Chapter 0.3, Problem 47E

(a)

To determine

To Graph:

The given functions $y_1$ and $y_2$ in the window $\left[-5,5\right]\text{ by }\left[-2,10\right]$ .

To State:

How many times the two graphs cross.

Answer to Problem 47E

The two graphs cross twice in the given window. They cross three times in total.

Explanation of Solution

Given:

  $y_1=x^2$ and $y_2=2^x$ .

Concepts Used:

Graphing given functions using a graphing calculator.

Calculations:

Graph $y_1=x^2$ and $y_2=2^x$ using a graphing calculator.

  

Conclusion:

The two graphs cross twice in the given window. They cross three times in total.

(b)

To determine

To Compare:

The corresponding changes in $y_1$ and $y_2$ as $x$ changes from $1$ to $2$ , from $2$ to $3$ and so on.

To State:

How large must $x$ be for the changes in $y_2$ to overtake the change in $y_1$ .

Answer to Problem 47E

The corresponding changes in $y_1$ are larger than that in $y_2$ till $x<3$ .

  $x$ must be larger than $3$ for the changes in $y_2$ to overtake the change in $y_1$ .

Explanation of Solution

Given:

  $y_1=x^2$ and $y_2=2^x$ .

Concepts Used:

Evaluating a function for various inputs and tabulating the results.

Making a difference table.

Calculations:

Tabulate the values of $y_1=x^2$ and $y_2=2^x$ for $1,2,3,4,\text{ and, }5$ .

$x$	$y_1$	$y_2$
1	1	2
2	4	4
3	9	8
4	16	16
5	25	32

Make a difference table for the two functions.

change in $x$	change in $y_1$	change in $y_2$
1-2	3	2
2-3	5	4
3-4	7	8
4-5	9	16

Conclusion:

The corresponding changes in $y_1$ are larger than that in $y_2$ till $x<3$ .

  $x$ must be larger than $3$ for the changes in $y_2$ to overtake the change in $y_1$ .

(c)

To determine

To solve:

The equation $x^2=2^x$ for $x$ .

Answer to Problem 47E

  $x\approx -0.76667$ , $x=2$ , and, $x=4$ .

Explanation of Solution

Given:

  $y_1=x^2$ and $y_2=2^x$ .

Concepts Used:

The solutions of the equation $f\left(x\right)=g\left(x\right)$ are the abscissae ( $x$ coordinates) of the intersection points of the graphs $y=f\left(x\right)$ and $y=g\left(x\right)$ .

Calculations:

Determine the intersection points of $y_1=x^2$ and $y_2=2^x$ using a graphing calculator.

  

Conclusion:

  $x\approx -0.76667$ , $x=2$ , and, $x=4$ are the solutions of the equation $x^2=2^x$ .

(d)

To determine

To solve:

The inequality $x^2<2^x$ for $x$ .

Answer to Problem 47E

  $x\in \left(-0.7667,3\right)$

Explanation of Solution

Given:

  $y_1=x^2$ and $y_2=2^x$ .

Concepts Used:

The solutions of the equation $f\left(x\right)<g\left(x\right)$ are the values of $x$ where the graph of $y=g\left(x\right)$ lies above the graph of $y=f\left(x\right)$ .

Calculations:

Determine the interval where the values of $x$ where the graph of $y=g\left(x\right)$ lies above the graph of $y=f\left(x\right)$ using a graphing calculator.

  

Conclusion:

The graph of $y=g\left(x\right)$ lies above the graph of $y=f\left(x\right)$ in the interval $\left(-0.7667,3\right)$ .