## Parametric Equation Questions and Answers

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Let x(t)=t+1 and y(t)=t^2 a. Graph (x(t),y(t) for -1 \leq 1 \leq 4 b. Find \frac {dx}{dt}, \frac {dy}{dt} the tangent slope \frac {dy}{dx} and speed when t=1 and t=4.

At time t minutes, robot A is at (t,2t+1) and robot B is at t^2,2t^2+1) a. Where is each robot when t=0 and t=1? b. Sketch the path each robot follows during the first minute. c. Find the slope of the tangent line \frac {dy}{dx}, to the path of each robot

Consider the parametric curve: x(t) = 2 cos(\sqrt 2 t) y(t) = 3 sin(\sqrt 3 t) 0 \leq t < n\pi As the value for n \to \infty, what set of points in the xy-plane will be it by the curve? Now change the coefficient of an angle: x(t) = 2 cos(\sqrt 2 t) y(t)

Hyperbolic coordinates (R, \theta) of a given point \mathcal{P} in the xy plane by requiring that the Cartesian description (x,y) of \mathcal {P} be given by the relationship: \begin {cases} x = R \cosh \theta \\ y = R \sinh \th

Let r(t) = x(t), y(t), z(t) , t \in [a, b], be a vector-valued function, where a < b are real numbers and the functions x(t), y(t), and z(t) are continuous. Explain why the graph of r is contained in some sphere centered at the origin.

The curve that is parametrized by x(t) = \cos^3 t and y(t) = \sin^3 t, with 0 \leq t \leq 2\pi, is called a hypocycloid. a. Show that the curvature is k(t) = \frac 1{|3\sin t \cos t|} . b. What is the minimum curvature and where on the curve does it occur

Draw the circle of radius 2 centered at (1,1,1) and lying on the plane x + y + z = 3. Parameterize the circle. (Hint: Find two orthogonal unit vectors which are parallel to the plane.)

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. (Enter your answers as a comma-separated list of ordered pairs.) x = t^3 - 3t, y = t^2 - 5 horizontal (x,y) =

Determine the initial point and terminal point, then determine the rectangular equation of the curve whose parametric equation are given, x = 2 sin (t), y = 4 cos (t), 0 less than or equal to t less than or equal to pi.

Parameterize the line through P = (8,-3) and Q = (10,0) so that the points P and correspond to the parameter values t = 5 and 8.

Find a set of parametric equations for the graphs of rectangular equation y = 2x^2 that satisfies the condition t = 3 at the point (3,18).

A laser is attached to the unit circle and located at position P(t)=(cos(t),sin(t)) at time t; assume 0 t \frac {\pi}{2}. The laser points in the direction indicated and hits the x-axis at the posit

Suppose a reflector is attached to a 26 inch diameter bike wheel, 8 inches from the center. If the bike wheel rolls without slipping, find parametric equations for the position of the reflector.

The equation for the location x (in meters) of a moving automobile at time t (in seconds) is x = 3t + 10. What is the slope of the equation (include units) and what is its physical meaning? What is

The path traveled by a car on the amusement park ride, the Wildcat, over the time interval 0 \leq t \leq 20.5 is modeled by the position vector h(t) = \langle 8t + 2\cos t - 2, (16-t)(1-\sin t)

Use the following parametric equations x = 1 + 2cost 0 \leq t \leq 2\pi x = - 2 + 2sint (a) Graph the parametric equations. Clearly label four points on the graph with the coordinates, (x, y), and the

The position vector of a particle moving in the xy-plane is given by vector r of t equals sin of 2t, 2 times cos of 2t, 0 less than or equal to t less than or equal to pi. i) Sketch this plane curve.

Find all points (if any) of horizontal and vertical tangency to the portion of the curve shown. Involute of a circle: x is equal to cos theta plus theta sin theta, y is equal to sin theta minus theta

At time t minutes, robot A is at (t, 2t + 1) and robot B is at (t^2, 2t^2 + 1). (a) Where is each robot when t = 0 and t = 1? (b) Find the slope of the tangent line, fraction {dy}{dx} , to the path of

Let lambda (t) = (t - 1, t^2 - 4 t + 1) with 0 less than or equal to t less than or equal to 2 (a) Find the rectangular equation for the curve C = Trace(lambda). (b) Sketch Trace (lambda).

