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


r = 4 + sin θ; (4, 0) and (3, 3π/2)

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 for \(x\) and \(y\) in terms of \(r\) and \(\theta\): \(x = r \cos \theta\) and \(y = r \sin \theta\).
2. Express \(x\) and \(y\) as functions of \(\theta\): \(x(\theta) = (4 + \sin \theta) \cos \theta\) and \(y(\theta) = (4 + \sin \theta) \sin \theta\).
3. Find the derivatives \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) by applying the product rule to each: \(\frac{dx}{d\theta} = \frac{d}{d\theta}[(4 + \sin \theta) \cos \theta]\) and \(\frac{dy}{d\theta} = \frac{d}{d\theta}[(4 + \sin \theta) \sin \theta]\).
4. Use the chain rule and product rule to compute these derivatives explicitly, remembering that \(\frac{d}{d\theta}(\sin \theta) = \cos \theta\) and \(\frac{d}{d\theta}(\cos \theta) = -\sin \theta\).
5. Calculate the slope of the tangent line at each given point by evaluating \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\) at the corresponding \(\theta\) values: \(\theta = 0\) for the point \((4,0)\) and \(\theta = \frac{3\pi}{2}\) for the point \((3, \frac{3\pi}{2})\).

Key Concepts

Polar Coordinates and Polar Curves

Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Polar curves are equations expressed as r = f(θ), describing how the radius changes with the angle. Understanding this system is essential to analyze points and curves given in polar form.

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 sin θ + r cos θ) / (dr/dθ cos θ - r sin θ). This formula relates derivatives of r with respect to θ to the slope in the xy-plane.

Differentiation with Respect to θ

To find the slope of the tangent line, you must differentiate r = 4 + sin θ with respect to θ. This involves applying basic differentiation rules to trigonometric functions, which is crucial for computing dr/dθ and subsequently the slope dy/dx.