Q. 4 - Precalculus Enhanced with Graphing Utilities

Question

Let $P=(x,y)$ be a point on the graph of $y=\frac{1}{x}$

(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?

Step-by-Step Solution

Answer

The distance $d$ is $d=\sqrt{x^2+\frac{1}{x^2}}$ and the graph is 

and the least value of $d$ is $$\begin{gathered} x=-1, \\ x=1 \end{gathered}$$

Part (a) Step 1: Given information

Given the point $P(x,y)$ and the curve $y=\frac{1}{x}$

Part (a) Step 2: Calculating the distance, we get

Calculating, we get

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

Part (b) Step 1: Given information

Given the equation $d\left(x\right)=\sqrt{x^2+\frac{1}{x^2}}$

Part (b) Step 2: Sketching the graph

Sketching the graph, we get

Part (c) Step 1: Given information

Given the equation $d\left(x\right)=\sqrt{x^2+\frac{1}{x^2}}$

Part (c) Step 2: Checking from the graph

Checking, we get that $d$ is smallest when $$\begin{gathered} x=-1, \\ x=1 \end{gathered}$$