Rates of Change in Polar Functions - AP Precalculus
Card 1 of 30
Determine $\frac{dy}{dx}$ for $r(\theta) = 2\sin(\theta)$ at $\theta = \frac{\pi}{2}$.
Determine $\frac{dy}{dx}$ for $r(\theta) = 2\sin(\theta)$ at $\theta = \frac{\pi}{2}$.
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Undefined. Vertical tangent occurs when $\frac{dx}{d\theta} = 0$ but $\frac{dy}{d\theta} \neq 0$.
Undefined. Vertical tangent occurs when $\frac{dx}{d\theta} = 0$ but $\frac{dy}{d\theta} \neq 0$.
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What is the polar equation for a limaçon with inner loop?
What is the polar equation for a limaçon with inner loop?
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$r = a + b\cos(\theta)$, $|a| < |b|$. When $|a| < |b|$, the curve creates an inner loop.
$r = a + b\cos(\theta)$, $|a| < |b|$. When $|a| < |b|$, the curve creates an inner loop.
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Find $\frac{dy}{d\theta}$ for $r(\theta) = 1 + \cos(\theta)$.
Find $\frac{dy}{d\theta}$ for $r(\theta) = 1 + \cos(\theta)$.
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$\frac{dy}{d\theta} = \cos(\theta) + \sin(\theta)$. Apply product rule with $r = 1 + \cos(\theta)$ and $\frac{dr}{d\theta} = -\sin(\theta)$.
$\frac{dy}{d\theta} = \cos(\theta) + \sin(\theta)$. Apply product rule with $r = 1 + \cos(\theta)$ and $\frac{dr}{d\theta} = -\sin(\theta)$.
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Find $\frac{dx}{d\theta}$ for $r(\theta) = 1 + \cos(\theta)$.
Find $\frac{dx}{d\theta}$ for $r(\theta) = 1 + \cos(\theta)$.
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$\frac{dx}{d\theta} = -\sin^2(\theta)$. Apply product rule with $r = 1 + \cos(\theta)$ and $\frac{dr}{d\theta} = -\sin(\theta)$.
$\frac{dx}{d\theta} = -\sin^2(\theta)$. Apply product rule with $r = 1 + \cos(\theta)$ and $\frac{dr}{d\theta} = -\sin(\theta)$.
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What is the polar equation for a spiral of Archimedes?
What is the polar equation for a spiral of Archimedes?
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$r = a\theta$. Linear relationship between radius and angle creates uniform spiral.
$r = a\theta$. Linear relationship between radius and angle creates uniform spiral.
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What is the derivative of $r = a\theta + b$ with respect to $\theta$?
What is the derivative of $r = a\theta + b$ with respect to $\theta$?
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$\frac{dr}{d\theta} = a$. Derivative of linear function is the coefficient of $\theta$.
$\frac{dr}{d\theta} = a$. Derivative of linear function is the coefficient of $\theta$.
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What is the polar equation for a spiral of Archimedes?
What is the polar equation for a spiral of Archimedes?
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$r = a\theta$. Linear relationship between radius and angle creates uniform spiral.
$r = a\theta$. Linear relationship between radius and angle creates uniform spiral.
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What is the formula for $\frac{dx}{d\theta}$ when $r = \sin(\theta)$?
What is the formula for $\frac{dx}{d\theta}$ when $r = \sin(\theta)$?
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$\frac{dx}{d\theta} = \cos(\theta)\cos(\theta) - \sin^2(\theta)$. Substitute $r = \sin(\theta)$ and $\frac{dr}{d\theta} = \cos(\theta)$ into formula.
$\frac{dx}{d\theta} = \cos(\theta)\cos(\theta) - \sin^2(\theta)$. Substitute $r = \sin(\theta)$ and $\frac{dr}{d\theta} = \cos(\theta)$ into formula.
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What is the formula for the rate of change of $x$ with respect to $\theta$?
What is the formula for the rate of change of $x$ with respect to $\theta$?
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$\frac{dx}{d\theta} = \frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)$. Product rule applied to $x = r\cos(\theta)$ with respect to $\theta$.
$\frac{dx}{d\theta} = \frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)$. Product rule applied to $x = r\cos(\theta)$ with respect to $\theta$.
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Find $\frac{dr}{d\theta}$ for $r(\theta) = 3\cos(2\theta)$.
