Polar Coordinates and Differentiation - AP Calculus BC

1. Convert the Cartesian equation $x^2 + y^2 = 4$ to polar form.

   $r = 2$. Circle centered at origin with radius $2$.

2. What is the formula for $r$ in terms of $x$ and $y$?

   $r = \text{sqrt}(x^2 + y^2)$. Pythagorean theorem applied to coordinates.

3. What is the formula for $\theta$ in terms of $x$ and $y$?

   $\theta = \text{tan}^{-1}(\frac{y}{x})$. Inverse tangent of y over x ratio.

4. What is the general form of a polar coordinate?

   $(r, \theta)$. Distance from origin and angle from positive x-axis.

5. Convert the Cartesian coordinate $(x, y)$ to polar coordinates.

   $(r, \theta) = (\text{sqrt}(x^2 + y^2), \text{tan}^{-1}(\frac{y}{x}))$. Distance formula and arctangent for angle.

6. State the formula for $x$ in terms of $r$ and $\theta$.

   $x = r \times \cos(\theta)$. Horizontal component uses cosine.

7. What is the expression for $dy/dx$ in polar coordinates?

   $\frac{dy}{dx} = \frac{r' \text{sin}(\theta) + r \text{cos}(\theta)}{r' \text{cos}(\theta) - r \text{sin}(\theta)}$. Chain rule applied to parametric equations.

8. What is the polar area formula for a sector?

   Area = $\frac{1}{2} \times r^2 \times \theta$. Half the product of radius squared and central angle.

9. Find the maximum value of $r = 2 + 3\text{cos}(\theta)$.

   Maximum $r = 5$. Maximum occurs when $\cos(\theta) = 1$.

10. Identify the polar form of a circle centered at the origin with radius $a$.

    $r = a$. Constant radius equals constant distance from origin.

11. What is the formula for the slope of a tangent line in polar coordinates?

    $\frac{dy}{dx} = \frac{r' \sin(\theta) + r \cos(\theta)}{r' \cos(\theta) - r \sin(\theta)}$. Derivative of y with respect to x in polar form.

12. Convert the polar coordinate $(r, \theta)$ to Cartesian coordinates.

    $(x, y) = (r \times \text{cos}(\theta), r \times \text{sin}(\theta))$. Use $x = r\cos(\theta)$ and $y = r\sin(\theta)$.

13. State the formula for arc length in polar coordinates.

    $L = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + (r')^2} \, d\theta$. Integrates $\sqrt{r^2 + \left( \frac{dr}{d\theta} \right)^2}$

14. What is the polar form of a limaçon with an inner loop?

    $r = a + b\text{cos}(\theta)$ where $a < b$. When $a < b$, creates inner loop.

15. What is the polar form of a spiral of Archimedes?

    $r = a + b\theta$. Spiral increasing linearly with angle.

16. State the formula for $y$ in terms of $r$ and $\theta$.

    $y = r \times \text{sin}(\theta)$. Vertical component uses sine.

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