Parametric Functions and Rates of Change - AP Precalculus
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State the parametric equations for a line with slope $m$ passing through $(x_0, y_0)$.
State the parametric equations for a line with slope $m$ passing through $(x_0, y_0)$.
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$x = x_0 + t, y = y_0 + mt$. Direction vector $(1, m)$ parameterizes the line.
$x = x_0 + t, y = y_0 + mt$. Direction vector $(1, m)$ parameterizes the line.
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What is the parametric representation for an ellipse?
What is the parametric representation for an ellipse?
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$x = a\text{cos}(t), y = b\text{sin}(t)$. General form with semi-axes $a$ and $b$.
$x = a\text{cos}(t), y = b\text{sin}(t)$. General form with semi-axes $a$ and $b$.
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Find the slope of the tangent line at $t = 1$ for $x(t) = t^2, y(t) = t^3$.
Find the slope of the tangent line at $t = 1$ for $x(t) = t^2, y(t) = t^3$.
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Slope is $\frac{3}{2}$. Evaluate $\frac{dy}{dx} = \frac{3t^2}{2t}$ at $t = 1$.
Slope is $\frac{3}{2}$. Evaluate $\frac{dy}{dx} = \frac{3t^2}{2t}$ at $t = 1$.
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Identify the parameter for $x(t) = \text{sin}(t), y(t) = \text{cos}(t)$.
Identify the parameter for $x(t) = \text{sin}(t), y(t) = \text{cos}(t)$.
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The parameter is $t$. The independent variable in the parametric representation.
The parameter is $t$. The independent variable in the parametric representation.
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Convert $x = 3t + 1, y = 2t - 4$ to Cartesian form.
Convert $x = 3t + 1, y = 2t - 4$ to Cartesian form.
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$y = \frac{2}{3}(x - 1) - 4$. Solve for $t$ from first equation, substitute into second.
$y = \frac{2}{3}(x - 1) - 4$. Solve for $t$ from first equation, substitute into second.
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Identify the parameter for $x(t) = \text{tan}(t), y(t) = \text{sec}(t)$.
Identify the parameter for $x(t) = \text{tan}(t), y(t) = \text{sec}(t)$.
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The parameter is $t$. The independent variable in these parametric equations.
The parameter is $t$. The independent variable in these parametric equations.
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Convert $x(t) = t, y(t) = t^2$ to a Cartesian equation.
Convert $x(t) = t, y(t) = t^2$ to a Cartesian equation.
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$y = x^2$. Direct substitution since $x = t$.
$y = x^2$. Direct substitution since $x = t$.
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What is the second derivative $\frac{d^2y}{dx^2}$ for parametric equations?
What is the second derivative $\frac{d^2y}{dx^2}$ for parametric equations?
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$\frac{d}{dt}(\frac{dy}{dx}) / \frac{dx}{dt}$. Uses chain rule twice for parametric second derivatives.
$\frac{d}{dt}(\frac{dy}{dx}) / \frac{dx}{dt}$. Uses chain rule twice for parametric second derivatives.
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What is the derivative $\frac{dy}{dx}$ for parametric equations $x(t)$ and $y(t)$?
What is the derivative $\frac{dy}{dx}$ for parametric equations $x(t)$ and $y(t)$?
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$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. Chain rule applied to parametric equations.
$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. Chain rule applied to parametric equations.
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Identify the parameter in: $x(t) = 3t + 2, y(t) = 4t - 1$.
Identify the parameter in: $x(t) = 3t + 2, y(t) = 4t - 1$.
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The parameter is $t$. The independent variable in parametric equations.
The parameter is $t$. The independent variable in parametric equations.
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Identify the type of curve: $x = 3 \text{cos}(t), y = 3 \text{sin}(t)$.
Identify the type of curve: $x = 3 \text{cos}(t), y = 3 \text{sin}(t)$.
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A circle with radius 3. Standard parametric form for a circle centered at origin.
A circle with radius 3. Standard parametric form for a circle centered at origin.
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State the parametric equations for a line with slope $m$ passing through $(x_0, y_0)$.
