11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 2θ; (π/2, π/4)

Verified step by step guidance

1. Recall that for a polar curve given by \(r = f(\theta)\), the slope of the tangent line \(\frac{dy}{dx}\) can be found using the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), and then applying the chain rule to find \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\).
2. Write down the expressions for \(x\) and \(y\) in terms of \(\theta\): \(x = r \cos(\theta) = 2\theta \cos(\theta)\) \(y = r \sin(\theta) = 2\theta \sin(\theta)\).
3. Differentiate both \(x\) and \(y\) with respect to \(\theta\) using the product rule: \(\frac{dx}{d\theta} = \frac{d}{d\theta} (2\theta \cos(\theta))\) \(\frac{dy}{d\theta} = \frac{d}{d\theta} (2\theta \sin(\theta))\).
4. Calculate the slope of the tangent line using the formula \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\).
5. Evaluate \(\frac{dy}{dx}\) at the given point \(\theta = \frac{\pi}{4}\) to find the slope of the tangent line at that point.

Key Concepts

Polar Coordinates and Curves

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how curves are defined in polar form, such as r = 2θ, is essential for analyzing their geometric properties and converting between coordinate systems.

Slope of Tangent Lines in Polar Coordinates

The slope of a tangent line to a polar curve at a point is found by converting the polar equation to Cartesian coordinates or using the formula dy/dx = (dr/dθ sinθ + r cosθ) / (dr/dθ cosθ - r sinθ). This allows determination of the tangent's slope without explicit Cartesian conversion.

Differentiation with Respect to θ

Calculating the slope requires differentiating r with respect to θ (dr/dθ). This derivative captures how the radius changes as the angle varies, which is crucial for applying the tangent slope formula in polar coordinates.