lim (x,y)→(3,2) x cos Ty = O 3 O 2 O 1 O 4

Q: y In x – x In y = 4, y' y In (x² + y² )= 2 arctan 2) = 2 arctan y'

A: This question can be solved using the concept of differentiation of implicit function.

Q: Let f(r, y, z) = ln(x² + z) + y²e#² – cos(2yz). Find fzyr- %3|

A: Consider,                                                                 f(x, y, z) = ln(x² + z) +…

Q: (a) z = x2 + y² + xy, x = sin t, y = e'. (b) z = x2 + y², x = cos t + sin t, y = cost – sin t. %3D

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Q: Let z(x, y) be the function implicitly defined as the solution to 2x + y + z + sin(xyz) = 4 + that…

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Q: Let v(x, y) = cos(4x)sinh(4y). Find f(z)with Im(f(z)) = v(x, y). I

A: Given function is v(x,y)

Q: f) y= xcos(xln x) g) y= 3(sin 2x – x* }** In. 3x'e** h) y= cosx

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Q: find ƒx , ƒy , and ƒz . ƒ(x, y, z) = tanh (x + 2y + 3z)

A: We have given fx,y,z=tanhx+2y+3z and we have to calculate fx,fy and fz

Q: (e) f(x) = sin(e*) + eco«(a) (f) f(x) = cos(x) In(5x) (x) S(z2) = ()" %3D - 1

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Q: 3. 다음 함수 /에 대하여 /w, /w 를 각각 구하고 비교하 여라. (1) f(x, y) = sin' xcosy (2) S(x, y) = 2x²+5xy+ y² %3D (3)…

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Q: Find y': Oy: Oy = Oy' = O y' y cos(xy)+1 x cos(xy)-1 x cos(xy+1) y cos(xy)-1 y cos(xy)+1 x cos(xy)-1…

A: To find the value of y' for the given function.

Q: 1) x dy = y(1 + In y – In x) dx 2) y' – y tan x + y? cos x = 0

A: Given differential equation isxdy=y(1+lny-lnx)dx We need to solve the equation. Note:- According to…

Q: Find a homogeneous (linear) DE with a following FSS: y1(x) := 1; y2(x) := sin(5x); y3(x) := cos(5x)

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Q: Evaluate: y +y = - cos X COSX A. y = 0, sinx + Ca cosx + x sin x 8. C, COSX + Ca casx t x cas X C. y…

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Q: (. 0) S(x, y) = In(x² + xy), at (4, 1) S(x, y) = sin (x²) + cos y, at

A: 19. f(x, y)=sin(x2)+cos y At π2, 0, fπ2, 0=sinπ22+cos 0                =sinπ4+cos 0…

Q: (x,() + x(x, t) = sin(t) u(n, 0) = cos(7), at %3D %3D nER.

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Q: dz at t = dt if z = f(x, y) = x² – 3xy+ 2y², x = x(t) = 3 sin 2t, y = y(t) = 4 cos 2t. Use chain…

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Q: find ƒx , ƒy , and ƒz . ƒ(x, y, z) = sinh (xy - z2)

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Q: -s Jatx? 3. 4-/ sinx coss dx |

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Q: (4,0,2) S sin y cos x dx + cos y sin x dy + 3 dz (1,0,0)

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Q: х. у 2) w=¨+2,x= cos“(t), y = sin°(1), z =-, at t = 3 Z Z t

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Q: uxx+uyy=0; u1(x,y)=cos x cosh y, u2(x,y)=ln(x2+y2)

A: Given:

Q: If U(x, y) = -e-Y sin x and f(x) = U(x, y) + iV(x, y) is analytic, then V(x, y) = Select one: O ey…

A: We have to find V(x, y) i.e.imaginary part of given analytic function,if we have real part is given.

