Q4. Prove that f(z) = Iz/2 never be holomorphic function at z = 0. %3D

Q: 1. Show that c(t) = (e¬t cos t, e-t sin t), 0 <t <∞ has finite length and calculate its valu

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Q: e Divergence Theorem to evalu (4x + 10y + z²) dS Js

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Q: 30. Find the continuous function f(t) satisfying csin z for all r. 2+ f(t)dt = sin2015 /3

A: According to Leibnitz rule:

Q: Find the linearization of the function f (x) = x at a = 1. a Va lal sin (a) L (1) Using this…

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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: 2 3 Evaluate the integral f S(x² + y)dydx. 0 1

A: Here given two variables in Integration

Q: Let z = cosh t. Find the differential dz when t = In 3 and dt = 0.2 O 4/3 O 4/15 O 8/15 O 1/3

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Q: 1- Find the derivative of ywith respect to x, t, or 0 as appropriate + 5. y = In 3x 6. y = In kx, k…

A: Derivative of the function

Q: Verify directly from (3) that the solution has derivatives of all orders in {z > 0}. Asume that h(x,…

A: Sol

Q: 3. Find the total differential of z = f(x, y) if z2 - y = (1– 2x)z.

A: Total differential of a function z=fx,y is dz=∂z∂xdx+∂z∂ydy

Q: Evaluate the double integral: ∬ 4y/(1+x2)dydx y runs from 1/2 to 2 x runs from 0 to 1

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Q: 10. Find the linearization, L(x,y,z) of the function f (x, y, z) = e* + cos(y + z) at Po (0,",").

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Q: Let z = f(x,y)= x^3+ xy+ y^4    (a)  Use differentials to estimate Δz as x changes from 1 to 1.05…

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Q: Evaluate the improper intergal 1 dx using the residue (x² + 16)² theorem.

A: We need to evaluate integral.

Q: Let z be defined implicitly as a function of x and y by the e e-zy əz find O UTM + cos (xy) = (4 –…

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Q: (0 if xE Q (1 if x E I If f (x) = show that f is not Reimann Integrable

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Q: et f be an integrable, odd periodic function witl t g(x) be its integral function.Prove that:

A: Some important rules of definite integral: Order of integration: ∫abf(x)dx=-∫baf(x)dxZero:…

Q: Find the local linear approximation of f(x) = In(5 + x) at xo = -4. %3D In(5 + x)

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Q: Show complete solution. Expand, that is, perform the indicated operations on the differential…

A: Given: xD(D+x)

Q: y = sech*x = In(_+ V1- x², sechx = In(- 1+ V1-x² Prove that .... ..

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Q: Use at least two methods to prove that dr/dt=0 when r=√x2 + y2, x=α cos t, and y=α sin t if α is a…

A: First method   r=x2+y2r=(α cost)2+α sint2            since x=α cost and y=α sint  =α2cos2t+α2sin2t…

Q: Express aw/ar and ôw/as as functions of r ands using chain rule w = = In(x² + y² + z²), x = res sinr…

A: Given problem is :

Q: 2. Let f(2) = 2°e* z + 0. What type of singularity does f(z) have at z 0? (Justify.) What is the…

A: the point z=0  essential singularity.

Q: 3(e). If it is true that the line integral F. dr 0 for all closed curves C₁ show that the line…

A: Given: 3C.For all closed curves C, line integral ∮CF.dr=0 and line intergral over a non-closed curve…

Q: 12.)Express lim E(3(x;,)² – 5x, + 2)Ax as a definite integral on the interval [0,1]. i=1

A: The given expression is: limn→∞ ∑i=1n 3xi2-5xi+2∆x    on interval [0, 1]   The definite integral  on…

Q: 2. Use the Definition of Residue to show that: dz -ni 3 Izl=1z sinh z Where u> |z| > 0

A: Prove,∫z=1dzz2 sinh z=-πi3 ∫z=1dzz2 sinh z=∫z=1fz dz .........equation 1 fz=1z2 sinh z , 0&lt;z&lt;π…

Q: 10. Find the bounds of integration a, b, c, d such that f (x, y) dydx f(x, y) dædy.

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Q: Let f(x, y, z) = ln(x² + z) + y²e#z – cos(2yz). Find fzya.

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Q: 1.6 Evaluate the line integral f x*dx + xydx where C is the triangle with vertices at (0, 0), (0,…

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Q: Compute the first-order central difference approximations ofO(h^4) of t function of y at x = 1h =…

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Q: of Let f (x, y) = tanh-¹(x-y) with x=e" and y= usinh (t). Then the value of dt (u,t) =(4, In 2) is…

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Q: 2. Suppose that z = f (x, y) and both x and y are functions of a single variable t such that dy t2 –…

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Q: Find the linearization L(x) of the function at a. f(x) = cos(x), a = 3n/2 %3D

A: We have to find the linearization L(x) of the given function.

Q: 3. Prove by using the Squeeze Theorem, or otherwise: 1 t2 dt is continuous at 0. T sin t (a) (b) 1+…

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: 1 Justify the Cauchy-Riemann equations for f(2) = e² where f(z) is holomorphic.

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Q: Find the Selution of (1-t +3t)4 (t) di - , (2) In which IL) denoter Bessel function

A: Leibniz integral rule : ddx∫bxaxFtdt=∫bxax∂∂xFtdt+Faxddxax-Fbxddxbx   Linear Volterra Equation of…

Q: if p> 1, (diverges; if p <1. 0o dx Prove that f p- 1 хР 1 Hence evaluate: dx. x6

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Q: Let f(x) = (cosh x)² be defined on (0, 0), find %3D d 1/2) . dy

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Q: v) ₁ √√¹-x² sin(x² + y²) dzdydx fő

A: Given is  ∫01∫01-x2∫01sinx2+y2dzdydx In order to solve this integral it is needed to first convert…

Q: 11. Use Green's Thm to evaluate the line integral if the curve C consists of 3 line segments, from…

A: Solution:-

Q: e In (x le) (2- secx dx to be integrable using the formula - se What should be the value of A for u"…

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Q: Assume x and y are functions of t. Evaluate dy / dt for 4xey = 8 - ln 64 + 6ln x, with the…

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Q: Evaluate the in tegral (x+ 2y) dy

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Q: Evaluate the integral: -2e 2е π/3 ₁² /³ tlnu tans ds du dt 0

A: In this triple integral we first integrate with respect to 's' then with respect to 'u' and in last…

Q: Show that {zn}nEN is Cauchy, where z, = sint dt. 1+t

A: Sol

Q: b- Use Chain Rule to compute w = 2y e* - Inz } əw at x = ln(t² + 1), at (t) equal to (1) for: y =…

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Q: Evaluate the improper intergal 1 dx using the residue (x² + 1)3 theorem.

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Q: 1. Prove: a. csch x In +c %3D b. sech x = 2 Arctan(e*) + c %3D

A: 1. (a). To prove: ∫cschx=lnex-1ex+1+C Proof: We know that, sinhx=ex-e-x2 The left hand side can be…

Q: 1 Find skewness and kurtosis for the RV X with PDF f(x) = 2T -∞ < x < x.

A: PDF of X,  fX(x) = 12πe-x22, -∞&lt;x&lt;∞ So, μ = E(X) = ∫-∞∞x f(x) dx=∫-∞∞x 12πe-x22 dx=0 [as…

Q: Let f(x) = e2sinx 1. Use the local linear approximation of fat x = 0 to approximate e2sin (0.01)

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