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Parametric Equations

Parametric equations are equations in which parameters are used to specify the values of quantities at points in space. In contrast, ordinary differential equations are equations in which the independent variable(s) and dependent variable(s) are both functions of time. The idea of a parametric equation in mathematics, physics, and engineering can be traced to the application of the concept of parameters to a function of a single variable in the 1630s by René Descartes. In the 1830s, the application of the parametric equation to the study of mechanical motion was pioneered by William Rowan Hamilton (1805–1865) in his "Dissertation on a Dynamical Theory of the Earth", which established the basis for modern kinematics. A parametric equation is a special kind of differential equation, and can be written as a differential equation in terms of a single variable and the parameters. More generally, a function of a single variable is a parametric function of a single variable. A common example of a parametric equation is the equation of a line, which is of the form: where "x" is the independent variable and "y" is the dependent variable. Another example is the equation of a circle: which can also be written as: Parametric equations are used in geometry to describe the behavior of a curve, and in calculus to describe the behavior of a function in terms of two variables. In physics, they are used to describe the behavior of a mechanical system when acted on by a force. In engineering, they are used to describe the behavior of mechanical systems, and to design mechanical systems. They are also used in computer graphics and computer aided design to specify a parametric curve or surface. The following example illustrates the definition of a parametric equation in two dimensions. Suppose a plane moves along a path described by the parametric equations: where "t" is time and "x" and "y" are the two independent variables. Then, the position of the plane at any time "t" = 0 is given by and the position of the plane is independent of the path taken to get there. This is because the path described by the equations is just a particular coordinate system (the x–y system) for measuring the plane's position. Consider the line defined by the parametric equations: where "x" and "x" are the two independent variables. The line is a tangent line to the circle of radius "a" at the point: and the line is a secant line to the circle of radius "a" at the point: The point is the "center" of the circle. Given the position of the plane at time "t", the parametric equations can be used to find the position of the plane at any other time "t" ? 0. Consider the parametric equations: where "x" and "y" are two independent variables. The line "y" = "x" is a tangent line to the circle of radius "a" at the point: The line "y" = "x" is a secant line to the circle of radius "a" at the point: The point is the "center" of the circle. The parametric equations can also be used to find the position of the plane at any other time "t" ? 0. Consider the parametric equations: Recall that the line "y" = "x" is a tangent line to the circle of radius "a" at the point and the line "y" = "x" is a secant line to the circle of radius "a" at the point The point is the "center" of the circle. The parametric equations can also be used to find the position of the plane at any other time "t" ? 0. Consider the function: where "a" and "b" are two independent variables. The line "y" = "a" is a tangent line to the circle of radius "a" at the point: The line "y" = "a" is a secant line to the circle of radius "a" at the point: The point is the "center" of the circle. The parametric equations can also be used to find the position of a line at any other time "t" ? 0. Consider the function: where "a" and "b" are two independent variables. The line "y" = "a" is a tangent line to the circle of radius "a" at the point:

Parametric Equations

548 Practice Problems
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00:50
Algebra and Trigonometry

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using $x(t)=a \cos t$ and $y(t)=b \sin t .$ Identify the curve.
Parameterize the line from (4,1) to (6,-2) so that the line is at (4,1) at $t=0,$ and at (6,-2) at $t=1$

Further Applications of Trigonometry
Parametric Equations
Christopher Stanley
00:47
Algebra and Trigonometry

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using $x(t)=a \cos t$ and $y(t)=b \sin t .$ Identify the curve.
$$\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$

Further Applications of Trigonometry
Parametric Equations
Christopher Stanley
00:49
Algebra and Trigonometry

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an $x-y$ table.
$$\left\{\begin{array}{l}
x(t)=4-t \\
y(t)=3 t+2
\end{array}\right.$$

Further Applications of Trigonometry
Parametric Equations
Christopher Stanley

Parametric Curves

199 Practice Problems
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00:55
Physics: A Conceptual World View

Top professional pitchers can throw fastballs at speeds of $100 \mathrm{mph}$. Given that $1 \mathrm{mph}=0.447 \mathrm{m} / \mathrm{s}$, what is this speed in meters per second?

