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


r = 4 cos 2θ; at the tips of the leaves

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 formula: \(\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}\)
2. Identify the given function: \(r = 4 \cos 2\theta\). Next, compute its derivative with respect to \(\theta\): \(\frac{dr}{d\theta} = \frac{d}{d\theta} (4 \cos 2\theta) = -8 \sin 2\theta\)
3. Determine the values of \(\theta\) at the tips of the leaves. The tips occur where \(r\) is at a maximum or minimum, which happens when \(\cos 2\theta = \pm 1\). Solve for \(\theta\) such that \(\cos 2\theta = \pm 1\) to find these points.
4. For each \(\theta\) found in the previous step, substitute \(r\), \(\frac{dr}{d\theta}\), \(\sin \theta\), and \(\cos \theta\) into the slope formula: \(\frac{dy}{dx} = \frac{(-8 \sin 2\theta) \sin \theta + (4 \cos 2\theta) \cos \theta}{(-8 \sin 2\theta) \cos \theta - (4 \cos 2\theta) \sin \theta}\)
5. Simplify the expression for each \(\theta\) to find the slope of the tangent line at the tips of the leaves. This will give you the slope values of the tangent lines at those points.

Key Concepts

Polar Coordinates and Polar Curves

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Polar curves are equations expressed as r = f(θ), describing shapes based on angle θ. Understanding how to interpret and plot these curves is essential for analyzing their properties, such as tangent lines.

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 curve to Cartesian coordinates and using the derivative dy/dx = (dy/dθ) / (dx/dθ). This requires differentiating x = r cos θ and y = r sin θ with respect to θ, then evaluating at the given point.

Finding Specific Points on Polar Curves (Tips of the Leaves)

The 'tips of the leaves' on a polar curve like r = 4 cos 2θ correspond to points where r reaches local maxima or minima, often where the derivative dr/dθ = 0. Identifying these points is crucial to evaluate the slope of the tangent line at those specific locations.