Suppose you start at the (10, 2, 10) point and move 4 units along the curve: x (t) = 2 t + 10, y (t) = {4 square root {t^3}} / {3} + 2, z (t) = {t^2} / {2} + 10 in the positive direction. Where are yo

Suppose a cat is chasing a ball around the floor, and its position is described by the parametric equations (x(t), y(t)) = (t^2 - 1, t - t^3). (a) The cat is following one of the parts from the prev

Set up but do not evaluate an integral in cylindrical coordinates for the mass of the solid lying between the paraboloids z=2-x^2-y^2 and z=x^2+y^2 if the density function is \delta(x,y,z)=z

Suppose that r(s) is an arc length parametrized curve such that \kappa (s) = 1 + s^2 and \tau (s) = \sin s . Find the length of the vector \frac{d^2 N}{ds^2} at s=1

Find parametric equations for the line of slope 5 that passes through the point (7, 4). (Use t as your parametrized variable).

Find parametric equations for the line through the points P=(-4,5,6) and Q=(3,-2,4)

Find parametric equations for the line through the points P=(-1,-1,3) and Q=(4, -3,6)

Find parametric equations that describe the line that passes through the points P= (3,-1,3) and Q=(4,4-1)

Write the following parametric equation in the form y=f(x) . \begin{alignat}{3} x = t^2 +2, \\ y= t^2 4. \end{alignat} .

For the given parametric curve, compute the following questions: x(t) = \tan t + 1 y(t) = \sec^2t - 3 a. Express the curve with an equation which relates x and y. b. Find the slope of the tangent line to the curve at the point t = \frac \pi 6 . c. State

Find a Cartesian equation for the parametric equation x=t, \quad y = 9t + 3

Find a parametric representation for the surface. The part of the ellipsoid x^{2} + 2y^{2} + 3z^{2} = 1 that lies to the left of the xz-plane

Determine A so that the curve y = 7x + 33 can be written in parametric form as x(t) = t - 5, y(t) = At - 2. 1. A = -5 2. A = 6 3. A = 7 4. A = 5 5. A = -7 6. A = -6

Find parametric equations for the line passing through the point (4,-7,-6) and orthogonal to the plane 2x-3y+z=5

Which is the polar form of the parametric equations x = 2t and y = t^2? a) r = 4 \tan^2 \theta \\ b) r = 4 \sec^2 \theta \\ c) r = 4 \sec \theta \\ d) r = 4 \tan \theta \sec \theta

Convert the parametric equations x= 3t, y= t + 7 into a Cartesian equation.

Find a parametrization, using \cos(t) and \sin(t), of the following curve: the intersection of the plane y=3 with the sphere x^2+y^2+z^2=73.

Use \cos(t) and \sin(t), with positive coefficients, to parametrize the intersection of the surfaces x^2+y^2=4 and z=6x^2.

The function \vec{r}(t) traces a circle. Determine the radius, center, and plane containing the circle \vec{r}(t)=1\vec{i}+(8\cos(t))\vec{j}+(8\sin(t))\vec{k}.

Find a vector parametrization equation \vec{r}(t) for the line through the points P=(0, 0, 5) and Q=(-2, -1, 2) for each of the given conditions on the parameter t. A) If \vec{r}(0)=(0, 0, 5) and \vec{r}(7)=(-2, -1, 2). B) If \vec{r}(5)=P and \vec{r}(8)=Q

Find a vector parametrization of the curve x=-4z^2 in the xz-plane.

Find an equation for the line tangent to the curve at the point defined by the given value of t x = csc t , y = 6 cot t , t = ? 3 A) y = ? 12 x + 6 ? 3 B) y = 2 ? 3 x ? 12 C) y = 12 x ? 6 ? 3 D) y = 12 x + 2 ? 3

Find an equation for the line tangent to the curve at the point defined by the given value of t x = 7 sin t , y = 7 cos t , t = 3 ? 4 A) y = 7 x + 7 ? 2 B) y = ? x + 7 ? 2 C) y = x ? 7 ? 2 D) y = 7 ? 2 x + 1 (Choose the correct one

Obtain the Cartesian equation of the curve by eliminating the parameter. x = t 2 , y = ? 7 + t 4 A) y = ? 7 + x B) y = ? 7 + x 4 C) y = ? 7 + x 2 D) y = ? 7 + 2 x (Choose the correct one)

Find parametric representation of the curve given by the intersection of the paraboloid z = x^2 + y^2 and the plane 6x + 2y + z = 15 .