State the parametric equations for a line with slope $m$ passing through $(x_0, y_0)$.
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$x = x_0 + t, y = y_0 + mt$. Direction vector $(1, m)$ parameterizes the line.
$x = x_0 + t, y = y_0 + mt$. Direction vector $(1, m)$ parameterizes the line.
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Find $\frac{dy}{dx}$ for $x(t) = \text{cos}(t), y(t) = \text{sin}(t)$.
Find $\frac{dy}{dx}$ for $x(t) = \text{cos}(t), y(t) = \text{sin}(t)$.
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$\frac{dy}{dx} = -\text{cot}(t)$. $\frac{dy}{dt} = \cos(t)$, $\frac{dx}{dt} = -\sin(t)$, so ratio is $-\cot(t)$.
$\frac{dy}{dx} = -\text{cot}(t)$. $\frac{dy}{dt} = \cos(t)$, $\frac{dx}{dt} = -\sin(t)$, so ratio is $-\cot(t)$.
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What is the parametric representation for an ellipse?
What is the parametric representation for an ellipse?
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$x = a\text{cos}(t), y = b\text{sin}(t)$. General form with semi-axes $a$ and $b$.
$x = a\text{cos}(t), y = b\text{sin}(t)$. General form with semi-axes $a$ and $b$.
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Determine the point at $t = 0$ for $x(t) = 2t + 1, y(t) = 3t - 2$.
Determine the point at $t = 0$ for $x(t) = 2t + 1, y(t) = 3t - 2$.
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Point is $(1, -2)$. Substitute $t = 0$ into both equations.
Point is $(1, -2)$. Substitute $t = 0$ into both equations.
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What is the parametric form for a hyperbola?
What is the parametric form for a hyperbola?
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$x = a\text{sec}(t), y = b\text{tan}(t)$. Standard parametrization using secant and tangent functions.
$x = a\text{sec}(t), y = b\text{tan}(t)$. Standard parametrization using secant and tangent functions.
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What is the geometric interpretation of $\frac{dy}{dt}$ and $\frac{dx}{dt}$?
What is the geometric interpretation of $\frac{dy}{dt}$ and $\frac{dx}{dt}$?
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Rates of change of $y$ and $x$ with respect to $t$. Components of velocity vector in parametric motion.
Rates of change of $y$ and $x$ with respect to $t$. Components of velocity vector in parametric motion.
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Convert $x = 2\text{cos}(t), y = 2\text{sin}(t)$ to Cartesian form.
Convert $x = 2\text{cos}(t), y = 2\text{sin}(t)$ to Cartesian form.
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$x^2 + y^2 = 4$. Use identity $\cos^2(t) + \sin^2(t) = 1$.
$x^2 + y^2 = 4$. Use identity $\cos^2(t) + \sin^2(t) = 1$.
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Find the slope of the tangent line at $t = 1$ for $x(t) = t^2, y(t) = t^3$.
Find the slope of the tangent line at $t = 1$ for $x(t) = t^2, y(t) = t^3$.
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Slope is $\frac{3}{2}$. Evaluate $\frac{dy}{dx} = \frac{3t^2}{2t}$ at $t = 1$.
Slope is $\frac{3}{2}$. Evaluate $\frac{dy}{dx} = \frac{3t^2}{2t}$ at $t = 1$.
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Identify the parameter for $x(t) = \text{sin}(t), y(t) = \text{cos}(t)$.
Identify the parameter for $x(t) = \text{sin}(t), y(t) = \text{cos}(t)$.
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The parameter is $t$. The independent variable in the parametric representation.
The parameter is $t$. The independent variable in the parametric representation.
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Find the point at $t = \frac{\text{π}}{2}$ for $x(t) = \text{cos}(t), y(t) = \text{sin}(t)$.
Find the point at $t = \frac{\text{π}}{2}$ for $x(t) = \text{cos}(t), y(t) = \text{sin}(t)$.
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Point is $(0, 1)$.. At $t = \frac{\pi}{2}$: $\cos(\frac{\pi}{2}) = 0$, $\sin(\frac{\pi}{2}) = 1$.