Describing Motion
Parametric Equations: Graphs
Prashant Bana
02:03
Calculus: Early Transcendental Functions

Replace $\theta$ with $\theta=\frac{\pi}{4}$ and determine the surface with parametric equations $x=\rho \cos \frac{\pi}{4} \sin \phi, y=\rho \sin \frac{\pi}{4} \sin \phi$ and $z=\rho \cos \phi.$

Vector-Valued Functions
Parametric Surfaces
Linh Vu
01:09
Calculus: Early Transcendental Functions

Sketch the curve and find any points of maximum or minimum curvature.
$y=4 x^{2}-3$

Vector-Valued Functions
Curvature

Conic Sections

1132 Practice Problems
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03:23
Introductory and Intermediate Algebra for College Students

The parabolic arch shown in the figure is 50 feet above the water at the center and 200 feet wide at the base. Will a boat that is 30 feet tall clear the arch 30 feet from the center?

Conic Sections and Systems of Nonlinear Equations
The Parabola; Identifying Conic Sections
Suman Saurav Thakur
0:00
Introductory and Intermediate Algebra for College Students

Determine whether each statement is true
or false. If the statement is false, make the necessary change(s) to produce a true statement.
The parabola whose equation is $x=2 y-y^{2}+5$ opens to the right.

Conic Sections and Systems of Nonlinear Equations
The Parabola; Identifying Conic Sections
00:59
Introductory and Intermediate Algebra for College Students

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
I graphed a parabola that opened to the right and contained a maximum point.

Conic Sections and Systems of Nonlinear Equations
The Parabola; Identifying Conic Sections
Suman Saurav Thakur

Orientation of Curves

19 Practice Problems
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01:31
Calculus: Early Transcendentals

Given the following vector fields and oriented curves $C,$evaluate $\int_{C} \mathbf{F} \cdot \mathbf{T} d s$.
$\mathbf{F}=\frac{\langle x, y\rangle}{x^{2}+y^{2}}$ on the line segment $\mathbf{r}(t)=\langle t, 4 t\rangle,$ for $1 \leq t \leq 10$

Vector Calculus
Line Integrals
Vikash Ranjan
02:08
Precalculus 10th

Use a graphing utility to graph the curve represented by the parametric equations.
$$\begin{aligned}&x=t\\&y=\sqrt{t}\end{aligned}$$

Topics in Analytic Geometry
Parametric Equations
Christy Galilei
02:55
Precalculus 10th

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary.
$$x=\sqrt{t+2}, \quad y=t-1$$

Topics in Analytic Geometry
Parametric Equations
Parker Gustafson

Derivative for Parametric Curve

100 Practice Problems
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00:13
Calculus: Early Transcendentals

Beautiful curves Consider the family of curves
$$\begin{array}{l}x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right) \\y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right)\end{array}$$
Plot a graph of the curve for the given values of $a, b,$ and $c$ with $0 \leq t \leq 2 \pi$.
$$a=7, b=4, c=1$$

Parametric and Polar Curves
Parametric Equations
Adnan Gill
01:43
Calculus: Early Transcendentals

Slopes of tangent lines Find all points at which the following curves have the given slope.
$$x=2+\sqrt{t}, y=2-4 t ; \text { slope }=-8$$

Parametric and Polar Curves
Parametric Equations
Adnan Gill
01:37
Calculus: Early Transcendentals

Derivatives Consider the following parametric curves.
a. Determine $dy/dx$ in terms of t and evaluate it at the given value of $t$.
b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of $t$.
$$x=2 t, y=t^{3} ; t=-1$$

Parametric and Polar Curves
Parametric Equations
Adnan Gill

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