Find T, N, B, K as functions of t if r(t) = (sint)i + (V2cost)j+ (sint)k.

Q: Q1// B: If u+v=e cos y and u-v=e'sin yfind the Jacobain of the a u, v at (x.y) (1,0) dx,y function

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Q: Assume that x = x (t) , y = y (t) and z = z (t) and differentiate the given equation with respect to…

A: Assume that x = x (t) , y = y (t) and z = z (t) We have to differentiate the given equation with…

Q: Q2/ Prove that 1 dx = v2 Jo 1+sin(x)

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Q: Find f.  f''(t)=sin t + cos t,    f(0)=2,    f'(0)=4f f(t)= ?

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Q: Q2/ find dz ond dZ as function ne U and v if of Z=tan (중), X=u Cos V , 9-u sinV

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Q: sin, ifx T 2л

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Q: The domain of the function 7 (t) = (In t, cos 2t, /t – 2) is

A: Given that function rt→=lnt,cos2t,t-2To find : Domain Solution: rt→=lnt,cos2t,t-2t>0,t-2≥0

Q: b. Show that 0s cos ay dA s where R= (0. G ) sin Ax

A: This is a problem of double integration and inequality.

Q: Find the best quadratic approximation to f on the interval [ 0, 1].                   f(x) = sin(…

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Q: Compute r' (t) and r'"'(t) for the following function. r(t) = (cos 4t, sin 3t, cos 2t) %3D

A: If, r(t) =<f(t),g(t),h(t)> , where f(t), g(t) and h(t) are differentiable functions,   then…

Q: If V = x² + y² where, x = a cos t, y= b sin t, find AP and verify the result. dt

A: x=acost y=bsint V=x2+y2 put the values of x and y V=acost2+bsint2 V=a2cos2t+b2sin2t

Q: Find the function if sin t T+1 1 tan 2x + 1

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Q: Find F(t) knowing that F'(t) = (sin(t), cos(t), 2t) and F(0) = (1,1,2).

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Q: Find ▼ · (V × F) if F(x, y, z) = sin xi + 7 cos(9x - 10y)j + 5zk.

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Q: (b) Show that, the function u = In(x² + y²) + 2 tan" satisfies Laplace's equation.

A: A Laplace equation is a 2nd-order partial differential equation. This is defined by,…

Q: Find F(t) knowing that F'(t) = (sin(t), cos(t), 2t) and F(0) = (1, 1, 2).

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Q: 1. f(t) = t3 – 12 +5t + 2 2. f(t) = cos25t 3. f(t) = -2cos6t + 5sin6t %3D

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Q: dw Use the Chain Rule to find , where w = sin 8x cos 4y, x= dt , and y = t. dw dt (Type an…

A: Given w= sin(8x)cos(4y), x=t/2 , y=t3

Q: GIVEN: f:0,2x]- -R, f(1) = sinz+ cos I %3D Find f (x) in factored form.

A: Differentiate the function fx with respect to x.

Q: Evaluate F. dr where F = (4z, – 2y, - x), and C is given by 7(t) = (t, sin(t), cos(t)), 0 < t < T

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Q: If f (x)= [a(«) =os* '(5) [1+ sin(t?)] dt find f 1 dt, where g(x) V1+t3 COS X

A: We need to find value of derivative.

Q: Express the function f in terms of the unit step function: sin(2t) 0st 2n.

