Chapter 2: Q. 2 (page 119)

2. Let $P = (x, y)$ be a point on the graph of $y = x^{2} - 8$ .
(a) Express the distance $d$ from $P$ to the point $(0, - 1)$ 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 is $d = \sqrt{x^{4} - 13 x^{2} + 49}$ and the values are $7, 6.08$ and the graph is

and $d$ is least when $x = - 2.55, x = 2.55$

Step by step solution

Part (a) Step 1: Given information

Given the point $P (x, y)$ and the curve $y = x^{2} - 8$

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

The distance of $P$ from $(0, - 1)$ is $d = \sqrt{x^{2} + {(y - 1)}^{2}}$ . So the required distance is

$d (x) = \sqrt{x^{2} + {(x^{2} - 8 + 1)}^{2}} d (x) = \sqrt{x^{2} + {(x^{2} - 7)}^{2}} d (x) = \sqrt{x^{2} + (x^{4} - 14 x^{2} + 49)} d (x) = \sqrt{x^{4} - 13 x^{2} + 49}$

Part (b) Step 1: Given information

Given the equation $d (x) = \sqrt{x^{4} - 13 x^{2} + 49}$

Part (b) Step 2: Substituting x=0 and calculating the value

Substituting, we get

$d (x) = \sqrt{x^{4} - 13 x^{2} + 49} d (0) = \sqrt{0^{4} - 13 {(0)}^{2} + 49} d (0) = \sqrt{49} d (0) = 7$

Part (c) Step 1: Given information

Given the equation $d (x) = \sqrt{x^{4} - 13 x^{2} + 49}$

Part (c) Step 2: Substituting x=1 and calculating the value

Substituting, we get

d (x) = \sqrt{x^{4} - 13 x^{2} + 49} d (1) = \sqrt{1^{4} - 13 {(1)}^{2} + 49} d (1) = \sqrt{37}

Part (d) Step 1: Given information

Given the equation $d (x) = \sqrt{x^{4} - 13 x^{2} + 49}$

Part (d) Step 2: Sketching the graph

The graph is

Part (e) Step 1: Given information

Given the equation $d (x) = \sqrt{x^{4} - 13 x^{2} + 49}$

Part (e): Checking the graph

The value of $d$ is least when $x = - 2.55, x = 2.55$

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Short Answer

Step by step solution

Part (a) Step 1: Given information

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

Part (b) Step 1: Given information

Part (b) Step 2: Substituting x=0 and calculating the value

Part (c) Step 1: Given information

Part (c) Step 2: Substituting x=1 and calculating the value

Part (d) Step 1: Given information

Part (d) Step 2: Sketching the graph

Part (e) Step 1: Given information

Part (e): Checking the graph

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2. Let P=(x,y)be a point on the graph of y=x2-8.(a) Express the distance dfrom Pto the point (0,-1)as a function of x.(b) What is dif x=0?(c) What is dif x=-1?(d) Use a graphing utility to graph d=d(x).(e) For what values of xis dsmallest?

Short Answer

Step by step solution

Part (a) Step 1: Given information

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

Part (b) Step 1: Given information

Part (b) Step 2: Substituting x=0 and calculating the value

Part (c) Step 1: Given information

Part (c) Step 2: Substituting x=1 and calculating the value

Part (d) Step 1: Given information

Part (d) Step 2: Sketching the graph

Part (e) Step 1: Given information

Part (e): Checking the graph

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Theoretical and Mathematical Physics

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