Defining and Differentiating Parametric Equations - AP Calculus BC
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What is a parametric representation of a circle?
What is a parametric representation of a circle?
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$x = \text{cos}(t), y = \text{sin}(t)$. Unit circle traced counterclockwise using trigonometric functions.
$x = \text{cos}(t), y = \text{sin}(t)$. Unit circle traced counterclockwise using trigonometric functions.
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Find $\frac{dy}{dx}$ for $x = t^2, y = t^3$.
Find $\frac{dy}{dx}$ for $x = t^2, y = t^3$.
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$\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}$. Apply formula: $\frac{dy}{dx} = \frac{3t^2}{2t}$ and simplify.
$\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}$. Apply formula: $\frac{dy}{dx} = \frac{3t^2}{2t}$ and simplify.
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What is the chain rule for parametric equations?
What is the chain rule for parametric equations?
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Relates $\frac{dy}{dt}$ and $\frac{dx}{dt}$ to $\frac{dy}{dx}$. Connects parametric derivatives to Cartesian slope using division.
Relates $\frac{dy}{dt}$ and $\frac{dx}{dt}$ to $\frac{dy}{dx}$. Connects parametric derivatives to Cartesian slope using division.
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Identify the curve: $x = a\text{cos}(t), y = b\text{sin}(t)$.
Identify the curve: $x = a\text{cos}(t), y = b\text{sin}(t)$.
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An ellipse. Standard parametric form of ellipse with semi-axes $a$ and $b$.
An ellipse. Standard parametric form of ellipse with semi-axes $a$ and $b$.
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What is the purpose of parametric equations?
What is the purpose of parametric equations?
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To describe curves in the plane using a parameter. Allows representation of complex curves that functions cannot describe.
To describe curves in the plane using a parameter. Allows representation of complex curves that functions cannot describe.
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What is the formula for the tangent line to a parametric curve?
What is the formula for the tangent line to a parametric curve?
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$y - y_1 = m(x - x_1)$ where $m = \frac{dy}{dx}$. Point-slope form where slope is the parametric derivative $\frac{dy}{dx}$.
$y - y_1 = m(x - x_1)$ where $m = \frac{dy}{dx}$. Point-slope form where slope is the parametric derivative $\frac{dy}{dx}$.
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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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$x^2 + y^2 = 4$. Circle with radius $2$ centered at origin using trigonometric identity.
$x^2 + y^2 = 4$. Circle with radius $2$ centered at origin using trigonometric identity.
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What does $\frac{dy}{dt} = 0$ indicate?
What does $\frac{dy}{dt} = 0$ indicate?
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Horizontal tangent line at that point. When vertical change is zero, tangent line becomes horizontal.
Horizontal tangent line at that point. When vertical change is zero, tangent line becomes horizontal.
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Find $\frac{dx}{dt}$ for $x = 4t^2 - 3t + 1$.
Find $\frac{dx}{dt}$ for $x = 4t^2 - 3t + 1$.
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$\frac{dx}{dt} = 8t - 3$. Differentiate with respect to $t$: derivative of $4t^2$ is $8t$.
$\frac{dx}{dt} = 8t - 3$. Differentiate with respect to $t$: derivative of $4t^2$ is $8t$.
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Convert $x = 5\text{cos}(t), y = 5\text{sin}(t)$ to Cartesian form.
Convert $x = 5\text{cos}(t), y = 5\text{sin}(t)$ to Cartesian form.
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$x^2 + y^2 = 25$. Circle with radius $5$ centered at origin using trigonometric identity.
$x^2 + y^2 = 25$. Circle with radius $5$ centered at origin using trigonometric identity.
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Identify the curve: $x = 2t, y = 3t^2$.
Identify the curve: $x = 2t, y = 3t^2$.
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A parabola. Linear $x$ and quadratic $y$ create a parabolic relationship.
A parabola. Linear $x$ and quadratic $y$ create a parabolic relationship.
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Find $\frac{d^2y}{dx^2}$ for $x = t^3, y = t^2$.
