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Parametric Equations

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Basic Derivative of Parametric Equations

For parametric equations $x = f(t)$ and $y = g(t)$ , the derivative $\frac{dy}{dx}$ is computed as $\frac{dy/dt}{dx/dt}$ provided $dx/dt \neq 0$ .

Second Derivative of Parametric Equations

The second derivative,

\frac{d^2y}{dx^2}

, for parametric equations can be found using the formula

\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}

, as long as

\frac{dx}{dt}\neq 0

Arc Length of a Parametric Curve

The arc length $L$ from $t=a$ to $t=b$ for the curve defined by $x=f(t)$ and $y=g(t)$ is given by the integral

L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt

Area Under Parametric Curve

For a parametric curve defined by $x=f(t)$ and $y=g(t)$ , the area under the curve from $t=a$ to $t=b$ is given by

A = \int_{a}^{b} y(t) \frac{dx}{dt} dt

Tangent Line to a Parametric Curve

The equation of the tangent line to a parametric curve at a point $t=t_0$ is given by $y-y_0 = \frac{dy}{dx}(t_0)\cdot(x-x_0)$ , where $x_0=f(t_0)$ and $y_0=g(t_0)$ .

Parametric Equation and Horizontal Tangents

A parametric curve defined by $x=f(t)$ and $y=g(t)$ has horizontal tangents where $\frac{dy}{dt} = 0$ as long as $\frac{dx}{dt} \neq 0$ at these points.

Parametric Equation and Vertical Tangents

A parametric curve has vertical tangents where $\frac{dx}{dt} = 0$ provided that $\frac{dy}{dt} \neq 0$ at these points.

Integration by Parts with Parametric Equations

To integrate products of derivatives of parametric functions (e.g., $u(t)\frac{dv}{dt}$ ), use integration by parts:

\int u \frac{dv}{dt} dt = uv - \int v \frac{du}{dt} dt

Parametric Equations and Area Between Curves

The area between two parametric curves $y_1=g_1(t)$ , $x_1=f_1(t)$ and $y_2=g_2(t)$ , $x_2=f_2(t)$ from $t=a$ to $t=b$ is

A = \int_{a}^{b} (y_2(t) - y_1(t)) \left(\frac{dx_2}{dt} - \frac{dx_1}{dt}\right)dt

Curve Tracing for Parametric Equations

Curve tracing involves plotting a parametric curve $x=f(t)$ and $y=g(t)$ by varying the parameter $t$ . Critical points, inflection points, and behavior at endpoints provide insights into the shape of the curve.

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