Find $\frac{dr}{d\theta}$ for $r(\theta) = 3\cos(2\theta)$.
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$\frac{dr}{d\theta} = -6\sin(2\theta)$. Chain rule: derivative of $3\cos(2\theta)$ is $-6\sin(2\theta)$.
$\frac{dr}{d\theta} = -6\sin(2\theta)$. Chain rule: derivative of $3\cos(2\theta)$ is $-6\sin(2\theta)$.
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What is the polar equation of a cardioid?
What is the polar equation of a cardioid?
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$r = a(1 + \cos(\theta))$. Heart-shaped curve with cusp at the origin.
$r = a(1 + \cos(\theta))$. Heart-shaped curve with cusp at the origin.
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What is the derivative of $r = a\theta + b$ with respect to $\theta$?
What is the derivative of $r = a\theta + b$ with respect to $\theta$?
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$\frac{dr}{d\theta} = a$. Derivative of linear function is the coefficient of $\theta$.
$\frac{dr}{d\theta} = a$. Derivative of linear function is the coefficient of $\theta$.
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What is the polar equation for a lemniscate?
What is the polar equation for a lemniscate?
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$r^2 = a^2\cos(2\theta)$. Figure-eight curve with equation involving $\cos(2\theta)$.
$r^2 = a^2\cos(2\theta)$. Figure-eight curve with equation involving $\cos(2\theta)$.
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Find $\frac{dy}{dx}$ for $r(\theta) = 1 + \sin(\theta)$ at $\theta = \frac{\pi}{2}$.
Find $\frac{dy}{dx}$ for $r(\theta) = 1 + \sin(\theta)$ at $\theta = \frac{\pi}{2}$.
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$0$. Horizontal tangent occurs when $\frac{dy}{d\theta} = 0$ at this angle.
$0$. Horizontal tangent occurs when $\frac{dy}{d\theta} = 0$ at this angle.
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What is the polar equation for an exponential spiral?
What is the polar equation for an exponential spiral?
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$r = ae^{b\theta}$. Exponential growth pattern with constant $a$ and growth rate $b$.
$r = ae^{b\theta}$. Exponential growth pattern with constant $a$ and growth rate $b$.
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What is the formula for $\frac{dy}{d\theta}$ when $r = \sin(\theta)$?
What is the formula for $\frac{dy}{d\theta}$ when $r = \sin(\theta)$?
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$\frac{dy}{d\theta} = \cos^2(\theta) + \sin(\theta)\cos(\theta)$. Substitute $r = \sin(\theta)$ and $\frac{dr}{d\theta} = \cos(\theta)$ into formula.
$\frac{dy}{d\theta} = \cos^2(\theta) + \sin(\theta)\cos(\theta)$. Substitute $r = \sin(\theta)$ and $\frac{dr}{d\theta} = \cos(\theta)$ into formula.
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What is the formula for $\frac{dx}{d\theta}$ when $r = \sin(\theta)$?
What is the formula for $\frac{dx}{d\theta}$ when $r = \sin(\theta)$?
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$\frac{dx}{d\theta} = \cos(\theta)\cos(\theta) - \sin^2(\theta)$. Substitute $r = \sin(\theta)$ and $\frac{dr}{d\theta} = \cos(\theta)$ into formula.
$\frac{dx}{d\theta} = \cos(\theta)\cos(\theta) - \sin^2(\theta)$. Substitute $r = \sin(\theta)$ and $\frac{dr}{d\theta} = \cos(\theta)$ into formula.
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Find the derivative $\frac{dr}{d\theta}$ for $r = 3\theta^2$.
Find the derivative $\frac{dr}{d\theta}$ for $r = 3\theta^2$.
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$\frac{dr}{d\theta} = 6\theta$. Power rule: derivative of $3\theta^2$ is $6\theta$.
$\frac{dr}{d\theta} = 6\theta$. Power rule: derivative of $3\theta^2$ is $6\theta$.
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What is the general polar form of an ellipse?
What is the general polar form of an ellipse?
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$r = \frac{ed}{1 + e\cos(\theta)}$. Standard form where $e$ is eccentricity and $d$ is directrix distance.
$r = \frac{ed}{1 + e\cos(\theta)}$. Standard form where $e$ is eccentricity and $d$ is directrix distance.
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Find $\frac{dr}{d\theta}$ for $r(\theta) = 2\theta + \sin(\theta)$.
