AP Calculus BC Flashcards: Polar Coordinates And Differentiation

What this deck covers

This deck focuses on Polar Coordinates And Differentiation, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Convert the Cartesian coordinate $(x, y)$ to polar coordinates.

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

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

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

Flashcard 3: State the formula for arc length in polar coordinates.

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

Flashcard 4: What is the general form of a polar coordinate?

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

Flashcard 5: Convert the Cartesian equation $x^2 + y^2 = 4$ to polar form.

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

Flashcard 6: What is the formula for $r$ in terms of $x$ and $y$?

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

Flashcard 7: What is the formula for $\theta$ in terms of $x$ and $y$?

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

Flashcard 8: What is the general form of a polar coordinate?

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

Flashcard 9: What is the polar area formula for a sector?

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

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

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

Flashcard 11: Convert the Cartesian coordinate $(x, y)$ to polar coordinates.

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

Flashcard 12: State the formula for $x$ in terms of $r$ and $\theta$.

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

Flashcard 13: What is the expression for $dy/dx$ in polar coordinates?

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

Flashcard 14: What is the polar area formula for a sector?

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

Flashcard 15: Find the maximum value of $r = 2 + 3\text{cos}(\theta)$.

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

Flashcard 16: What is the expression for $dy/dx$ in polar coordinates?

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

Flashcard 17: Identify the polar form of a circle centered at the origin with radius $a$.

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

Flashcard 18: What is the formula for the slope of a tangent line in polar coordinates?

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

Flashcard 19: Convert the Cartesian equation $x^2 + y^2 = 4$ to polar form.

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

Flashcard 20: Convert the polar coordinate $(r, \theta)$ to Cartesian coordinates.

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

Flashcard 21: What is the formula for the slope of a tangent line in polar coordinates?

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

Flashcard 22: What is the formula for $r$ in terms of $x$ and $y$?

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

Flashcard 23: State the formula for $y$ in terms of $r$ and $\theta$.

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

Flashcard 24: Convert the polar coordinate $(r, \theta)$ to Cartesian coordinates.

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

Flashcard 25: State the formula for arc length in polar coordinates.

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

Flashcard 26: What is the formula for $\theta$ in terms of $x$ and $y$?

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

Flashcard 27: What is the polar form of a spiral of Archimedes?

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

Flashcard 28: What is the polar form of a limaçon with an inner loop?

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

Flashcard 29: What is the polar form of a limaçon with an inner loop?

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

Flashcard 30: State the formula for $x$ in terms of $r$ and $\theta$.

Answer: $x = r \times \text{cos}(\theta)$. Horizontal component uses cosine.

Flashcard 31: What is the polar form of a spiral of Archimedes?

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

Flashcard 32: State the formula for $y$ in terms of $r$ and $\theta$.

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