Chapter 2.6, Problem 2AYU

2. $P\text{ = }\left(x,y\right)$ be a point on the graph of $y\text{ = }{x}^2-\text{ 8}$.

(a) Express the distance d from P to the point $\left(0,-1\right)$ as a function of x.

(b) What is d if $x\text{ = 0}$?

(c) What is d if $x\text{ = 1}$?

(d) Use a graphing utility to graph $d = d\left(x\right)$.

(e) For what values of x is d smallest?

To determine

To find:

a. To express the distance $d$ from $p$ to the point $\left(0,-1\right)$ as a function of $x$.

Answer to Problem 2AYU

Solution:

a. $\sqrt{x^4-13x^2+49}$

Explanation of Solution

Given:

$y=x^2-8$

Calculation:

a. To express the distance $d$ from $P$ to the point $\left(0,-1\right)$ as a function of $x$:

$d=\sqrt{{\left(x-0\right)}^2+{\left(y+1\right)}^2}=\sqrt{x^2+y^2+1+2y}$

Since $P$ is a point on the graph $y=x^2-8$. Substitute $x^2-8$ for $y$. Then

$d\left(x\right)=\sqrt{x^2+{\left(x^2-8\right)}^2+1+2\left(x^2-8\right)}$

$=\sqrt{x^2+x^4-16x^2+64+1+2x^2-16}$

$=\sqrt{x^4-13x^2+49}$

To determine

To find:

b. To find $d$ if $x=0$.

Answer to Problem 2AYU

Solution:

b. 7

Explanation of Solution

Given:

$y=x^2-8$

Calculation:

b. To find $d$ if $x=0$

$d\left(0\right)=\sqrt{49}=7$

To determine

To find:

c. To find $d$ if $x=-1$.

Answer to Problem 2AYU

Solution:

c. $6.08$

Explanation of Solution

Given:

$y=x^2-8$

Calculation:

c. To find $d$ if $x=-1$:

$d\left(1\right)=\sqrt{1-13+49}=\sqrt{37}=6.08$

To determine

To find:

d. To graph $d=d\left(x\right)$.

Answer to Problem 2AYU

Solution:

c.

Explanation of Solution

Given:

$y=x^2-8$

Calculation:

d. To graph $d=d\left(x\right)$:

To determine

To find:

e. To find values of $x$ is $d$ smallest.

Answer to Problem 2AYU

Solution:

e. $x\approx -2.55 \text{or} 2.55$

Explanation of Solution

Given:

$y=x^2-8$

Calculation:

e. To find values of $x$ is $d$ smallest:

From the graph it can be seen that when $x=-2.55 \text{or} 2.55, d\left(=2.60\right)$ attains its minimum.

Therefore, $d$ is smallest when $x\approx -2.55 \text{or} 2.55$.