Chapter 3.1, Problem 43E

(a)

To determine

To find: the two number from 1 to 10.

Answer to Problem 43E

The numbers are $2$ and $7$ .

Explanation of Solution

The required two number can be select randomly between the 1 and 10.

Suppose selected numbers are $2$ and $7$ .

Therefore, the numbers are $2$ and $7$ .

(b)

To determine

To find: the graph of the obtained equation.

Answer to Problem 43E

The required graph is shown below.

Explanation of Solution

Given:

The general equation is $y=\left(x-a\right)\left(x+b\right)$ .

As the two selected numbers are $2$ and $7$ .

Substitute $2$ for $x$ and $7$ for $y$ in the equation $y=\left(x-a\right)\left(x+b\right)$ .

  $y=\left(x-2\right)\left(x+7\right)$

Sketch the graph of the equation $y=\left(x-2\right)\left(x+7\right)$ .

  

Therefore, the required graph is shown above.

(c)

To determine

To find: the graph of the derivative of the function.

Answer to Problem 43E

The required graph is shown below.

Explanation of Solution

Given:

The function is $y=\left(x-2\right)\left(x+7\right)$ .

Consider the function.

  $y=\left(x-2\right)\left(x+7\right)$

Differentiate the function with respect to $x$ .

  $\begin{array}{c}y=x^2+7x-2x-14\\ \frac{dy}{dx}=\frac{d}{dx}\left[x^2+7x-2x-14\right]\\ =2x\end{array}$

Graph of the equation is drawn as,

  

Therefore, the required graph is shown above.

(d)

To determine

To find: the $y$ intercept of the graph.

Answer to Problem 43E

The $y$ intercept is $5$ .

Explanation of Solution

Given:

The function is $y=\left(x-2\right)\left(x+7\right)$ .

Consider the graph.

  

As it is clear from the graph that the graph of derivative of the function cuts the vertical line $5\text{units}$ . So, the $y$ intercept is $5$ .

Therefore, the $y$ intercept is $5$ .

(e)

To determine

To find: a simple way to predict the $y$ intercept.

Answer to Problem 43E

The simple way to find the $y$ intercept is $\left(-a+b\right)$ .

Explanation of Solution

Given:

The function is $y=\left(x-a\right)\left(x+b\right)$ .

Consider the function.

Expand the function with algebraic properties.

  $\begin{array}{c}y=x^2-ax+bx+ab\\ =x^2+x\left(-a+b\right)+ab\end{array}$

Here, $\left(-a+b\right)$ is the simplest way to find the $y$ intercept.

Substitute $2$ for $a$ and $7$ for $b$ in the equation $\left(-a+b\right)$ .

  $\begin{array}{c}\left(-a+b\right)=-2+7\\ =5\end{array}$

So, result is verified.

Therefore, the simple way to find the $y$ intercept is $\left(-a+b\right)$ .