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`5) Find an equation of a line that is tangent to $y=2 \sin x$ and whose slope is a palue.`

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

To find the slope of the tangent line to the curve $y=2\sin x$, we need to differentiate $y$ with respect to $x$

step 2

The derivative of $y=2\sin x$ with respect to $x$ is $y'=2\cos x$

step 3

Since we are looking for a tangent line with a specific slope, let's denote this slope as $m$

step 4

Set the derivative equal to the slope $m$: $2\cos x = m$

step 5

To find the $x$-coordinate(s) where the slope of the tangent is $m$, solve the equation $2\cos x = m$

step 6

Once the $x$-coordinate(s) are found, use the original equation $y=2\sin x$ to find the corresponding $y$-coordinate(s)

step 7

With the point $(x, y)$ and the slope $m$, use the point-slope form of the equation of a line: $y - y_1 = m(x - x_1)$

step 8

Substitute the known values into the point-slope form to get the equation of the tangent line

Answer

The equation of the line that is tangent to $y=2\sin x$ with slope $m$ is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point of tangency on the curve and $m$ is the given slope.

Key Concept

Finding the equation of a tangent line to a curve at a point with a given slope

Explanation

To find the equation of a tangent line with a given slope to a curve defined by a function, we first find the derivative of the function to determine the slope of the tangent at any point. Then we solve for the specific point(s) where the derivative equals the given slope, and use the point-slope form of a line to write the equation of the tangent line.

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