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Q: a) Demonstrate Gra phically the contimuity /Discontinuity of the tunction fx)- * ,0くxく」 1. 1+x)…

A: Authoring guidelines:…

Q: lim sin (2 – t) cot² (t² – t – 2) t+2

A: We are given: limt→2sin2(2-t) cos2(t2-t-2) We need to evaluate the given limit This can be evaluated…

Q: Integrate H(x, y, z) = x2sqrt(5 - 4z), over the parabolic dome z = 1 - x2 - y2, z>=0.

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Q: If f(r, y, z) does lim f(r, y, z) exist? %3D 1² + 2y? + z² ' (z,y,z)¬(0,0,0)

A: To find its limit we have to first convert the function into polar form.

Q: (a) Find the solution of the limit using polar coordinate system. x² + y² x² + y2 lim(x.y)→(0,0)

A: Solution:Given to the find the limitlim(x,y)→(0,0)x2+y2x2+y2In polar co ordinates x=r cos θ   and y=…

Q: Jerify Divergence Theorem for F = = (x + y') i – 2 xỷ + 2 yzk and the volu

A: Given: F⇀=(x+y2)i^-2xj^+2yzk^The plane, 2x+y+2z=6

Q: solid Q bounded by the surfaces S1:x² + he first octant

A: We have to choose the correct answer from the given option according to the given condition

Q: rea under the curve y = 2x – x² from x = 1 to x = 2 with n = 4 %3D tonglon

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Q: Determine the surface of revolution of arc y³+y=x²+6x on the interval [-5,-1] if it is revolved…

A: Use the formula of arc length to determine its length. Now use to it determine the volume generated…

Q: Exercise 11. iz Let f(z) and CR be the upper half circle with center (0,0) and radius R z5+1 taken…

A: Given problem is :

Q: (a) Find the solution of the limit using polar coordinate system. x² + y2 lim (x,y)-(0,0) x2 + y?…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 55. A point moves along the intersection of the elliptic parab- oloid z = x+ 3y and the plane y = 1.…

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Q: F - dr where F(x, y, z) = (4.25yz cos(xyz), 4.25xz cos(xyz), 4.25zy cos(ryz)) Evaluate п 1 and Cis…

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Q: - The region bounded by y = 2 -x, y = 0, x = 1, x = 2 is rotated about the x axis. 197/12

A: volume generated by revolution of the solid is 19π12 detailed solution with explanation is given…

Q: Tangent to the parabola z = x² + 3y² , x = 1 , at point ( 1, 2, 13

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: bounded by Y=x3, V=1x i rotare about X=1

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Q: 2. Let F = 3.xyz i – 4xyz j + 2xyz k . Find div F and curl F. -

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Q: ди ди + x - Əx %| ду 0 =

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Q: 6. Let f(x,y) = sin(x + y) + arcsin (xy) a. Find the linearization L(x,y) at the point (0, ")

A: We have to find linearization of given function f(x,y): Concept: The linearization of a function…

Q: Utt Urr 100, x > 0, t > 0. u (x, 0) = u (x, 0) = u (0, t) = 0 lim u (x, t) = 0, x → 00

A: 1. Laplace transform is an integral transform that converts a function of real variable t to a…

Q: Find a formula for a functionf(x, y, z) whose level surface f= 3 is a cylinder of radius 5 centered…

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Q: Find all point on the plane x+ yt z= 5 in the first octant at which f(x, y,z) xy'z' has a maximum…

A: We have to find all points on the plane x+y+z=5 in the first octant at which fx,y,z=xy2z2 has a…

Q: tion z2 – (x2 + y2)2 and let p = (0, 0, 0). ermine the set Tp(Zf ) of all tangent prs to Zf at p. Is…

A: Sol

Q: Sketch the region enclosed by the given curves. y = tan(7x), y = 2 sin(7x), -77/21 sxs 7/21 y y 3 3…

A: For sketching the region y=tan7x, y=2sin7x, -π21≤x≤π21. Let y1(x)=tan7x, y2(x)=2sin7x, -π21≤x≤π21 We…

Q: au %3D ax at u(x, 0) = 0 ди (x, 0) = 0 at u(0, t) = sin? t u(1, г) — 0,

A: Given,           ∂2u∂t2-∂2u∂x2=0                       for 0<x<1, t>0      ...(1)u(x,…

Q: tor field F = 21 + 4j + 3k across the surface S.

