Question

Find the equation of the tangent line to the function f(x)=csc(x) at x=4π. Use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct, and include a sketch of this graph.

Answer 1

To find the equation of the tangent line to the function f(x) = csc(x) at x = 4π, we need to find the derivative of the function first and then substitute the values into the point-slope form of a line.

Step-by-step explanation:

To find the equation of the tangent line to the function f(x) = csc(x) at x = 4π, we need to find the derivative of the function first. The derivative of csc(x) is -csc(x)cot(x). Evaluating this derivative at x = 4π, we find that the slope of the tangent line is -csc(4π)cot(4π). Now, we can use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the point where the tangent line intersects the curve. Substitute the values y - csc(4π) = -csc(4π)cot(4π)(x - 4π). Simplifying this equation gives us the equation of the tangent line to the function f(x) = csc(x) at x = 4π.