right)\)is obtained as\(g'\left( x \right) = {e^x} - 6x\).Draw the graph of the function\(g'\left( x \right) = {e^x} - 6x\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\({e^x} - 6X\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of graph of the function \(g'\left( x \right) = {e^x} - 6x\) is shown below:

The graph in part (b) is an approximation of this graph, which is actual one.