Find $\frac{d^2y}{dx^2}$ for $x = t^3, y = t^2$.
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$\frac{d^2y}{dx^2} = \frac{2}{3t^2}$. Use formula: $\frac{d}{dt}(\frac{dy}{dx}) \div \frac{dx}{dt}$ for second derivative.
$\frac{d^2y}{dx^2} = \frac{2}{3t^2}$. Use formula: $\frac{d}{dt}(\frac{dy}{dx}) \div \frac{dx}{dt}$ for second derivative.
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Find $\frac{dy}{dx}$ for $x = e^t, y = \text{ln}(t)$.
Find $\frac{dy}{dx}$ for $x = e^t, y = \text{ln}(t)$.
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$\frac{dy}{dx} = \frac{\frac{1}{t}}{e^t}$. Apply formula with $\frac{dx}{dt} = e^t$ and $\frac{dy}{dt} = \frac{1}{t}$.
$\frac{dy}{dx} = \frac{\frac{1}{t}}{e^t}$. Apply formula with $\frac{dx}{dt} = e^t$ and $\frac{dy}{dt} = \frac{1}{t}$.
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What does differentiating parametric equations yield?
What does differentiating parametric equations yield?
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The slope of the tangent to the curve. Gives the slope of the tangent line at any point on the curve.
The slope of the tangent to the curve. Gives the slope of the tangent line at any point on the curve.
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Identify the curve: $x = a\text{cos}(t), y = b\text{sin}(t)$.
Identify the curve: $x = a\text{cos}(t), y = b\text{sin}(t)$.
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An ellipse. Standard parametric form of ellipse with semi-axes $a$ and $b$.
An ellipse. Standard parametric form of ellipse with semi-axes $a$ and $b$.
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What are parametric equations?
What are parametric equations?
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Equations that express coordinates as functions of a parameter. Uses a third variable (parameter) to define both $x$ and $y$ coordinates.
Equations that express coordinates as functions of a parameter. Uses a third variable (parameter) to define both $x$ and $y$ coordinates.
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Identify the parameter in $x = 3t + 2, y = 2t - 1$.
Identify the parameter in $x = 3t + 2, y = 2t - 1$.
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The parameter is $t$. The parameter is the independent variable in parametric equations.
The parameter is $t$. The parameter is the independent variable in parametric equations.
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Find $\frac{dy}{dt}$ for $y = 5\text{sin}(t)$.
Find $\frac{dy}{dt}$ for $y = 5\text{sin}(t)$.
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$\frac{dy}{dt} = 5\text{cos}(t)$. Differentiate with respect to $t$: derivative of $\sin(t)$ is $\cos(t)$.
$\frac{dy}{dt} = 5\text{cos}(t)$. Differentiate with respect to $t$: derivative of $\sin(t)$ is $\cos(t)$.
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Convert $x = \text{cos}(t), y = \text{sin}(t)$ to Cartesian form.
Convert $x = \text{cos}(t), y = \text{sin}(t)$ to Cartesian form.
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$x^2 + y^2 = 1$. Use trigonometric identity: $\cos^2(t) + \sin^2(t) = 1$.
$x^2 + y^2 = 1$. Use trigonometric identity: $\cos^2(t) + \sin^2(t) = 1$.
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Find $x(0)$ and $y(0)$ for $x = t^2 - 2t, y = \text{ln}(t + 1)$.
Find $x(0)$ and $y(0)$ for $x = t^2 - 2t, y = \text{ln}(t + 1)$.
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$x(0) = 0, y(0) = 0$. Substitute $t = 0$ into both parametric equations.
$x(0) = 0, y(0) = 0$. Substitute $t = 0$ into both parametric equations.
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Identify the curve: $x = a\text{cosh}(t), y = b\text{sinh}(t)$.
Identify the curve: $x = a\text{cosh}(t), y = b\text{sinh}(t)$.
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A hyperbola. Standard parametric form using hyperbolic functions $\cosh$ and $\sinh$.