For the parametric equation: \begin{Bmatrix} x=\tan t\\ y= 4\sin^2 t 2 \end{Bmatrix} with \frac{ \pi}{4} \leq t \leq \frac{\pi}{4} a. Sketch a graph, b. Indicate start and end points, c. Place a dot at t= the leas value and write start nea

For the parametric equation: \begin{Bmatrix} x=\cos t\\ y= 2\sin t\end{Bmatrix} with \frac{ \pi}{4} \leq t \leq \frac{3\pi}{4} a. Sketch a graph, b. Indicate start and end points, c. Place a dot at t= the leas value and write start nearby,

For the parametric equation: \begin{Bmatrix} x=\cos t\\ y= \sin t \end{Bmatrix} with \pi \leq t \leq \frac{3\pi}{4} a. Sketch a graph, b. Indicate start and end points, c. Place a dot at t= the leas value and write start nearby, d. Place

For the parametric equation: \begin{Bmatrix} x=\cos t\\ y= \sin t\end{Bmatrix} with \frac{ \pi}{2} \leq t \leq \frac{\pi}{2} a. Sketch a graph, b. Indicate start and end points, c. Place a dot at t= the leas value and write start nearby, d

For the parametric equation: \begin{Bmatrix} x=t+1\\ y=\frac{1}{t} 2 \end{Bmatrix} with 3 \leq t \leq 3 a. Sketch a graph, b. Indicate start and end points, c. Place a dot at t= the leas value and write start nearby, d. Place an arrow on

For the parametric equation: \left\{ x='sqrt t \\ y=4-1 \right\} with 0 \leq t \leq 9 a. Sketch a graph, b. Indicate start and end points, c. Place a dot at t=the leas value and write start nearby, d. Place an arrow on the curve to show the positive dire

A curtate cycloid is given by the parametric equations: x=t-\sin t \quad y=1-\cos t A) Find the value of t, 0 less than or equal to t less than or equal to 2\pi, at which the graph has a maximum. B) Find \frac{d^2y}{dx^2}.

Let x(t) = 3sin(2t) + 4cos(2t) and let y(t) = 4sin(2t) + 3\cos(2t) (a) What is the period of and y? (Give an exact answer based on your knowledge of the sine and cosine functions) (b) Plot x and y on the same axes. Estimate the amplitude from the

Determine the parametric equations of the path that travels the circle: \\ (x-2)^2 + (y-1)^2 = 36 on a time interval from 0 \leq t \leq 2\pi \\ if the particle makes one full circle starting at the point (2, 7) traveling clockwise.

Consider the graph of the equation y = x^2 - 16. The x-intercept(s) is(are): The y-intercept(s) is (are): Is the graph symmetric with respect to the x-axis? Is the graph symmetric with respect to the y-axis? Is the graph symmetric with respect to the

Parametric equations, but I don't understand why \frac {d^2y}{dx^2} is only taking the first derivative of x twice.

(a) Sketch the curve with vector function: r (t) = t i + cos pi t J + sin pi t k t greater than or equal to 0. (b) Find r' (t) and r'' (t).

Find the equation in x and y for the line tangent to the curve given parametrically by x = 8 \tan (\frac{t}{24}) , y = 6 \sec (\frac{t}{24}) at the point on the curve associated with t = 4 \pi. Write the equation of the tangent line in the form y = ___.

The curve given by x(t) = 9 \sin (t) , y(t) = 2 \sin (t + 8 \sin (t)) has two tangent lines at the point (x, y) = (0, 0). List both of them in order of increasing slope. Your answers should be in the form of y = f(x). a) Line with smaller slope: y = ___

Determine whether the points P and Q lie on the given surface r(u,v) = 6u + v, 1 + u-3v, sqrt{uv} P(33, -22, 6) Q(15, -3, 2)