Q. 28 Approximate the arc length of f ... [FREE SOLUTION]

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Chapter 6: Q. 28 (page 539)

Approximate the arc length of f (x) on [a, b], using the approximation ${鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x$ with the given value of n. In each problem list the values of $未 y_{k}$ for $k = 0, 1, 2 . . . . . . n$ .
$f (x) = e^{x}, [a, b] = [- 1, 3], n = 4$

Short Answer

Expert verified

The arc length is $6 + \frac{1}{e^{2}} - \frac{2}{e} + 2 e^{2} - 2 e - 2 e^{3} + e^{4}$ .

Step by step solution

Step 1. Given information .

Consider the given function $f (x) = e^{x}, [a, b] = [- 1, 3], n = 4$ .

Step 2. Formula used .

Arc length of $f (x) = {鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x$ .

Step 3. Find the arc length .

$f (x) = {鈭�}_{k = 1}^{4} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x$

$未 x = \frac{b - a}{n} = \frac{4}{4} = 1 x_{0} = a + k 路未 x = - 1 x_{1} = - 1 + 1 = 0, x_{2} = - 1 + 2 = 1 x_{3} = - 1 + 3 = 2, x_{4} = - 1 + 4 = 3 未 y_{k} = f (x_{k}) - f (x_{k - 1}) 未 y_{1} = 1 - \frac{1}{e}, 未 y_{2} = e - 1 未 y_{3} = e^{2} - e, 未 y_{4} = e^{3} - e^{2}$

$A r c l e n g t h = {鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{未 y_{1}}{未 x})}^{2}} 路未 x + \sqrt{1 + {(\frac{未 y_{2}}{未 x})}^{2}} 路未 x + \sqrt{1 + {(\frac{未 y_{3}}{未 x})}^{2}} 未 x + \sqrt{1 + {(\frac{未 y_{4}}{未 x})}^{2}} 路未 x = \sqrt{1 + {(1 - \frac{1}{e})}^{2}} 路 1 + {\sqrt{1 + (e - 1)}}^{2} 路 1 + \sqrt{1 + {(e^{2} - e)}^{2}} 路 1 + \sqrt{1 + {(e^{3} - e^{2})}^{2}} 路 1 = 6 + \frac{1}{e^{2}} - \frac{2}{e} + 2 e^{2} - 2 e - 2 e^{3} + e^{4}$

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91影视

Approximate the arc length of f (x) on [a, b], using the approximation ${鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x$ with the given value of n. In each problem list the values of $未 y_{k}$ for $k = 0, 1, 2 . . . . . . n$ .
$f (x) = e^{x}, [a, b] = [- 1, 3], n = 4$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the arc length .

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91影视

Approximate the arc length of f (x) on [a, b], using the approximation 鈭�k=1n1+未yk未x2路未xwith the given value of n. In each problem list the values of 未ykfor k=0,1,2......n.fx=ex,a,b=-1,3,n=4

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the arc length .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Calculus

Logic and Functions

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Approximate the arc length of f (x) on [a, b], using the approximation ${鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x$ with the given value of n. In each problem list the values of $未 y_{k}$ for $k = 0, 1, 2 . . . . . . n$ .
$f (x) = e^{x}, [a, b] = [- 1, 3], n = 4$