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Chapter 5: Problem 61

Suppose \(|x|\) is small but nonzero. Explain why the slope of the line containing the point \((x, \sin x)\) and the origin is approximately \(1 .\)

Short Answer

Expert verified
When \(|x|\) is small but nonzero, the slope of the line containing the point \((x, \sin x)\) and the origin is approximately 1 because as \(x\) approaches 0, the limit \(\lim_{x\to0}\frac{\sin x}{x}\) can be simplified using the small-angle approximation \(\sin x \approx x\). This results in the slope being approximately equal to 1.

Step by step solution

01

Finding the slope of the line

The slope of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the following formula: Slope = \(\frac{y_2 - y_1}{x_2 - x_1}\) In our case, the two points are \((x, \sin x)\) and the origin \((0,0)\). Plugging these values into the formula, we get: Slope = \(\frac{\sin x - 0}{x - 0} = \frac{\sin x}{x}\)
02

Analyzing the behavior of the slope for small values of x

Now, we will analyze the behavior of the slope as \(x\) approaches \(0\). This can be done using the limit: \[\lim_{x\to0}\frac{\sin x}{x}\] Using the small-angle approximation, we know that \(\sin x \approx x\) when \(|x|\) is small. Thus, the slope can be simplified as: Slope \(\approx \frac{x}{x}\) Simplifying this expression, we find that the slope is approximately equal to 1: Slope \(\approx 1\) In conclusion, when \(|x|\) is small but nonzero, the slope of the line containing the point \((x, \sin x)\) and the origin is approximately 1.

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