Q. 21 Approximate the arc length of f ... [FREE SOLUTION]

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Chapter 6: Q. 21 (page 539)

Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .
$f (x) = I n x, [a, b] = [1, 3], n = 4$

Short Answer

Expert verified

The arc length is 2.3 .

The graph is sketched as below:

Step by step solution

Step 1. Given information .

Consider the given function $f (x) = I n x$ .

Step 2. Formula used .

Arc length $= \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{鈭� y_{k}}{鈭� x})}^{2}} 路鈭� x$

Step 3. Find arc length .

$f (x) = I n x, a = 1, b = 3 鈭� x = \frac{b - a}{n} = \frac{3 - 1}{4} = \frac{1}{2} x_{k} = a + k 鈭� x x_{0} = 1, x_{1} = 1 + \frac{1}{2} = \frac{3}{2} x_{3} = 1 + 2 脳 \frac{1}{2} = 2, x_{4} = 1 + 4 脳 \frac{1}{2} = 3 鈭� y_{1} = f (x_{1}) - f (x_{0}) = \ln (\frac{3}{2}) - \ln (1) = 0 路 4055 - 0 = 0 路 4055 鈭� y_{2} = f (x_{2}) - f (x_{1}) = \ln (2) - \ln (\frac{3}{2}) = 0 路 28768 鈭� y_{3} = f (x_{3}) - f (x_{2}) = \ln (\frac{5}{2}) - \ln (2) = 0 路 22314 鈭� y_{4} = f (x_{4}) - f (x_{3}) = \ln (3) - \ln (\frac{5}{2}) = 0 路 1823$

Arc length $= [\sqrt{1 + {(\frac{0 路 405}{0 路 5})}^{2}} + \sqrt{1 + {(\frac{0 路 2876}{0 路 5})}^{2}} + \sqrt{1 + {(\frac{0 路 22314}{0 路 5})}^{2}} + \sqrt{1 + {(\frac{0 路 182}{0 路 5})}^{2}}] 脳 0 路 5 = [\sqrt{1.6561} + \sqrt{1 路 3308} + \sqrt{1.199} + \sqrt{1.132}] 脳 0 路 5 = 4.59 脳 0.5 = 2.3$

Step 4. Sketch of the function fx=In x .

The sketch of the given function is shown below .

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91影视

Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .
$f (x) = I n x, [a, b] = [1, 3], n = 4$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find arc length .

Step 4. Sketch of the function fx=In x .

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91影视

Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .fx=Inx,a,b=1,3,n=4

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find arc length .

Step 4. Sketch of the function fx=In x .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .
$f (x) = I n x, [a, b] = [1, 3], n = 4$