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Q: Find f. f(t) =|| f"(t) - 3et+ 5 sin t, f(0) = 0, f(T) = 0

A: To find f.f''t=3et+5 sin t, f0=0, fπ=0

Q: d(t ) = (2t, 4 sin(t), cos(2t)) 5(0) = (– 4, 1, – 2) F(0) = (1, 2, 0) F(t) =

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Q: Find f. f(t) = 5e¹ - 6 sin(t) — 5 f"(t) = 5et + 6 sin t, f(0) = 0, f(π) = X 0

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Q: 2 f(t) = ( tv4t2 + 1, sin(t) cos² (t), t2 + 1

A: As you ask parts b and c. in part b we need to find the equation of the tangent line to the curve at…

Q: Ex2: Solve by using Laplace transform y'-x = et • x'+y = e=t, where, x = 0, y = 0,

A: The given system is, where initial condition is x=0, y=0. Now taking Laplace transform of equation…

Q: Let z be defined implicitly as a function of r and y by the equation dz 4+ In (ry) – cos (cy – 2) =…

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Q: 1) f(t) =t cos(@t) 2) f(t)=tsin(@t) 3) f(t) =t cosh(at)

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Q: ди ди then x +y = u. 31. If u = Jy² - x2 sin-

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Q: Find a function f that satisfies the equation | f(t) dt = 2 cos x + 3x – 2. %3D -

A: Derivative of a definite integral is: ddx∫0xf(t)dt=f(x) Formulae used: ddxcosx=-sinxddxx=1ddxC=0…

Q: Find f. f '(t) = 10 cos(t) + sec2(t)

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Q: Let C be defined by 7(t) = |cos (t), sin (2t), t²| for 0 < t < n 2 sin(Vz) 9x*dx + 2zdy + + 2y ) dz…

A: Using the result that  ∫cF·dr=∫cPdx+Qdy+Rdz=∫P(r→(t))dxdt+Q(r→(t))dydt+R(r→(t))dzdt dt Here,…

Q: Evaluate: L{t f -t e2t Cos t dt}.

A: Evaluate: Lt∫0te2t costtdt First of all we find    L(cost)=ss2+1L(e2tcost) =…

Q: Show that U(,y) is Harmonic function and find its conjugate for U(x, y) = 2x – x³ + 3xy² -

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Q: Find the differential of each function. (a) y = tan(v7t) dy = 4 - v2 %3D y = 4 + v2 (b) dy = %3D

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Q: 2Tt Prove that a" sinax cos bx dx = 0 0.

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Q: The residue of the function sinz is (z+i)3

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Q: z{cos (: -)u(: - )} = a) 3 3

A: We find the given Laplace transform. Formula:  L{cos(at)}=s/(s2+a2)......(1)…

Q: Compute r" (t) and r"(t) for the following function. r(t) = (cos 3t, sin 6t, cos 2t)

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Q: Consider the function g(x) = 5 S cos t dt – x?. Then g' (x) is

A: Given gx=12∫x4x2costdt  -x2

Q: If u(x; y) = e sin x - x. Show that u(x, y) is harmonic and find its harmonic conjugate v(x, y) to…

A: Given that ux,y=e-ysinx-x The partial derivatives of u are,…

Q: Calculate when t = dr given that = (t – sin t) y = (1– cos t)

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Q: The two functions y1(t) = sint cos t and y2 (t) = sin 2t are linearly Independent. True False

A: →Two functions y1 and y2 are said to be linearly independent,                 if, y1 ≠k y2 , where…

Q: (f) If w=4x² +4y² +z² ,x= p sinø cos0, y= psinøsin0 , z =pcosø, Əw find and ap

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Q: Find a function f such that ∇f = ex cos yi − ex sin yj + zk.

A: Given, ∇f = ex cos yi − ex sin yj + zk.

Q: Evaluate F. dr where F = (- 3z, 4y, x), and C is given by - 7(t) (t, sin(t), cos(t)), 0 <t < ™

A: To evaluate : ∫cF→. dr→ Given: F→=-3z, -4y, xand r→t=t, sint, cost 0≤t≤π Now,  r't=1, cost, -sint

Q: Find the differential of each function. (a) y = tan(v7t) dy = 5 - v2 5 + v2 (b) y= dy

A: a) Given function is By using chain rule of differentiation , we have

Q: Determine the value of a for which the function F(x,y) equation Fr - aFyy= 0. 8. cos(2x) cos(6y)…

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