9. T: R² → R², T(x, y) = (x + 2y, y – x) 10 T. D2

Q: 2x :S dr f(x, y) dy. f (x, y) dy. •1-x² f (x, y) dy. -V1-r

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Q: Compute DF(, ) of F(1, y) = (e" cos(ry), r² + y) %3D

A: Given function is F(x,y)=eycos(xy), x2+y Let  f1(x, y)=eycos(xy)f2(x, y)=x2+y

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

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Q: dz dz Calcula y dy z = x In(y) az = In(y) 1

A: Given : z = x ln( y ) To Find : ∂z∂x, ∂z∂ywe have given four options to determine the correct option…

Q: Find dY at x = -1ifY zu%5V=4 3. and u= X²+ X

A: Given that y=u3+5u-4 and u=x2+x We need to find dydx at x=-1 Take y=u3+5u-4 Substitute "u"…

Q: Let f (x, y) = 2x3 + x2y - y2. Compute d2f/dx2, d2f/dy2, and d2f/dxdy at (x, y) = (1, 2).

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Q: Sx, ifx 1 f(x) = Show that f is R- integrable.

A: R- integrable function

Q: Q2/ by using chain Rule to find dw at t = 1 if dt w = 2ye* – Inz ,x = In(t? + 1), y = tan 1t,z = et

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Q: Let f (x, y) = xy2 + 5. Evaluate df/dy at (x, y) = (2, -1) and interpret your result.

A: Given: f (x, y) = xy2 + 5.

Q: dy Find at t = a where dt 1 7. у %3D = 3s + 2 and s t2 when a =1. + 4

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Q: If f(z, v) – 4z°y+ 8zy, then the total differential of f at z-1 and y - 1is: O df - 17 dz + 22 dy O…

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Q: dw Use an appropriate form of the chain rule to find dt w = In(10x² – 4y + 5z³); c = t', y = tỉ, z =…

A: From the given function: w=ln10x2-4y+5z3 ; x=t12 ; y=t23 ; z=t-2 As we know that: The Power law:…

Q: b Let y = x²sin(y) + yeXy, where x = s + 2t and y = st dv when s (ii) dt = o and t =1 is [d]

A: Find your answer below

Q: -Find the D.f ofr (x, y) = (x² – y)' at Po(3, 1) with

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Q: Q5/ Determine the total differential If Z = f(x,y) = x²y², x=2rcose, y=2rsinė

A: this is a problem of differential. Based on the general formula and concepts we will answer the…

Q: Evaluate fSS 6e(²+y²+z²)/² dV over B where B= {(x, y, 2) : x² + y² + z² < 1}. Take T = 3.

A: Since we have    x2 + y2 + z2  expression in the integral and region B is the sphere of radius 1…

Q: dz Suppose that z = x³ – 3y?, x = 4(t³ – 1), and y = Int. Find using the Chain Rule. dt %3D -

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Q: Suppose that u and v are differentiable at x = 2 and that u(2) = 4, v(2) = -9, u'(2) = 5, and v'(2)…

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Q: Show that y" + ay' = b, wherein a and b are constants, can be reduced to y' =+ C₁e-ax using…

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Q: 2z Forz> 0, 순(/"m(P + 1) a) In (t³ + 1) dt For x > 0,

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Q: Q4) Find y" + 3y' 5y at x-5 where y = xsin(1/escx))

A: To find the y"+3y'×5y at x=5  Given y=xsin(1/cscx)

Q: Find the linear approximation L (x) to ƒ (x) = xcosx at a = T.

A: Given that, f(x) = x cosx and we have to find linear approximation L(x) at a = π

Q: dx If x2 + y2 + z2 = 9, dt dy 6, and dt dz when (x, у, z) %3D (2, 2, 1). dt 3, find %D %3D dz dt II

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Q: 3. Let X(t) = t2W (t) – 2 f sW(s)ds, Is X(t) a martingale?

A: X(t)=t2W(t)-2∫0tsW(s)dsWe know that, X(t) is a martingale, if,EX(t+1)=X(t)

Q: when (x, y, z) = (2, 2, 1). dt dz dy If x2 + y2 + z2 = 9, dx 8, and dt 4, find dt %3D %3D %3D dz dt

A: differentiate the given function with respect to 't' 2x dx/dt +2y dy/dt +2z dz/dt =0 Now substitute…

Q: 교꾸가고 (a, y) + (0,0) (x, y) = (0,0) f(x, y) = x2+y2 is discontinuous at (0,0).

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Q: e solution of the differen uation dy/dt 3 y with in ndition y(0) = 5 is %3D t

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Q: dy t² +3 21. Find 9 if y = In dt /1-t,

A: We need to find derivative of given function with respect to t

Q: 18. Use the Linear Approximation to f(x, y) = /x/y at (9,4) to estimate 19.1/3.9.

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Q: Suppose that z = xy² and x and y are differentiable func- tions of t with x =2,y = when t = -1. Then…

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Q: 1. Use the Chain Rule to find u and ou for w = x²y3 where x = dw at s2 +t2 and y = ds

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Q: If x, y, and z are lengths of the edges of a rectangularbox, then the common length of the box’s…

A: Given- s=x2+y2+z2 To find- dsdt in terms of dydtand dzdt, if x is constant. Concept Used- As the…

Q: I the function z(x, y) = ex sin(3y) and x = ut-t², y = √u² + t², Find : dz dz and 2- (by chain rule)…

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Q: 2 2 If z = f(x, y), where x = s² - t² and y = t² s² and z is differentiable, then - Select one: True…

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Q: dy (3) If y = | (t – 1)²dt for x > 0, then compute dz In z

A: fundamental calculus theorem 2 states that if f is continuous on [a, b], then  ∫abf(x)dx=F(b)-F(a)…

Q: d²s ds + dt 4t + 2cost where s = 0, ds/dt = 0, t= 0 dt2

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Q: - Find the total differential df of f (x, y) = (x² + y²) 3/2.

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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: Let y(t) satisfies dy 3ty-t-t for t20, dt 1+t2 and y(t 0) 1. Show that y(t) S1+t² for t2 0.

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Q: 22. u = Vr² + s², r = y + x cos t, s=x + y sin t; ди ди ди əx’dy when x = 1, y = 2, t = 0 dt

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Q: Q3) Compute for y(x) = csc3(4x) + 2x – 3 and x(t) = 4sin t² at t=1 dt %3D

A: Let y(x)=csc3(4x)+2x-3 and x(t)=4sint2 Differentiate y(x) with respect to x…

Q: find (In t)² dt dr ds (rst-space) 18. se In r T/4 cln sec v 21 19. e* dx dt dv (tux-space) -00 4-q²…

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Q: (ii)If y=(x+Jx* +1. prove that y'= Vx +1

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Q: ) Fnl the Vdume of the Selid enclesed by the curres "about the x-axS. y=e^, y=0 x =©,'x=/

A: To find out the volume of solid of revolution.

Q: Suppose that f(0) = 1, ƒ(0.5) = 2.5, f(1) = 2, and f(0.25) = f (0.75) = a. Using trapezoidal rule…

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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: Find the linearization of the function f (x) = =° at z = 1. L (1) = Using this linearlization, ƒ…

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Q: Q1: Use Green's theorem to evalute S. (xy + y2)dx + x²dy by the curve y y = x2.

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