# How do you find an equation of the tangent line to the curve at the given point # y=sin(2x)# and #x=pi/2#?

See below.

First we need to differentiate

Plugging in

Equation Is:

Graph:

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To find the equation of the tangent line to the curve y = sin(2x) at the point x = π/2, we need to find the slope of the tangent line and the coordinates of the given point.

First, we find the derivative of the function y = sin(2x) using the chain rule. The derivative is given by dy/dx = 2cos(2x).

Next, we substitute x = π/2 into the derivative to find the slope of the tangent line at that point. Substituting x = π/2 into the derivative, we get dy/dx = 2cos(2(π/2)) = 2cos(π) = -2.

Now, we have the slope of the tangent line, which is -2.

To find the coordinates of the given point, substitute x = π/2 into the original function y = sin(2x). Substituting x = π/2, we get y = sin(2(π/2)) = sin(π) = 0.

Therefore, the coordinates of the given point are (π/2, 0).

Using the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept, we can substitute the slope (-2) and the coordinates of the given point (π/2, 0) to find the equation of the tangent line.

Substituting the values, we have 0 = -2(π/2) + b. Simplifying, we get 0 = -π + b. Solving for b, we find b = π.

Therefore, the equation of the tangent line to the curve y = sin(2x) at the point x = π/2 is y = -2x + π.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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