Polar Coordinates and Differentiation - AP Calculus BC
Card 1 of 30
State the formula for arc length in polar coordinates.
State the formula for arc length in polar coordinates.
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$L = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + (r')^2} , d\theta$. Integrates $\sqrt{r^2 + \left( \frac{dr}{d\theta} \right)^2}$
$L = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + (r')^2} , d\theta$. Integrates $\sqrt{r^2 + \left( \frac{dr}{d\theta} \right)^2}$
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State the formula for $x$ in terms of $r$ and $\theta$.
State the formula for $x$ in terms of $r$ and $\theta$.
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$x = r \times \cos(\theta)$. Horizontal component uses cosine.
$x = r \times \cos(\theta)$. Horizontal component uses cosine.
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What is the general form of a polar coordinate?
What is the general form of a polar coordinate?
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$(r, \theta)$. Distance from origin and angle from positive x-axis.
$(r, \theta)$. Distance from origin and angle from positive x-axis.
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What is the formula for the slope of a tangent line in polar coordinates?
What is the formula for the slope of a tangent line in polar coordinates?
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$\frac{dy}{dx} = \frac{r' \sin(\theta) + r \cos(\theta)}{r' \cos(\theta) - r \sin(\theta)}$. Derivative of y with respect to x in polar form.
$\frac{dy}{dx} = \frac{r' \sin(\theta) + r \cos(\theta)}{r' \cos(\theta) - r \sin(\theta)}$. Derivative of y with respect to x in polar form.
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What is the polar form of a spiral of Archimedes?
What is the polar form of a spiral of Archimedes?
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$r = a + b\theta$. Spiral increasing linearly with angle.
$r = a + b\theta$. Spiral increasing linearly with angle.
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What is the formula for $r$ in terms of $x$ and $y$?
What is the formula for $r$ in terms of $x$ and $y$?
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$r = \text{sqrt}(x^2 + y^2)$. Pythagorean theorem applied to coordinates.
$r = \text{sqrt}(x^2 + y^2)$. Pythagorean theorem applied to coordinates.
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Find the maximum value of $r = 2 + 3\text{cos}(\theta)$.
Find the maximum value of $r = 2 + 3\text{cos}(\theta)$.
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Maximum $r = 5$. Maximum occurs when $\cos(\theta) = 1$.
Maximum $r = 5$. Maximum occurs when $\cos(\theta) = 1$.
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Convert the Cartesian coordinate $(x, y)$ to polar coordinates.
Convert the Cartesian coordinate $(x, y)$ to polar coordinates.
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$(r, \theta) = (\text{sqrt}(x^2 + y^2), \text{tan}^{-1}(\frac{y}{x}))$. Distance formula and arctangent for angle.
$(r, \theta) = (\text{sqrt}(x^2 + y^2), \text{tan}^{-1}(\frac{y}{x}))$. Distance formula and arctangent for angle.
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What is the formula for $\theta$ in terms of $x$ and $y$?
What is the formula for $\theta$ in terms of $x$ and $y$?
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$\theta = \text{tan}^{-1}(\frac{y}{x})$. Inverse tangent of y over x ratio.
$\theta = \text{tan}^{-1}(\frac{y}{x})$. Inverse tangent of y over x ratio.
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What is the polar form of a limaçon with an inner loop?
What is the polar form of a limaçon with an inner loop?
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$r = a + b\text{cos}(\theta)$ where $a < b$. When $a < b$, creates inner loop.
$r = a + b\text{cos}(\theta)$ where $a < b$. When $a < b$, creates inner loop.
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Convert the polar coordinate $(r, \theta)$ to Cartesian coordinates.
Convert the polar coordinate $(r, \theta)$ to Cartesian coordinates.
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$(x, y) = (r \times \text{cos}(\theta), r \times \text{sin}(\theta))$. Use $x = r\cos(\theta)$ and $y = r\sin(\theta)$.
$(x, y) = (r \times \text{cos}(\theta), r \times \text{sin}(\theta))$. Use $x = r\cos(\theta)$ and $y = r\sin(\theta)$.
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What is the expression for $dy/dx$ in polar coordinates?
What is the expression for $dy/dx$ in polar coordinates?
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$\frac{dy}{dx} = \frac{r' \text{sin}(\theta) + r \text{cos}(\theta)}{r' \text{cos}(\theta) - r \text{sin}(\theta)}$. Chain rule applied to parametric equations.
$\frac{dy}{dx} = \frac{r' \text{sin}(\theta) + r \text{cos}(\theta)}{r' \text{cos}(\theta) - r \text{sin}(\theta)}$. Chain rule applied to parametric equations.
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Identify the polar form of a circle centered at the origin with radius $a$.
Identify the polar form of a circle centered at the origin with radius $a$.
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$r = a$. Constant radius equals constant distance from origin.
$r = a$. Constant radius equals constant distance from origin.
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Convert the Cartesian equation $x^2 + y^2 = 4$ to polar form.
Convert the Cartesian equation $x^2 + y^2 = 4$ to polar form.
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$r = 2$. Circle centered at origin with radius $2$.
$r = 2$. Circle centered at origin with radius $2$.
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What is the polar area formula for a sector?
What is the polar area formula for a sector?
