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 $= \lim_{n \to \infty} \sum_{k = 1}^{n} \sqrt{1 + {(\frac{\partial y_{k}}{\partial x})}^{2}} \cdot \partial x$

Step 3. Find arc length .

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

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

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

The sketch of the given function is shown below .

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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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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

Applied Mathematics

Pure Maths

Geometry

Calculus

Mechanics Maths

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 .
$f (x) = I n x, [a, b] = [1, 3], n = 4$