Parametric equations

Algebra and Trigonometry 1st

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

00:47

Algebra and Trigonometry 1st

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

00:49

Algebra and Trigonometry 1st

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

Parametric Curves

199 Practice Problems

00:55

Physics: A Conceptual World View 7th

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

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

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

1131 Practice Problems

03:23

Intermediate Algebra for College Students 6e

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

0:00

Intermediate Algebra for College Students 6e

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

Intermediate Algebra for College Students 6e

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

Orientation of Curves

19 Practice Problems

01:31

Calculus Early Transcendentals 3rd Edition

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

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

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

Derivative for Parametric Curve

100 Practice Problems

00:13

Calculus Early Transcendentals 3rd Edition

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

01:43

Calculus Early Transcendentals 3rd Edition

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

01:37

Calculus Early Transcendentals 3rd Edition

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