Consider the surface S: √x + √y + √z = a with a> 0. If P (x0, y0, z0) is a point on the surface that is not on either axis, show that: a) An equation of the plane tangent to S at point P is x/√x0 + y/√y0 + z/√z0 = a  b) The sum of the lengths of the segments determined by the tangent plane in the coordinate axes is constant equal to a2.

Q: Calculate the distance from point A(1, 2, 1) to the plane of equation 3x-4y + 5z = 1.

A: To find distance of point to plane using the formulae: Given point is and plane equation is .

Q: 8.) Find the planes tangent to the following surfaces indicated points: a:) x² + 2y ²2/3 X ² = 10₁…

A:

Q: Find a point on the surface x - yz = 11 that is closest to the origin. (a, ± √b, √c), where a ,b= =…

A: We have to find the point on the surface x-yz=11 that is closest to the origin.

Q: let l={(x,y) 7x+4y=1} be a line in Cartesian model for Euclidean geometry. find a coordinate…

A: Given: I=x, y 7x+4y=1 represents a line in Cartesian model.   To find: A coordinate function using…

Q: Verify Stokes' theorem for the helicoid (1, 0) = First, compute the surface integral: = (VF)…

A: Given surface is helicoid whose vector parametric equation is \)" data-mce-style="cursor:…

Q: Sketch the surface given by the equation x2+z2/3+y=3 indicating the points where the surface…

A: Sketch the surface given by the equation x2+z2/3+y=3   indicating the points where the   surface…

Q: Suppose C is the line y = 5-2x in the xy-plane. Determine parametric and Cartesian equations of the…

A: Given the line equation C in xy-plane is y=5-2x     put x = k,  in above line . k be a parameter…

Q: e'y x+y² a) Find the equation of the tangent plane to the surface at the point P(0, -2). Present…

A:

Q: 19. Find the distance from the point (3, -2,7) to the plane 4x – 6y + z = 5. Show all necessary work…

A: A detailed solution is given below.

Q: If (x1, yı) and (x2, y2) are distinct points in the plane, show that y 1 1 =0 X1 x2 y2 is the…

A: If (x1, y1) and (x2, y2) are distinct points in the plane, to show that  xy1x1y11x2y21 =0 is the…

Q: Consider the curve C which is the intersection of the surfaces shown in the attached figure.

A: To evaluate the below integral. ∫Cx−y−2zds

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.