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)

검증된 단계별 안내

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 \(\frac{dx}{d\theta}\).

Write down the expressions for \(x\) and \(y\) in terms of \(\theta\): \(x = r \cos(\theta) = 2\theta \cos(\theta)\) \(y = r \sin(\theta) = 2\theta \sin(\theta)\).

Differentiate both \(x\) and \(y\) with respect to \(\theta\) using the product rule: \(\frac{dx}{d\theta} = \frac{d}{d\theta} (2\theta \cos(\theta))\) \(\frac{dy}{d\theta} = \frac{d}{d\theta} (2\theta \sin(\theta))\).

Calculate the slope of the tangent line using the formula \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\).

Evaluate \(\frac{dy}{dx}\) at the given point \(\theta = \frac{\pi}{4}\) to find the slope of the tangent line at that point.

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주요 개념

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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.

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)