Q2. If w = f(u, v) , u = cosx and v = sinx, show that: (30 Marks) d?w = v? fuu + u² fvv – 2uvfuv – ufu – vfp dx2

Q: 37. ; with r = 1 andh = ;: 0, p2 sin¬! V1-h² /r²-h/r2-n? (0,, 33 47-3/3 (0, 0.71)

A: Let's simplify in steps Let's take the numerator first:

Q: Hw. Determine t posdise red nd d. Sinx a x 2 In(X):07, de. os, Xa. 2, 3-tealion by e position fndla)…

A: NOTE:- By bartleby rules, only question (1) is answered

Q: If z = f(x, y), where x = r cos 0 and y = r sin 0, show that a²z 1 a°z 1 əz r дr дх? dy? дr?

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Q: aw and ar Q2) Find if as w = 2x + 4y - z, r+s X =r tan (s), y= sin (r - s), z=e**s.

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Q: Q.1B) If w = f(u,v) where u = tan"12 &v=In(x² + y²), the show that: dw Х- +- дх дw dw <-2- ду ду

A: Since the student has posted multiple questions and does not mention any specific question, so we…

Q: If W = 2x3 + 3y², a 5t2, y = 2t³ %3D = %3D dW then find at t= 1 dt O 1912 O None O 1273 O 1572 2212

A: A detailed solution is given below

Q: Give an e xam Ple of (T, m') ameasurable SPa ce F; x → T {E & CX : fCE) ET3 m is not

A: Given, Τ, m a mesurable space fi:X→Τ , m=E⊆X: f(E)∈Τm is not a σ-algebra on X.

Q: If w= XZe' y=Va²–x² , z=Sinx , then ( dw is |

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Q: dw when r= 3 and s = - ar 3 if w = (x+ y + z)“, x=r-s, y= cos (r + s), z= sin (r + s). Find

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Q: 2. Given that z and w are independent variable and dependent variablerespectively, obtain dz/dw a.…

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Q: Q.1B) If w = f(u,v) where u = tan-12 & v = In(x² + y²), the show that: %3D dw aw aw X +y 2- ax ду dv

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Q: If f(x, y) = x²e + cosy af af then find dy

A: We will use partial derivative to solve this.

Q: EX: Verify that z= I+ V3i satis fies the 2. eq.

A: the given equation is z2-z+1=0 given the value of z is z=1+3i2 substitute the value of z in the…

Q: OL. If w = f(u, v), u = cosx and v = sinx, show that: (30 Marks) %3D d?w dr v² fau + u?f- 2uvfuv -…

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Q: Q.1B) If w = f(u, v) where u = tan-12 & v = In(x² + y²),the show that: %3D dw aw aw +y- = 2 ax ду dv

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Q: 24yıy2, 0< yı < 1, 0 < y2 < (0, fV1, y2) = %3D elsewhere Find the cov(Y,,Y2) is it possible for Y,…

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Q: Be w = (u2 + v2)3/2, where u = e-t sin(θ), v = e-t cos(θ), calculate: ∂w/∂t and ∂w/∂θ

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Q: s) Given u(r, s) =r2 - 2s2, wheter= and s = y? - r, compute for du

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Q: Q1. If w = f(u, v), u = cosx and v = sinx, show that: d?w v2 fuu + u² fov – 2uv fuv - ufu - vf dx2

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Q: 2. Given that z and w are independent variable and dependent variable dz respectively, obtain dw a.…

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Q: Q.IB) If w=f(u, v) where u = tan ¹2 &v=ln(x² + y²), the show that: Əw dw aw +y ax dy 2- ду x-

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Q: az Find when u = 0 and v dv 117 if z(x, y) = sin x + cos y, x = u •v, and y =u + v.

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Q: Q3. If w = f(u, v) , u = cosx and v = sinx, show that: d?w v² funu + u? fov – 2uvfuv – ufu – vfp %3D…

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Q: . If U = z sin y/x where x= 3r2 + 2s, y= 4r – 2s², z = 2r? – 3s², find a) Or b)

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Q: 10. If z = x + f(u), where u = xy, show that дz dz х дх = x. ду

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Q: 63. r(и, v) %3D Зи сos vi + Зи sin vj + 18u'k ||

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Q: Gy Let Px) = x²/3, x. thens Rx) = x2/3 ×シ。 then: %3D O Find ê cx) ? 2 Gruphs off R and f

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Q: Ex 15: Prove that : d du a) tanh u = sech² u dx dx

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Q: Q.1B) If w = f(u,v) where u = tan-12 & v = In(x² + y²), the show that: %3D dw aw aw X +y¬ 2- ax ду…

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Q: f(z) =e? is contiuous in C

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Q: 01. If w f(u, v), u = cosx and v = sinx, show that: d'w dx2 = v? fuu + u? frv - 2uvfur - ufu - vf,

A: Given w=f(u,v), u=cos x and v=sin xso we see that dudx=-sin x =-v   and dvdx=cos x=u.By chain rule,…

Q: To evaluate y-csc²(y³)cor²(y³)dy, we let u =cse (v³) A True B) False | csch2zdz =-In|csch2z –…

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Q: Q9. If W = f(u)=u' + sinu+Inu, and a=ax+by, where a, and b are constants, show that aw aw b. ax

A: We have to prove the given statement.

Q: 2- FIND f (t) or fis) AS APPRO PRIATE A) L}Je costdt COS ) y+2メ=8(t-)0)00)=イ

A: Here we use basic definition of Laplace transform

Q: Q9. If W = f(u)=u³ +sin ¹u+lnu, and u= ax +by, where a, and b are constants, show that aw Əw b oy Əx

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Q: if y³ - y = 2x Then, d'y -24y prove that dx (3y -1) |

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Q: Q3. If w = f(u, v) , u = cosx and v = sinx, show that: d?w v² fuu + u?fov – 2uv fuv – ufu – vfp dx2

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Q: Q2)f f is Cont then Ifl is cont. IS

A: See the attachment

Q: Q12) If w=f(a,b,c)=cos(ab)sin(b)+c, what is at ( -,6 )?

A: Given ,fa,b,c=Cos abSin b+CPut 12,π3,6i.ea=12 ; b=π3 ; C=6

Q: f(z) = e? is contiuous in C

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Q: If f(x,y,z)=exp(3x)cos(-4yz), find Vf. %3D

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Q: where: %3D Solue ulX, o)=2 X •a u (0,t)=o 1 ノ

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Q: if y' - y =2x Then, d'y -24y prove that dx (3y –1)

A: Given that: y3-y=2x

Q: If x"y" = (x + y)"** m. m+n Then, P rove that dx?

A: Double derivative

Q: aw and ar dw O Find if as

A: In this question, the concept of the derivative is applied. Derivative  A a derivative is used to…

Q: dy _ 2y 4. If tan(ry) + 5= ry then show that dr?

A: We have to do differentiate with respect to x.

Q: Find Vf(x, y) if f(x,y) = e** cos y ..... ...

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Q: If f(2) = sin(z?) = U(x, y) + iV (x, y), determine U and V. Verify that the C-R

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