Point is $(0, 1)$.. At $t = \frac{\pi}{2}$: $\cos(\frac{\pi}{2}) = 0$, $\sin(\frac{\pi}{2}) = 1$.
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Convert $x = 3t + 1, y = 2t - 4$ to Cartesian form.
Convert $x = 3t + 1, y = 2t - 4$ to Cartesian form.
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$y = \frac{2}{3}(x - 1) - 4$. Solve for $t$ from first equation, substitute into second.
$y = \frac{2}{3}(x - 1) - 4$. Solve for $t$ from first equation, substitute into second.
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Find $\frac{dy}{dx}$ for $x(t) = 2t^2, y(t) = 3t^2$.
Find $\frac{dy}{dx}$ for $x(t) = 2t^2, y(t) = 3t^2$.
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$\frac{dy}{dx} = \frac{3}{2}$. $\frac{dy}{dt} = 6t$, $\frac{dx}{dt} = 4t$, so ratio is constant.
$\frac{dy}{dx} = \frac{3}{2}$. $\frac{dy}{dt} = 6t$, $\frac{dx}{dt} = 4t$, so ratio is constant.
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State the parametric equations for a line through $(1, 2)$ with slope 3.
State the parametric equations for a line through $(1, 2)$ with slope 3.
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$x = 1 + t, y = 2 + 3t$. Point $(1,2)$ with direction vector $(1,3)$.
$x = 1 + t, y = 2 + 3t$. Point $(1,2)$ with direction vector $(1,3)$.
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Identify the curve: $x = 4\text{cos}(t), y = 4\text{sin}(t)$.
Identify the curve: $x = 4\text{cos}(t), y = 4\text{sin}(t)$.
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A circle with radius 4. Parametric circle with radius 4 centered at origin.
A circle with radius 4. Parametric circle with radius 4 centered at origin.
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Convert $x = 5\text{cos}(t), y = 3\text{sin}(t)$ to Cartesian form.
Convert $x = 5\text{cos}(t), y = 3\text{sin}(t)$ to Cartesian form.
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$(\frac{x}{5})^2 + (\frac{y}{3})^2 = 1$. Ellipse with semi-major axis 5 and semi-minor axis 3.
$(\frac{x}{5})^2 + (\frac{y}{3})^2 = 1$. Ellipse with semi-major axis 5 and semi-minor axis 3.
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Find the point at $t = \text{π}$ for $x(t) = \text{cos}(t), y(t) = \text{sin}(t)$.
Find the point at $t = \text{π}$ for $x(t) = \text{cos}(t), y(t) = \text{sin}(t)$.
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Point is $(-1, 0)$.. At $t = \pi$: $\cos(\pi) = -1$, $\sin(\pi) = 0$.
Point is $(-1, 0)$.. At $t = \pi$: $\cos(\pi) = -1$, $\sin(\pi) = 0$.
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Convert $x = 2t + 3, y = 4t - 5$ to a Cartesian equation.
Convert $x = 2t + 3, y = 4t - 5$ to a Cartesian equation.
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$y = 2x - 11$. From $x = 2t + 3$, get $t = \frac{x-3}{2}$, substitute.
$y = 2x - 11$. From $x = 2t + 3$, get $t = \frac{x-3}{2}$, substitute.
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What is the parametric form for a line segment from $(x_0, y_0)$ to $(x_1, y_1)$?
What is the parametric form for a line segment from $(x_0, y_0)$ to $(x_1, y_1)$?
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$x = x_0 + t(x_1-x_0), y = y_0 + t(y_1-y_0)$. Linear interpolation between two points using parameter $t$.
$x = x_0 + t(x_1-x_0), y = y_0 + t(y_1-y_0)$. Linear interpolation between two points using parameter $t$.
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Identify the parameter for $x(t) = \text{tan}(t), y(t) = \text{sec}(t)$.
Identify the parameter for $x(t) = \text{tan}(t), y(t) = \text{sec}(t)$.
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The parameter is $t$. The independent variable in these parametric equations.
The parameter is $t$. The independent variable in these parametric equations.
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