Rates of Change in Polar Functions - AP Precalculus

1. What is the formula for $\frac{dx}{d\theta}$ when $r = \sin(\theta)$?

   $\frac{dx}{d\theta} = \cos(\theta)\cos(\theta) - \sin^2(\theta)$. Substitute $r = \sin(\theta)$ and $\frac{dr}{d\theta} = \cos(\theta)$ into formula.

2. Find $\frac{dy}{d\theta}$ for $r(\theta) = \theta^2$.

   $\frac{dy}{d\theta} = 2\theta \sin(\theta) + \theta^2 \cos(\theta)$. Substitute $r = \theta^2$ and $\frac{dr}{d\theta} = 2\theta$ into the formula.

3. What is the polar equation of a cardioid?

   $r = a(1 + \cos(\theta))$. Heart-shaped curve with cusp at the origin.

4. Find $\frac{dy}{d\theta}$ for $r(\theta) = 1 + \cos(\theta)$.

   $\frac{dy}{d\theta} = \cos(\theta) + \sin(\theta)$. Apply product rule with $r = 1 + \cos(\theta)$ and $\frac{dr}{d\theta} = -\sin(\theta)$.

5. What is the formula for the derivative of $r(\theta)$ with respect to $\theta$?

   $\frac{dr}{d\theta}$. Standard notation for the derivative of polar radius function.

6. Find $\frac{dy}{dx}$ for $r(\theta) = 1 + \sin(\theta)$ at $\theta = \frac{\pi}{2}$.

   $0$. Horizontal tangent occurs when $\frac{dy}{d\theta} = 0$ at this angle.

7. Find $\frac{dx}{d\theta}$ for $r(\theta) = \theta^2$.

   $\frac{dx}{d\theta} = 2\theta \cos(\theta) - \theta^2 \sin(\theta)$. Substitute $r = \theta^2$ and $\frac{dr}{d\theta} = 2\theta$ into the formula.

8. What is the polar equation for a line through the origin at an angle $\alpha$?

   $\theta = \alpha$. Constant angle creates a ray from the origin.

9. What is the formula for the instantaneous rate of change of $r$?

   $\frac{dr}{d\theta}$. Represents the instantaneous rate of change of radius with angle.

10. What is the formula for the rate of change of $x$ with respect to $\theta$?

    $\frac{dx}{d\theta} = \frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)$. Product rule applied to $x = r\cos(\theta)$ with respect to $\theta$.

11. Find $\frac{dx}{d\theta}$ for $r(\theta) = 1 + \cos(\theta)$.

    $\frac{dx}{d\theta} = -\sin^2(\theta)$. Apply product rule with $r = 1 + \cos(\theta)$ and $\frac{dr}{d\theta} = -\sin(\theta)$.

12. What is the derivative of $r = a\theta + b$ with respect to $\theta$?

    $\frac{dr}{d\theta} = a$. Derivative of linear function is the coefficient of $\theta$.

13. Find $\frac{dr}{d\theta}$ for $r(\theta) = 3\cos(2\theta)$.

    $\frac{dr}{d\theta} = -6\sin(2\theta)$. Chain rule: derivative of $3\cos(2\theta)$ is $-6\sin(2\theta)$.

14. Find the derivative $\frac{dr}{d\theta}$ for $r = 3\theta^2$.

    $\frac{dr}{d\theta} = 6\theta$. Power rule: derivative of $3\theta^2$ is $6\theta$.

15. Convert the polar function $r = 3\sin(\theta)$ to Cartesian coordinates.

    $x^2 + (y - \frac{3}{2})^2 = (\frac{3}{2})^2$. Circle with center $(0, \frac{3}{2})$ and radius $\frac{3}{2}$.

16. Identify the formula for $\frac{dy}{dx}$ in polar coordinates.

    $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$. Chain rule connecting Cartesian and polar derivatives.

17. What is the formula for $\frac{dy}{d\theta}$ when $r = \sin(\theta)$?

    $\frac{dy}{d\theta} = \cos^2(\theta) + \sin(\theta)\cos(\theta)$. Substitute $r = \sin(\theta)$ and $\frac{dr}{d\theta} = \cos(\theta)$ into formula.

18. What is the formula for arc length $L$ of a polar curve $r(\theta)$?

    $L = \int_{a}^{b} \sqrt{(\frac{dr}{d\theta})^2 + r^2} \, d\theta$. Integrates the speed element in polar coordinates over the interval.

19. What is the formula for the derivative of $y = r\sin(\theta)$ with respect to $\theta$?

    $\frac{dy}{d\theta} = \frac{dr}{d\theta}\sin(\theta) + r\cos(\theta)$. Product rule applied to the Cartesian conversion formula.

20. What is the polar equation of a rose curve with 4 petals?

    $r = a\cos(2\theta)$. Four-petaled rose has period $\pi$ in the cosine function.

21. What is the polar equation for a spiral of Archimedes?

    $r = a\theta$. Linear relationship between radius and angle creates uniform spiral.

22. What is the formula for the rate of change of $y$ with respect to $\theta$?

    $\frac{dy}{d\theta} = \frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)$. Product rule applied to $y = r\sin(\theta)$ with respect to $\theta$.

23. What is the polar equation for an exponential spiral?

    $r = ae^{b\theta}$. Exponential growth pattern with constant $a$ and growth rate $b$.

24. Determine $\frac{dy}{dx}$ for $r(\theta) = 2\sin(\theta)$ at $\theta = \frac{\pi}{2}$.

    Undefined. Vertical tangent occurs when $\frac{dx}{d\theta} = 0$ but $\frac{dy}{d\theta} \neq 0$.

25. Identify the formula for $\frac{dx}{d\theta}$ in polar coordinates.

    $\frac{dx}{d\theta} = \frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)$. Standard formula for horizontal rate of change in polar coordinates.

26. Identify the formula for $\frac{dy}{d\theta}$ in polar coordinates.

    $\frac{dy}{d\theta} = \frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)$. Standard formula for vertical rate of change in polar coordinates.

27. What is the polar equation for a circle centered at the origin with radius 2?

    $r = 2$. Constant radius distance from origin defines a circle.

28. What is the general polar form of an ellipse?

    $r = \frac{ed}{1 + e\cos(\theta)}$. Standard form where $e$ is eccentricity and $d$ is directrix distance.

29. What is the polar equation for a lemniscate?

    $r^2 = a^2\cos(2\theta)$. Figure-eight curve with equation involving $\cos(2\theta)$.

30. What is the formula for the derivative of $x = r\cos(\theta)$ with respect to $\theta$?

    $\frac{dx}{d\theta} = \frac{dr}{d\theta}\cos(\theta) - r\sin(\theta)$. Product rule applied to the Cartesian conversion formula.

31. Find $\frac{dr}{d\theta}$ for $r(\theta) = 2\theta + \sin(\theta)$.

    $\frac{dr}{d\theta} = 2 + \cos(\theta)$. Derivative of each term: $2$ from $2\theta$ and $\cos(\theta)$ from $\sin(\theta)$.

32. What is the polar equation for a limaçon with inner loop?

    $r = a + b\cos(\theta)$, $|a| < |b|$. When $|a| < |b|$, the curve creates an inner loop.

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