Chapter 1.3, Problem 47E

(a)

To determine

To find: The graphs of the function $y_1=x^2$ and $y_2=2^x$ in $\left[-5,5\right]$ by $\left[-2,10\right]$ and write the number of times both graphs cross each other.

Answer to Problem 47E

The graphs of the function $y_1=x^2$ and $y_2=2^x$ in $\left[-5,5\right]$ by $\left[-2,10\right]$ are shown in Figure (1). Both the graph the functions cross each other two times but one more time the graphs cross outside the graph window.

Explanation of Solution

Given information: The functions are $y_1=x^2$ and $y_2=2^x$ .

Calculation:

To graph the functions $y_1=x^2$ and $y_2=2^x$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y_1=x^2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{^}\boxed{2}$ . Now, press enter right hand side of the equation $y_2=2^x$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{2}\boxed{^}\boxed{\text{X}}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\text{X}^2\\ {\text{Y}}_2=2^\text{X}\end{array}$

Press the $\boxed{\text{WINDOW}}$ and adjust the window to $\left[-5,5\right]$ by $\left[-2,10\right]$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

In the graph window it appears that functions cross each other twice. But actually functions cross each other three times.

(b)

To determine

To find: Comparison of the change in values of $y_1$ and $y_2$ as $x$ moves from $1$ to $2$ , $2$ to $3$ and so on and also check the value of $x$ such that changes in $y_2$ is greater than changes in $y_1$ .

Answer to Problem 47E

The below table shows the changes in the values of $y_1$ and $y_2$ as $x$ moves from $1$ to $2$ , $2$ to $3$ and so on. The minimum value of $x$ for the changes in $y_2$ to overtake the changes in $y_1$ is $4$ .

$x$	$y_1$	$y_2$	Change in values of $y_1$	Change in values of $y_2$
$1$	$1$	$2$	$3$ $5$ $7$ $9$	$2$ $4$ $8$ $16$
$2$	$4$	$4$
$3$	$9$	$8$
$4$	$16$	$16$
$5$	$25$	$32$

Explanation of Solution

Given information:

The functions are $y_1=x^2$ and $y_2=2^x$ .

Calculation:

The functions are given as $y_1=x^2$ and $y_2=2^x$ .

The below table shows all the data of the curves:

$x$	$y_1$	$y_2$	Change in values of $y_1$	Change in values of $y_2$
$1$	$1$	$2$	$3$ $5$ $7$ $9$	$2$ $4$ $8$ $16$
$2$	$4$	$4$
$3$	$9$	$8$
$4$	$16$	$16$
$5$	$25$	$32$

The minimum value of $x$ for the changes in $y_2$ to overtake the changes in $y_1$ is $4$ .

(c)

To determine

To find: The values of $x$ such that $x^2=2^x$ .

Answer to Problem 47E

The values of $x$ such that $x^2=2^x$ are $x=-0.7667$ , $x=2$ and $x=4$ .

Explanation of Solution

Given information:

The functions are $y_1=x^2$ and $y_2=2^x$ .

Calculation:

To graph the functions $y_1=x^2$ and $y_2=2^x$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y_1=x^2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{^}\boxed{2}$ . Now press enter right hand side of the equation $y_2=2^x$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{2}\boxed{^}\boxed{\text{X}}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\text{X}^2\\ {\text{Y}}_2=2^\text{X}\end{array}$

Press the $\boxed{\text{WINDOW}}$ and adjust the window to $\left[-5,5\right]$ by $\left[-2,10\right]$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

It is observed from graph that both the curves intersect at three points $x=-0.7667,2,4$ .

Therefore, the values of $x$ such that $x^2=2^x$ are $x=-0.7667$ , $x=2$ and $x=4$ .

(d)

To determine

To find: The values of $x$ such that $x^2<2^x$ .

Answer to Problem 47E

The values of $x$ such that $x^2<2^x$ is the interval $\left(-0.7667,2\right)\cup \left(4,\infty \right)$ .

Explanation of Solution

Given information:

The functions are $y_1=x^2$ and $y_2=2^x$ .

Calculation:

To graph the functions $y_1=x^2$ and $y_2=2^x$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y_1=x^2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{^}\boxed{2}$ . Now press Enter. Enter right hand side of the equation $y_2=2^x$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{2}\boxed{^}\boxed{\text{X}}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\text{X}^2\\ {\text{Y}}_2=2^\text{X}\end{array}$

Press the $\boxed{\text{WINDOW}}$ and adjust the window to $\left[-5,5\right]$ by $\left[-2,10\right]$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

It is observed from graph that the curve $y=x^2$ lies below the curve $y=2^x$ in the interval $\left(-0.7667,2\right)\cup \left(4,\infty \right)$ .

Therefore, the values of $x$ such that $x^2<2^x$ is the interval $\left(-0.7667,2\right)\cup \left(4,\infty \right)$ .