Chapter 6: Q. 22 (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.
$f (x) = e^{x}, [a, b] = [- 1, 3], n = 4$

Short Answer

Expert verified

The arc length is $20.6825$ .

The graph is sketched as below:

Step by step solution

Step 1. Given information .

Consider the given function $f (x) = e^{x}$ .

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 .

$f (x) = e^{x}, a = - 1, b = 3, n = 4 \partial x = \frac{b - a}{n} = \frac{3 + 1}{4} = 1 x_{k} = a + k \partial x x_{0} = - 1 + 0 = - 1, x_{1} = - 1 + 1 = 0 x_{2} = - 1 + 2 = 1, x_{3} = - 1 + 3 = 2 x_{4} = - 1 + 4 = 3 \partial y_{1} = f (0) - f (1) = e^{0} - e^{- 1} = 1 - 0.368 = 0.632 \partial y_{2} = f (1) - f (0) = e - 1 = 2.718 - 1 = 1.718 \partial y_{3} = f (2) - f (1) = e^{2} - e = 4.6707 \partial y_{4} = f (3) - f (2) = e^{3} - e^{2} = 12.6964$

$A r c l e n g t h = [\sqrt{1 + 0.399} + \sqrt{1 + 2.951} + \sqrt{1 + 21.8154} + \sqrt{1 + 161.198}] \times 1 = 1.1827 + 1.9877 + 4.7765 + 12.7856 = 20.6825$

Step 4. Sketch of the function fx=ex .

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

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find arc length .

Step 4. Sketch of the function fx=ex .

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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=ex,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=ex .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Calculus

Statistics

Pure Maths

Probability and Statistics

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