Find all points on the curve y = tan x, −π/2 < x < π/2, where the tangent line is parallel to the line y = 2x. Sketch the curve and tangent lines together, labeling each with its equation.

Verified step by step guidance

1. Identify the slope of the given line y = 2x. The slope is 2.
2. Find the derivative of the curve y = tan(x) to determine the slope of the tangent line at any point x. The derivative is given by \( \frac{dy}{dx} = \sec^2(x) \).
3. Set the derivative equal to the slope of the given line to find the x-values where the tangent line is parallel to y = 2x. Solve \( \sec^2(x) = 2 \) for x.
4. Solve the equation \( \sec^2(x) = 2 \) to find the x-values within the interval \( -\frac{\pi}{2} < x < \frac{\pi}{2} \). This involves solving \( \cos(x) = \pm \frac{1}{\sqrt{2}} \).
5. Substitute the x-values found into the original equation y = tan(x) to find the corresponding y-values, giving the points on the curve where the tangent is parallel to y = 2x.

Key Concepts

Derivative of a Function

The derivative of a function at a point gives the slope of the tangent line to the curve at that point. For the function y = tan(x), the derivative is y' = sec^2(x). Understanding how to compute and interpret derivatives is crucial for finding where the tangent line is parallel to a given line.

Parallel Lines

Two lines are parallel if they have the same slope. In this problem, the line y = 2x has a slope of 2. To find points on the curve y = tan(x) where the tangent is parallel to y = 2x, we need to set the derivative of y = tan(x) equal to 2 and solve for x.

Trigonometric Functions and Their Properties

Understanding the properties of trigonometric functions, such as the tangent function, is essential. The function y = tan(x) is periodic and has vertical asymptotes at x = ±π/2. Recognizing these properties helps in sketching the curve and understanding the behavior of the function within the given interval.