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Chapter 2: Problem 62

Slope of a Tangent Line In Exercises 61 and 62 , find the slope of the tangent line to the sine function at the origin. Compare this value with the number of complete cycles in the interval \([0,2 \pi] .\)

Short Answer

Expert verified
The slope of the tangent line to the sine function at the origin is 1. This matches the number of complete cycles of the function in the interval \([0, 2 \pi]\).

Step by step solution

01

Find the Derivative

Start by finding the derivative of the sine function. If \(y = \sin(x)\), then the derivative \(y' = \cos(x)\).
02

Evaluate the Derivative at the Origin

Evaluate the derivative at the origin by placing \(x = 0\). So, \(y'(0) = \cos(0) = 1\). The slope of the tangent line to the sine function at the origin is 1.
03

Compare with the Number of Cycles

In the interval \([0,2 \pi] \), the sine function completes exactly one cycle, from 0 to \(2 \pi\). So the number of complete cycles in the interval is 1, which is the same as the slope of the tangent line at the origin.

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