Textbook Question
Area in Polar Coordinates
Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.
Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ
8
views
16. Parametric Equations & Polar Coordinates
Calculus in Polar Coordinates
4:45 minutesProblem 12.3.14
Textbook Question
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 guidance1
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 \(y = r \sin \theta\).
Express \(x\) and \(y\) as functions of \(\theta\): \(x(\theta) = (4 + \sin \theta) \cos \theta\) and \(y(\theta) = (4 + \sin \theta) \sin \theta\).
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
\(\frac{dy}{d\theta} = \frac{d}{d\theta}[(4 + \sin \theta) \sin \theta]\).
Use the chain rule and product rule to compute these derivatives explicitly, remembering that \(\frac{d}{d\theta}(\sin \theta) = \cos \theta\) and \(\frac{d}{d\theta}(\cos \theta) = -\sin \theta\).
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 \((3, \frac{3\pi}{2})\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
05:32
Intro to Polar Coordinates
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.
Recommended video:
05:13Slopes of Tangent Lines
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.
Recommended video:
05:53Finding Differentials
Related Videos
Related Practice