## Three Dimensional Graph Questions and Answers

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What does the pair of equations y = 3, z = 7 represent? in other words, describe the set of points (x, y, z) such that y = 3 and z = 7.

What does the equation y = 3 represent in r^3? What does z = 5 represent? What does the pair of equation y = 3, z = 5 represent?

What type of surface is x^2 + y^2 = (e^{-z})^2? What is the generating curve?

1. By setting one variable equal to a constant (either x=c, y=c, z=c for some constant ), find a plane that intersects the graph of z=3y^2 - 2x^2 +5 in a: a. Parabola opening upward. b. Pair of i...

Determine the y-intercept(s) of the level curve, where f(x, y) = 2e^(10*sqrt(x^2 + y^2)) and z = 4. (Answer starting with the smallest y and round your answer to 3 decimal places.)

For the surfaces z=x^{2}+y^{2};x^{2}+y^{2}=2y; (a) Describe the surfaces (b) Find a vector form for the curve of intersection. The answer is below I just dont know how they got part A: (a) Paraboloid

Sketch the following surfaces whose equations. a. r = \theta b. z = \theta c. \rho = \phi

Identify the surface of the given vector equations. a) \vec{r}(u,v) = 2 sin(u) \hat{i} + 3 cos(u) \hat{j} + v \hat{k}, 0 \leq v \leq 2. b) \vec{r}(s,t) = \langle s, t, t^{2} - s^{2} \rangle.

(1) Neatly sketch the curve with the vector equation \vec{r}(t)= \langle -t^{2}, 4, t \rangle. Identify any special points on its graph and indicate with an arrow the orientation of this curve. (2)...

Graph the curve. r(t) = cos(pi*t) i + sin(pi*t) j + e^(-t) k, t greater than or equal to 0.

Sketch this spherical/cylindrical coordinate \{(x,y,z) | \rho,r \geq 0 & 0 \leq \Theta \leq 2 \pi & z= \Theta \}

A) Let f(x, y) = y^2 - x^2 and let P be the point P = (3, 5). In the xy-plane sketch the level curve that contains the point P. B) Find the equation of the intersection of z = y^2 - x^2 with the y...

How to find the intersection of a cylinder and a plane?

Does it matter which variable I set equal to 0 when finding the intersection of planes?

Sketch traces of the surface given by equation below.Then describe and sketch the surface x^{2}+y^{3}+z^{2}=0.

Consider the function f(x, y) = 7-3xy^2/4 . Which graph below corresponds to the following traces: ......1. The trace for x = 1.4 ......2. The trace for x = -1.4 ......3. The trace for x = -0.3 ....

1. The temperature on a triangular plate (1^st quadrant, bounded by equations y = x , y = 3 and x = 0 ) is T ( x , y ) = x 2 + y 2 ? 2 x ? 4 y + 20 . Find all hot and cold spots (showing the te...

How can three unit vectors be orthogonal?

1. An equation involving two variables, such as 2x^2-y =3, generally defines a curve in R^2 (obtained by plotting all of the solutions (x,y) of the equation). However, it also naturally defi...

Consider the function f(x,y)=2x^{2}+y+1.Sketch the surface in \mathbb{R}^{3} from the perspective of the first octant.

The equations: 2x + y = 7; 3x-z = 11; 3y + 2z=-1 a) Represent a certain line as the intersection of 3 planes each of which is parallel to one of the coordinate axes in space. True or False ? b) Det...

The integral of a region in the order dx dz dy is \int_0^2 \int_0^{2-y} \int_0^{4 - y^2} dx dz dy . For the volume given by the integral, sketch the solid that is formed from the integral with labe...

Sketch the region \{ (x,y,z) | 0 \leq \rho \leq 2, 0 \leq \theta \leq \frac{\pi}{2} \}

Sketch the two surfaces given in cylindrical coordinates as S_1 : z = sin theta and S_2 : r = 1 Then identify the curve of intersection..

First is giving me the space curve of r(t)= and t ranges over the close interval [0,2 \pi] Find an equation of the form y=f(x) for the projection of the curve into...

Consider the surface S defined by the equation xy = z. a)True or False: the traces of S parallel to the xy-plane are lines. b)True or False: the traces of S parallel to the yz-plane are lines. c...

Use Traces to sketch and Identify the surface: 4x^2 + 9y^2 + z = 0.

Given the triple integral \int_0^1 \int_0^{1 - x^2} \int_0^{1 - x} f(x, y, z) dy dz dx Sketch the solid (Label the axis, functions, and Intersection:)

Consider the vector-valued function defined by Vector r (t) = (4 cos (2t), 3 sin (2t), g(t)), for t greater than or equal to 0 i) On what cylindrical surface must this space curve lie? (Appropriate...

Sketch the surface with equation y^2 + 4z^2 = 9 . Identify the coordinates of all intercepts. Give a detailed description of all the horizontal and vertical traces of this surface. The description...

Suppose that x(t)= t^{2} andy (t)=\sin (\frac{\pi t}{2});then the vector-valued function \vec{r}(t)=\left \langle x(t),y(t) \right \rangle determines the trajectory of an Asian tiger mosquito,for 0...

