Calculus of Polar Curves - Calc BC - Calculator Allowed

A polar curve is a curve represented in polar coordinates, where each point on the curve is defined by a distance from a reference point (the pole) and an angle from a reference direction.

The slope of a polar curve is given by the formula: $$\frac{dy}{dx} = \frac{r \frac{d\theta}{dt} \text{sin}(\theta) + \frac{dr}{d\theta} \text{cos}(\theta)}{r \frac{d\theta}{dt} \text{cos}(\theta) - \frac{dr}{d\theta} \text{sin}(\theta)}$$ .

A cardioid is a heart-shaped curve represented in polar coordinates by the equation $$r = a + b \text{cos}(\theta)$$ or $$r = a + b \text{sin}(\theta)$$ , where a and b are constants.

The area A enclosed by a polar curve from angle θ=a to θ=b is given by the formula: $$A = \frac{1}{2} \times \text{integral from a to b} (r(\theta))^2 d\theta$$ .

The area enclosed by the cardioid $$r = 2 + 2\text{cos}(\theta)$$ is $$A = 6\text{π}$$ .

To find the points of intersection between two polar curves, set their equations equal to each other and solve for θ.

In polar coordinates, the angle θ represents the direction of the point from the pole, measured from the positive x-axis.

The area A between two polar curves $$r_1(\theta)$$ and $$r_2(\theta)$$ from θ=a to θ=b is given by: $$A = \frac{1}{2} \times \text{integral from a to b} (r_1^2 - r_2^2) d\theta$$ .

A four-leaved rose is represented in polar coordinates by the equation $$r = a \text{cos}(n\theta)$$ or $$r = a \text{sin}(n\theta)$$ , where n is an even integer.

The slope of the four-leaved rose $$r = \text{cos}(2\theta)$$ is given by: $$\frac{\text{cos}(2\theta)\text{cos}(\theta) - 2\text{sin}(2\theta)\text{sin}(\theta)}{-\text{cos}(2\theta)\text{sin}(\theta) - 2\text{sin}(2\theta)\text{cos}(\theta)}$$ .