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Area = $\frac{1}{2} \times r^2 \times \theta$. Half the product of radius squared and central angle.
Area = $\frac{1}{2} \times r^2 \times \theta$. Half the product of radius squared and central angle.
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State the formula for $y$ in terms of $r$ and $\theta$.
State the formula for $y$ in terms of $r$ and $\theta$.
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$y = r \times \text{sin}(\theta)$. Vertical component uses sine.
$y = r \times \text{sin}(\theta)$. Vertical component uses sine.
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What is the formula for $\theta$ in terms of $x$ and $y$?
What is the formula for $\theta$ in terms of $x$ and $y$?
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$\theta = \text{tan}^{-1}(\frac{y}{x})$. Inverse tangent of y over x ratio.
$\theta = \text{tan}^{-1}(\frac{y}{x})$. Inverse tangent of y over x ratio.
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State the formula for $x$ in terms of $r$ and $\theta$.
State the formula for $x$ in terms of $r$ and $\theta$.
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$x = r \times \text{cos}(\theta)$. Horizontal component uses cosine.
$x = r \times \text{cos}(\theta)$. Horizontal component uses cosine.
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What is the formula for the slope of a tangent line in polar coordinates?
What is the formula for the slope of a tangent line in polar coordinates?
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$\frac{dy}{dx} = \frac{r' \text{sin}(\theta) + r \text{cos}(\theta)}{r' \text{cos}(\theta) - r \text{sin}(\theta)}$. Derivative of y with respect to x in polar form.
$\frac{dy}{dx} = \frac{r' \text{sin}(\theta) + r \text{cos}(\theta)}{r' \text{cos}(\theta) - r \text{sin}(\theta)}$. Derivative of y with respect to x in polar form.
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What is the polar form of a spiral of Archimedes?
What is the polar form of a spiral of Archimedes?
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$r = a + b\theta$. Spiral increasing linearly with angle.
$r = a + b\theta$. Spiral increasing linearly with angle.
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Find the maximum value of $r = 2 + 3\text{cos}(\theta)$.
Find the maximum value of $r = 2 + 3\text{cos}(\theta)$.
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Maximum $r = 5$. Maximum occurs when $\cos(\theta) = 1$.
Maximum $r = 5$. Maximum occurs when $\cos(\theta) = 1$.
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Convert the Cartesian coordinate $(x, y)$ to polar coordinates.
Convert the Cartesian coordinate $(x, y)$ to polar coordinates.
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$(r, \theta) = (\text{sqrt}(x^2 + y^2), \text{tan}^{-1}(\frac{y}{x}))$. Distance formula and arctangent for angle.
$(r, \theta) = (\text{sqrt}(x^2 + y^2), \text{tan}^{-1}(\frac{y}{x}))$. Distance formula and arctangent for angle.
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What is the polar form of a limaçon with an inner loop?
What is the polar form of a limaçon with an inner loop?
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$r = a + b\text{cos}(\theta)$ where $a < b$. When $a < b$, creates inner loop.
$r = a + b\text{cos}(\theta)$ where $a < b$. When $a < b$, creates inner loop.
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Convert the polar coordinate $(r, \theta)$ to Cartesian coordinates.
Convert the polar coordinate $(r, \theta)$ to Cartesian coordinates.
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$(x, y) = (r \times \text{cos}(\theta), r \times \text{sin}(\theta))$. Use $x = r\cos(\theta)$ and $y = r\sin(\theta)$.
$(x, y) = (r \times \text{cos}(\theta), r \times \text{sin}(\theta))$. Use $x = r\cos(\theta)$ and $y = r\sin(\theta)$.
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What is the formula for $r$ in terms of $x$ and $y$?
What is the formula for $r$ in terms of $x$ and $y$?
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$r = \text{sqrt}(x^2 + y^2)$. Pythagorean theorem applied to coordinates.
$r = \text{sqrt}(x^2 + y^2)$. Pythagorean theorem applied to coordinates.
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State the formula for $y$ in terms of $r$ and $\theta$.
State the formula for $y$ in terms of $r$ and $\theta$.
Tap to reveal answer
$y = r \times \text{sin}(\theta)$. Vertical component uses sine.
$y = r \times \text{sin}(\theta)$. Vertical component uses sine.
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What is the polar area formula for a sector?
What is the polar area formula for a sector?
Tap to reveal answer
Area = $\frac{1}{2} \times r^2 \times \theta$. Half the product of radius squared and central angle.
Area = $\frac{1}{2} \times r^2 \times \theta$. Half the product of radius squared and central angle.
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What is the general form of a polar coordinate?
What is the general form of a polar coordinate?
Tap to reveal answer
$(r, \theta)$. Distance from origin and angle from positive x-axis.
$(r, \theta)$. Distance from origin and angle from positive x-axis.
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Convert the Cartesian equation $x^2 + y^2 = 4$ to polar form.
Convert the Cartesian equation $x^2 + y^2 = 4$ to polar form.
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$r = 2$. Circle centered at origin with radius $2$.
$r = 2$. Circle centered at origin with radius $2$.
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Identify the polar form of a circle centered at the origin with radius $a$.
Identify the polar form of a circle centered at the origin with radius $a$.
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$r = a$. Constant radius equals constant distance from origin.
$r = a$. Constant radius equals constant distance from origin.
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