Consider the curve: \boldsymbol{r}(t) = \langle 3 \cos(t/5), 3 \sin(t/5), ( 4t)/( 5) \rangle. a) Draw the curve. Plot at least three specific points, but also make the overall shape of the curve...

Consider the following surface given in cylindrical coordinates: z = r. Sketch its graph and show the xz-trace, the yz-trace and the level curve z = 3.

(2) When two surfaces intersect, they generally intersect in a curve. Find a parametrization of the intersection curve for the following pairs of surfaces by eliminating one of the variables and th...

Find the intersection of the surface z=x^2+y^2+2 and the plane z=2y+10 Draw a rough 2D sketch of the surface with the intersection embedded within it. Write a parameterization for intersection...

Sketch the following surfaces in \mathbb{R}^3 .(i) 3x + 5y + 15z = 15 (ii) z = 25 - x^2 - y^2 (iii) z=\sqrt(9-x^2-y^2)(Hint : this surface is part of sphere .what sphere ?what part?) (iv) z=\sqrt(...

Try to sketch by hand the curve of intersection of the parabolic cylinder y = x^2 and the top half of the ellipsoid x^2 + 3y^2 + 3z^2 = 9. Then find parametric equations for this curve and us...

Find the intersection of the surface z = x^2 + y^2 + 2 and the plane z + 2y = 10. Draw a rough 3D sketch of the surface with the intersection embedded within it. Write a parameterization for inte...

Give a geometric description of the set of points (x,y,z) satisfying the pair of equation z = {x}^2 and y = 0. Sketch a figure of this set of points.

Find a vector valued function whose graph is the surface given by x^{2} + y^{2} = a^{2} z^{2}. Also identify the type of surface.

(1) Describe the level curves of the function. Sketch the level curves for the given c values. z = xy, c = 1 (2) Use two paths to show that following limit does not exist. \lim_{(x, y) \to (0, 0)}...

Multivariate calculus Write the definition of a cone in \mathbb{R}^{n} and give an example of a cone in \mathbb{R}^{3}.

Part 1: Find the plane containing the vectors (4,1,-2) and (-1, 3, 0). Find a parallel plane passing through the point (2, 1, -1). Part 2: Consider the plane with normal vector (2, 1, -2) and pas...

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t)= t ^{2}i + tj + 2k

A curve C has the parametrization x = a sint cos alpha , y = b sint sin alpha , z = c cost, t greater than or equal to 0 , where a, b , c, alpha are all positive constants. Show that C lies on th...

Let f(x,y) = y^2 \sqrt x + x^2 - 10x - y^2. Find four points in S = \{(x,y) |0 \leq x \leq 9, -1 \leq y \leq 6\}

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function. Identify the surface and sketch its graph. r(u, v) = 6 cos(v) cos(u)i + 6 cos(v) sin(u)...

Determine a scalar equation for the plane that passes through the point (2, 0, -1) and is perpendicular to the line of intersection of the planes 2x + y - z + 5 = 0 and x + y + 2z + 7 = 0.

Parametrize the portion of the hyperboloid of two sheets z^2 = x^2 = y^ = 1 that lies below and on the plane z = 6 and above the xy-plane.

Find: Below is a topographical map of a hill mapped as the contour map of a continuous function, f(x,y) .Use this to answer the following questions. Image src='annotation_2019-07-29_162130-44969...

Suppose we have the planes z=y and x-y+z= 2. Explain whether these planes are parallel or if they intersect. If they intersect, find and describe the intersection of these two planes as well as the...

Use a CAS to plot the surface identify the type of quadric surface from your graph. 5x^2 = z^2 - 2y^2 . Choose the best graph of the equation below. What type of quadric surface is this?

Find: Traditionally, the earth's surface has been modeled as a sphere, but the World Geodetic System of 1984 (WGS-84) uses an ellipsoid as a more accurate model. It places the center of the earth a...

Find the rectangular equation for the surface by eliminating the parameters from the vector valued function and sketch its graph. r(u,v) = [2sin(u)] i + [2cos(u)] j + 3v k , 0 < u < 2 pi , 0 < v < 5

The vector function r(t) = (3 \cos(t), 2 \sin (t)), 0 \leq t \leq 2 \pi , gives the x and y coordinates of my path on the tilted plane z = x - 2y + 1. Draw the trace of my path(the curve given by...

A sphere of radius 3 is centered at the origin. It may be viewed as a parametrzed surface r(\theta, \phi) = (3\cos \theta \sin \phi, 3\sin \theta \sin \phi, 3\cos \phi) , a level surface of the...

Let r(t) = r(t) = (t^2, t+1, 2t^2 - 3t) describe the trajectory of the particle. (a) The trajectory of a particle. The trajectory's underlying curve can be seen as the intersection of a cylinder...

Sketch the surface given by the equation z = 16 - 4 x^4 - y^4 in the first octant only and label the intercepts and traces.

Let r(t) = (2-t^3 , 2t-1 , ln t) be a position vector for a curve C , t > 0 . (a) Find C's point of intersection with the xy-plane . (b) Find the parametric equation of the tangent line to C at (1,...

Suppose the space curve r(t) = (3t,t ln(t), \sqrt t) gives the position of a particle as a function of time t. a) Find parametric equations for the line which is tangent to this curve at the point...

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