Chapter 11.1, Problem 14E

Solution

To evaluate: The graph of the function $f\left(x\right)=x\cos x^2$ in radian mode on graphing calculator. Also, check whether $f^{\prime }\left(2\right)$ is positive, negative or zero by the slope of the graph.

The graph of the function $f\left(x\right)=x\cos x^2$ is shown in figure (1) and $f^{\prime }\left(2\right)$ is positive.

Given information: The given function is $f\left(x\right)=x\cos x^2$ .

Calculation:

The ratio of a square viewing window on the graphing calculator is $1.5:1$ . Consider a square viewing window $\left[-7.5,7.5\right]$ by $\left[-5,5\right]$ for the given function.

The derivative $f^{\prime }\left(2\right)$ describe the slope of the tangent line of the function at $x=2$ .

To graph the function $f\left(x\right)=x\cos x^2$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key. Enter the function $f\left(x\right)=x\cos x^2$ after the symbol ${\text{Y}}_1$ . Press the $\boxed{\text{GRAPH}}$ key to product the graph of the function in a square viewing window $\left[-7.5,7.5\right]$ by $\left[-5,5\right]$ .

Now, to draw the tangent line at $x=2$ select $\text{Tangent}($ under the $\text{DRAW}$ key and enter $2$ . Press $\boxed{\text{ENTER}}$ key to produce the tangent line to the graph as shown below in Figure (1).

  

Figure (1)

From the graph, it can be observed that the tangent line rises from left to right. So, $f^{\prime }\left(2\right)$ is positive.

Thus, the graph of the function $f\left(x\right)=x\cos x^2$ is shown in figure (1) and $f^{\prime }\left(2\right)$ is positive.