Q: 22. u- Vr? + s, r- y + x cos 1, s -x+ y sin t; ди ди ди when x= 1, y = 2, 1 = 0 ar ay at

A: Given: u=r2+s2  r=y+x cos t and  s=x+y sin t By using Chain Rule we have to find: ∂u∂x, ∂u∂y ,and…

Q: F(t, x, y, 2) = [0, cos(t), sin(t)] x 7, 0<t<

A: Solution

Q: sin(xz – 1) In(6+ 3xz + 4yz) = 0 determines z as function of x and y in a neighbourhood of the point…

A: Claim: at (x,y)=(1,1) for the equation  Take partial derivative with respect to x Substitute…

Q: Find the potential P(x,y) attached to the exact ODE 2x sin(y) +x² cos(y) y' = 0

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Q: 2. Let L:= D2 - x-D+4x2. Verify that Ly 0 where y(x) = sin(x). %3D %3D %3D

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Q: 3. Find dy where; (x+y) =2 o) cos(xy)- 주+ ers + b) e+ u= xlax (use chain rule)

A: We have to find the derivative

Q: Find fa, fy, and fz. f(z, y, 2) = z In (a*y cos(52)) z In (a*y* cos(52)) fr = fy = f2 =

A: First order Partial differentials

Q: 32. Il sin(x + y) = x,'then dx A 1- cos(x+y) (cos(x +,pv) IB 1+ :aas(x + y) cos(x + v) C cos(x + y)…

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Q: A) If w = 2 y et* – In z where x = In(t² + 1), | dw y = tan-1 3t and z= e3t. Find by Chain Rule dt

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Q: e) Suppose that W= √√x² + y² + z², x= cos 0, y = sin 0, z = tan 0. dw T Use the chain rule to find…

A: According to our company's guidelines I can only answer (e), since (e) and (f) are not inter…

Q: Vhat is the general solution of (x + y) dx + tan ady O xsinx + cos x + ysinx = C O sinx + y cos a +…

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Q: The conjugate harmonic function v(x, y) of u(x, y) = In (x2 + y2) is *() + 1. 2tan -1 2tan-1 + C ()…

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Q: Let z = x*y + 3xy* where x = sin 2t and y dz find = cos t. Use the chain rule to %3D when t = 0. dt

A: z=x2y+3xy4where x = sin 2t and y = cos t We need to find dzdt when t =0 Let's use the chain rule :…

Q: If Z = /xy + y, x= cos 0 , y= sin 0 when 0 = 2 dz By using chain rule find de

A: According to chain rule of differentiation .According to product rule of differentiation . According…

Q: i. y = x-% % -4Vx ii. y = x+Inx+ sin(3x) iii. y = x° cOS X y = x° cos x

A: Use product rule of differentiation

Q: Example H-W:_for f(x,y) = cos(xy) – x³ + y* compute fxyy, fxxyy %3D

A: We have to find the partial derivative

Q: Find fr, fy, and fz. f (x, y, 2) = z In(x*y³ cos(52)) 4.3 fa = fy = fz =

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Q: dz Use the Chain Rule to find : z = = cos(x + 4y),x = 5t*, y = - %3D dt

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Q: Let z(x, y) be the function implicitly defined as the solution to 2x + y + z + sin(xyz) = 4 + that…

A: ddxsinnx =cosnxddxnx =ncosnx

Q: Resuelve (y+xcotJdr- xdy = 0. In cos .Im지 +c O. y xcos-= C O. In]x{ = sec J+c O Ninguna de las…

A: We will first rearrange the given equation to some suitable  form and then use substitution  to…

Q: h of the DE's =)y = 2 sin² x ; y(0) = 0 , y'(0) =-2

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Q: A:= {(x, y) € R² :1< log(x) < 2}. = {(x, y, z) E R³: sin(xy² + «/2) · cos(z) = 1} . (a) Decide…

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Q: x²+y²` du du =+y dx If u=sin| ,x+y #0,Prove that x =tan u x+y dy

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Q: Ôz 12. If z = cos(x y)+ y cos x , where x=u +v and y=u-v use the chain rule to find and ди

A: Given: z=cosxy+ycosxx=u2+vy=u-v2 Recall the chain rule,  ∂z∂u=∂z∂x·∂x∂u+∂z∂y·∂y∂u…

Q: the initial value DE y' - ytanx = sinx, y(0) = 1 cos 2x 5-cos 2x %3D 4 sin x 5-cos 3x 5-cos 2x В) у…

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Q: If U(x, y) = -e-Y sin x andf(z) = U(x, y) + iV(x, y) is analytic, then V(x, y) = %3D %3D -ey cos x…

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Q: Show that g(x) = cos 2 =x+3 ™K(x, t) g(t) dt 0 || sin x cos t, 0≤x≤t Y sin texr

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Q: Find of /dy if f(x, y) = y sin(xy) 1 + xycos(xy)csc(xy) csc(xy) B) xy cos(xy) + sin(x) A) O

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