Q. 77 Prove, in two ways, that the arc... [FREE SOLUTION]

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Chapter 6: Q. 77 (page 540)

Prove, in two ways, that the arc length of a linear function $f (x) = m x + c$ on an interval $[a, b]$ is equal to $(b - a) \sqrt{1 + m^{2}}$ : (a) by using the distance formula; (b) by using Theorem 6.7.

Short Answer

Expert verified

Arc length of a linear function $f (x) = m x + c$ on an interval $[a, b]$ is equal to $(b - a) \sqrt{1 + m^{2}}$ .

Step by step solution

Part (a) Step 1. Given information

A linear function, $f (x) = m x + c$ .

Part (a) Step 2. Proof by using the distance formula

$The distance from (a, f (a)) = (a, m a + c) to (b, f (b)) = (b, m b + c) is given by, D = \sqrt{{(b 鈭� a)}^{2} + ((m b + c) 鈭� {(m a + c))}^{2}} D = \sqrt{{(b 鈭� a)}^{2} + m^{2} (b 鈭� {a)}^{2}} D = \sqrt{{(b 鈭� a)}^{2} (1 + m^{2})} D = (b 鈭� a) \sqrt{1 + m^{2}} Since the graph of f (x) = m x + c is a line, this is the exact arc length$

Part (b) step 1. Proof by using Theorem 6.7

$The arc length of f (x) from x = a to x = b can be represented by the definite integral : {鈭�}_{a}^{b} \sqrt{1 + (f' {(x))}^{2}} d x For the given linear function, f (x) = mx + c, f' (x) = m Therefore, D = {鈭�}_{a}^{b} \sqrt{1 + {(m)}^{2}} d x D = (\sqrt{1 + m^{2}}) {鈭�}_{a}^{b} 1 d x D = (\sqrt{1 + m^{2}}) [x_{a}^{b}] D = (\sqrt{1 + m^{2}}) (b - a)$

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91影视

Prove, in two ways, that the arc length of a linear function $f (x) = m x + c$ on an interval $[a, b]$ is equal to $(b - a) \sqrt{1 + m^{2}}$ : (a) by using the distance formula; (b) by using Theorem 6.7.

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) Step 2. Proof by using the distance formula

Part (b) step 1. Proof by using Theorem 6.7

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91影视

Prove, in two ways, that the arc length of a linear function f(x)=mx+con an interval a,bis equal to (b-a)1+m2: (a) by using the distance formula; (b) by using Theorem 6.7.

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) Step 2. Proof by using the distance formula

Part (b) step 1. Proof by using Theorem 6.7

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Geometry

Decision Maths

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Prove, in two ways, that the arc length of a linear function $f (x) = m x + c$ on an interval $[a, b]$ is equal to $(b - a) \sqrt{1 + m^{2}}$ : (a) by using the distance formula; (b) by using Theorem 6.7.