Chapter 6: Q. 28 (page 539)

Approximate the arc length of f (x) on [a, b], using the approximation $\sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$ with the given value of n. In each problem list the values of $δ y_{k}$ 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 $f (x) = \sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$ .

Step 3. Find the arc length .

$f (x) = \sum_{k = 1}^{4} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$

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

$A r c l e n g t h = \sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{1}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{2}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{3}}{δ x})}^{2}} δ x + \sqrt{1 + {(\frac{δ y_{4}}{δ x})}^{2}} \cdot δ x = \sqrt{1 + {(1 - \frac{1}{e})}^{2}} \cdot 1 + {\sqrt{1 + (e - 1)}}^{2} \cdot 1 + \sqrt{1 + {(e^{2} - e)}^{2}} \cdot 1 + \sqrt{1 + {(e^{3} - e^{2})}^{2}} \cdot 1 = 6 + \frac{1}{e^{2}} - \frac{2}{e} + 2 e^{2} - 2 e - 2 e^{3} + e^{4}$

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Approximate the arc length of f (x) on [a, b], using the approximation $\sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$ with the given value of n. In each problem list the values of $δ y_{k}$ 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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Approximate the arc length of f (x) on [a, b], using the approximation ∑k=1n1+δykδx2·δxwith the given value of n. In each problem list the values of δykfor 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

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