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Q: 1.13 Evaluate the line integral $ x*dx + xydx where C is the triangle with vertices at (0, 0), (0,…

A: The given function is x4dx+xydy By green's theorem, the integral becomes, ∫06∫06∂M∂y-∂N∂xdxdy where…

Q: 9. Evaluate the line integral / (2x – 3y) ds; where C is the line segment from (-1,0) to (0, 1)…

A: 9.

Q: Show Z=arctan(y/x) satisfies Zxx+Zyy=0.

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Q: Use the Chain Rule to find az/ðs and ôz/dt. z = arcsin(x – y), x = s² + t?, y = 2 - 2st as az at II

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Q: 1 4. Let f(=) =- s · Show that J (z)dz = 0 , where C is a simple closed contour not passing…

A: Moreras theorem

Q: The tangent plane to the graph of the function f(x,y)=x+xy+y2, which is parallel to the z=1+2x+3y…

A: Here given surface is fx,y=x+xy+y2 or x+xy+y2-z=0......1 Then, consider Fx,y,z=x+xy+y2-z

Q: (a) Find the solution of the limit using polar coordinate system. x² + y² x2 + y2 lim (x,y)→(0,0)

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 1_2 Can you please help me with this example in a step by step guide. Thank you very much!!

A: To calculate the coordinates (x,y,z) and also define the absolute extremum for the provided surface…

Q: 9-x²-y² is the upper √y half of the disc centered at origin with radius r = 3 The domain of f(x, y):…

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Q: rsin z cos t dt lim 2.r

A: Use L'Hospital rule and fundamental theorem of derivatives to solve it.

Q: 8. Evaluate the line integral x dx + xedy where C is the closed path along the triangle in the…

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Q: At what points (x, y, z) in space are the functions in continuous? ƒ(x, y, z) = x2 + y2 - 2z2

A: Solution:-  At what points (x, y, z) in space are the functions in continuous? Let (m,n,p) is the…

Q: ry lim (z.v)¬(0,0) + y² exists.

A: Please refer the attached image for complete solution.

Q: about the line x = -2 ソ=2× ソ=2° 1 2

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Q: ây _ở: x>0, t> 0 where y (0, t) = 1, y (x, 0) = 0. Find the bounded solution of

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Q: Show that 1 y - X 1. Z-y X-Z = 1. %3D alx-y) alz - x)

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Q: Find the divergence of F(x, y, z) =(a -1 Y, y? + 1' 3xz2 + 2y ) at (2,0, – x COS

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Q: 4| ind Volumes of のnen reo リュー! )ス"0) reten iegTon 1- dv - dzdy chy 1 Limik of 2:0> y2 Limik of y:…

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Q: 5) Find the region in the xy-plane for which (sin x – y)y' = vx- y satisfies the nonlinear existence…

A: A differential equation  y' =f(x,y) will have at least one solution in the region where f(x,y) is…

Q: find div and curl F F(x,y,z)= 7y3z2i- 8x2z5j-3xy4k F(x,y,z) = exyi-cosy j +sin2 z k

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Q: 5xy sin ³²x + 3y² 0, f(x, y) (a) Prove that lim T if (x,y) + (0,0) if (x,y) = (0,0) f(x,y) does not…

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Q: 8. Evaluate the line integral fx'dx + xe 'dy where C is the closed path along the triangle in the…

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Q: 3.4.6 Evaluate the line integral [sin y dx + (x- cos y) dy] taken in the anticlockwise sense, where…

A: Green's Theorem States that  Let C be the positively oriented, smooth, and simple closed curve in a…

Q: Let f(x, y, z) = x arctan z. Then solution.] af Əzəy (2,3,1)= a In 2 where a = [Show

A: Find the value of a

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as: