Chapter 2.6, Problem 4AYU

4. $P\text{ = }\left(x,y\right)$ be a point on the graph of $y\text{ = }\frac{1}{x}$.

(a) Express the distance d from P to the origin as a function of x.

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

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

To determine

To find:

a. To express the distance $d$ from $p$ to the origin as a function of $x$.

Answer to Problem 4AYU

Solution:

a. $\sqrt{\frac{x^4+1}{x^2}}$

Explanation of Solution

Given:

$y=\frac{1}{x}\cdot n$

Calculation:

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

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

Since $P$ is a point on the graph $y=\frac{1}{x}$. Substitute $\frac{1}{x}$ for $y$. Then

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

$=\sqrt{\frac{x^4+1}{x^2}}$

To determine

To find:

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

Answer to Problem 4AYU

Solution:

b.

Explanation of Solution

Given:

$y=\frac{1}{x}\cdot n$

Calculation:

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

To determine

To find:

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

Answer to Problem 4AYU

Solution:

c. $-1\text{ or }1$

Explanation of Solution

Given:

$y=\frac{1}{x}\cdot n$

Calculation:

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

From the graph it can be seen that when $x=-1 \text{or} 1, d$ attains its minimum value $1.414$. Therefore, $d$ is smallest when $x=-1 \text{or} 1$.