A hyperbola. Standard parametric form using hyperbolic functions $\cosh$ and $\sinh$.
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Find the parametric equations for a line: $y = 2x + 3$.
Find the parametric equations for a line: $y = 2x + 3$.
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$x = t, y = 2t + 3$. Set $x = t$ as parameter, then $y = 2t + 3$ follows directly.
$x = t, y = 2t + 3$. Set $x = t$ as parameter, then $y = 2t + 3$ follows directly.
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Identify the parameter interval for an ellipse: $0 \text{ to } 2\text{π}$.
Identify the parameter interval for an ellipse: $0 \text{ to } 2\text{π}$.
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Completes one full revolution around the ellipse. Parameter traces the entire ellipse once as $t$ goes from $0$ to $2\pi$.
Completes one full revolution around the ellipse. Parameter traces the entire ellipse once as $t$ goes from $0$ to $2\pi$.
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Convert $x = 3t, y = 4t$ to Cartesian form.
Convert $x = 3t, y = 4t$ to Cartesian form.
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$y = \frac{4}{3}x$. Both coordinates are proportional to $t$, creating a straight line.
$y = \frac{4}{3}x$. Both coordinates are proportional to $t$, creating a straight line.
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Find $\frac{dy}{dx}$ for $x = 4\text{cos}(t), y = 3\text{sin}(t)$.
Find $\frac{dy}{dx}$ for $x = 4\text{cos}(t), y = 3\text{sin}(t)$.
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$\frac{dy}{dx} = -\frac{3}{4}\text{tan}(t)$. Apply formula: $\frac{dy/dt}{dx/dt} = \frac{3\cos(t)}{-4\sin(t)}$.
$\frac{dy}{dx} = -\frac{3}{4}\text{tan}(t)$. Apply formula: $\frac{dy/dt}{dx/dt} = \frac{3\cos(t)}{-4\sin(t)}$.
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What is the significance of $\frac{dy}{dx} = 0$?
What is the significance of $\frac{dy}{dx} = 0$?
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Indicates a horizontal tangent. Zero slope means the tangent line is perfectly horizontal.
Indicates a horizontal tangent. Zero slope means the tangent line is perfectly horizontal.
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Find $\frac{d^2y}{dx^2}$ for $x = t, y = t^3$.
Find $\frac{d^2y}{dx^2}$ for $x = t, y = t^3$.
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$\frac{d^2y}{dx^2} = 6t$. Apply second derivative formula: $\frac{d}{dt}(3t^2) \div 1 = 6t$.
$\frac{d^2y}{dx^2} = 6t$. Apply second derivative formula: $\frac{d}{dt}(3t^2) \div 1 = 6t$.
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What is the formula for the second derivative in parametric form?
What is the formula for the second derivative in parametric form?
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$\frac{d^2y}{dx^2} = \frac{d}{dt}(\frac{dy}{dx})/\frac{dx}{dt}$. Differentiate $\frac{dy}{dx}$ with respect to $t$, then divide by $\frac{dx}{dt}$.
$\frac{d^2y}{dx^2} = \frac{d}{dt}(\frac{dy}{dx})/\frac{dx}{dt}$. Differentiate $\frac{dy}{dx}$ with respect to $t$, then divide by $\frac{dx}{dt}$.
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Find $\frac{dy}{dx}$ for $x = 2t + 1, y = 3t^2$.
Find $\frac{dy}{dx}$ for $x = 2t + 1, y = 3t^2$.
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$\frac{dy}{dx} = \frac{6t}{2} = 3t$. Apply formula: $\frac{dy}{dx} = \frac{6t}{2} = 3t$.
$\frac{dy}{dx} = \frac{6t}{2} = 3t$. Apply formula: $\frac{dy}{dx} = \frac{6t}{2} = 3t$.
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What is a cycloid?
What is a cycloid?
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A curve generated by a point on the rim of a rolling circle. Classic curve traced by a point on a wheel rolling along a line.
A curve generated by a point on the rim of a rolling circle. Classic curve traced by a point on a wheel rolling along a line.
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