Find $\frac{dr}{d\theta}$ for $r(\theta) = 2\theta + \sin(\theta)$.
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$\frac{dr}{d\theta} = 2 + \cos(\theta)$. Derivative of each term: $2$ from $2\theta$ and $\cos(\theta)$ from $\sin(\theta)$.
$\frac{dr}{d\theta} = 2 + \cos(\theta)$. Derivative of each term: $2$ from $2\theta$ and $\cos(\theta)$ from $\sin(\theta)$.
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What is the polar equation for a line through the origin at an angle $\alpha$?
What is the polar equation for a line through the origin at an angle $\alpha$?
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$\theta = \alpha$. Constant angle creates a ray from the origin.
$\theta = \alpha$. Constant angle creates a ray from the origin.
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What is the polar equation for a circle centered at the origin with radius 2?
What is the polar equation for a circle centered at the origin with radius 2?
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$r = 2$. Constant radius distance from origin defines a circle.
$r = 2$. Constant radius distance from origin defines a circle.
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Identify the formula for $\frac{dy}{dx}$ in polar coordinates.
Identify the formula for $\frac{dy}{dx}$ in polar coordinates.
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$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$. Chain rule connecting Cartesian and polar derivatives.
$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$. Chain rule connecting Cartesian and polar derivatives.
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What is the formula for arc length $L$ of a polar curve $r(\theta)$?
What is the formula for arc length $L$ of a polar curve $r(\theta)$?
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$L = \int_{a}^{b} \sqrt{(\frac{dr}{d\theta})^2 + r^2} , d\theta$. Integrates the speed element in polar coordinates over the interval.
$L = \int_{a}^{b} \sqrt{(\frac{dr}{d\theta})^2 + r^2} , d\theta$. Integrates the speed element in polar coordinates over the interval.
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Convert the polar function $r = 3\sin(\theta)$ to Cartesian coordinates.
Convert the polar function $r = 3\sin(\theta)$ to Cartesian coordinates.
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$x^2 + (y - \frac{3}{2})^2 = (\frac{3}{2})^2$. Circle with center $(0, \frac{3}{2})$ and radius $\frac{3}{2}$.
$x^2 + (y - \frac{3}{2})^2 = (\frac{3}{2})^2$. Circle with center $(0, \frac{3}{2})$ and radius $\frac{3}{2}$.
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What is the formula for the instantaneous rate of change of $r$?
What is the formula for the instantaneous rate of change of $r$?
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$\frac{dr}{d\theta}$. Represents the instantaneous rate of change of radius with angle.
$\frac{dr}{d\theta}$. Represents the instantaneous rate of change of radius with angle.
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Find $\frac{dy}{d\theta}$ for $r(\theta) = \theta^2$.
Find $\frac{dy}{d\theta}$ for $r(\theta) = \theta^2$.
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$\frac{dy}{d\theta} = 2\theta \sin(\theta) + \theta^2 \cos(\theta)$. Substitute $r = \theta^2$ and $\frac{dr}{d\theta} = 2\theta$ into the formula.
$\frac{dy}{d\theta} = 2\theta \sin(\theta) + \theta^2 \cos(\theta)$. Substitute $r = \theta^2$ and $\frac{dr}{d\theta} = 2\theta$ into the formula.
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Find $\frac{dx}{d\theta}$ for $r(\theta) = \theta^2$.
Find $\frac{dx}{d\theta}$ for $r(\theta) = \theta^2$.
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$\frac{dx}{d\theta} = 2\theta \cos(\theta) - \theta^2 \sin(\theta)$. Substitute $r = \theta^2$ and $\frac{dr}{d\theta} = 2\theta$ into the formula.
$\frac{dx}{d\theta} = 2\theta \cos(\theta) - \theta^2 \sin(\theta)$. Substitute $r = \theta^2$ and $\frac{dr}{d\theta} = 2\theta$ into the formula.
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Identify the formula for $\frac{dy}{d\theta}$ in polar coordinates.
Identify the formula for $\frac{dy}{d\theta}$ in polar coordinates.
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$\frac{dy}{d\theta} = \frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)$. Standard formula for vertical rate of change 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.
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Identify the formula for $\frac{dx}{d\theta}$ in polar coordinates.
Identify the formula for $\frac{dx}{d\theta}$ in polar coordinates.
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$\frac{dx}{d\theta} = \frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)$. Standard formula for horizontal rate of change 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.
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