/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q. 1 Let P=(x,聽y)聽be a point on the... [FREE SOLUTION] | 91影视

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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 2: Q. 1 (page 119)

Let $P = (x, y)$ be a point on the graph of $y = x^{2} - 8$ .
(a) Express the distance $d$ from $P$ to the origin as a function of $x$
(b) What is $d$ if $x = 0$ ?
(c) What is $d$ if $x = 1$ ?
(d) Use a graphing utility to graph $d = d (x)$ .
(e) For what values of $x$ is $d$ smallest?

Short Answer

Expert verified

The distance $d$ is $d (x) = \sqrt{x^{4} - 15 x^{2} + 64}$ and the values are $8, 7.07$ and the graph is

and $d$ is smallest when $x = - 2.74$ or

$x = 2.74$

Step by step solution

01

Part (a) Step 1: Given information

Given the point P on the graph of $y = x^{2} - 8$

02

Part (a) Step 2: Calculate the distance using the distance formula

The distance of $P$ from origin is $d = \sqrt{x^{2} + y^{2}}$ . So the required distance is

$d (x) = \sqrt{x^{2} + {(x^{2} - 8)}^{2}} d (x) = \sqrt{x^{2} + (x^{4} - 16 x^{2} + 64)} d (x) = \sqrt{x^{4} - 15 x^{2} + 64}$

03

Part (b) Step 1: Given information

Given the equation $d (x) = \sqrt{x^{4} - 15 x^{2} + 64}$

04

Part (b) Step 2: Substituting and calculating the value

Substituting $x = 0$ , we get

role="math" $d (x) = \sqrt{x^{4} - 15 x^{2} + 64} d (0) = \sqrt{0^{4} - 15 {(0)}^{2} + 64} d (0) = \sqrt{64} d (0) = 8$

05

Part (c) Step 1: Given information

Given the equation $d (x) = \sqrt{x^{4} - 15 x^{2} + 64}$

06

Part (c) Step 2: Substituting and calculating the value

Substituting, we get

$d (x) = \sqrt{x^{4} - 15 x^{2} + 64} d (1) = \sqrt{1^{4} - 15 {(1)}^{2} + 64} d (1) = \sqrt{1 - 15 + 64} d (1) = \sqrt{50} d (1) = 7.07$

07

Part (d) Step 1: Given information

Given the equation $d (x) = \sqrt{x^{4} - 15 x^{2} + 64}$

08

Part (d) Step 2: Sketching the graph

Sketching, we get

09

Part (e) Step 1: Given information

Given the equation $d (x) = \sqrt{x^{4} - 15 x^{2} + 64}$

10

Part (e) Step 2: Checking from graph

Checking, we get that $d$ is smallest when $x = - 2.74$ or

$x = 2.74$

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