dw at (u,v) = (5,2) for the function w(x.y) = xy² - In x; x= e u+v Evaluate ', y = uv. dv O A. 200e' - 1 O B. 150e '-1 Oc. 300e7 -1 O D. -1

Q: Find fr and fy- ray f(r, v) = [" 3e 3e" dt f„(x, y) = fy(x, y)

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Q: dx 1s+ kıl35-x) +kaly-x) dy %3D dt k, , k. k3 are Con Stants.

A: Given system is dxdt=15+k135-x+k2y-xdydt=k2x-y+k335-y Here, 1k1=5⇒k1=151k2=10⇒k2=1101k3=5⇒k3=15 So,…

Q: Compute div F, where F(x, y, z) = (x² + 3y, y? – 4z, sin r cos y – 22). A. (4 – sin r sin y, – cos r…

A: Given : F(x, y, z)=(x²  + 3y ,  y² -4z ,  sin(x )cos(y) - z2 )  Now we find div F = ?

Q: The value of local acceleration in the x direction for flOW with all wrong u = 2(1 + t) is given by…

A: Given that  u=2(1+t)  To find the local acceleration in the x direction of flow

Q: The maximum directional derivative of f (x, y, z) =r sin(zy)-yz + 10 %3D at (1, -1, 0) is O v3 O 2…

A: For the scalar function f(x,y,z), gradient is the maximum rate of change.  So, first we will find…

Q: Compute Scy dx + xy dy if C goes from (0,0) to (1,2) along the line parameterized as x = t, y = 2t.

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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: 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: Let y = tan(2z + 3). Find the differential dy when z = 2 and dæ = 0.4 Find the differential dy when…

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Q: dy Find at x = dt dy dt || 4 if y = 3x² + 5 and dx dt = 1.

A: Let's find dy/dt value at x=4.

Q: dy dz Find at z = 1 if y = 4z +4 and = 2. dt

A: Given: y=4x2+4

Q: Use differentials to estimate the change in f (x, y, z) = x² In(5yz + 1) as (x, y, z) changes from…

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Q: Find fr and fy- S(x, y) = /"9e" dt f.(r, y) : %3D f,(x, y) =

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Q: Choose the gradient of the function f defined by f(x,v,z) = cos(xy +9xz + 13yz). O sin(xy O -cos(…

A: Take gradient and solve

Q: Evaluate the line integral / (7x + 3y)dx + (2x – 8y)dy along the curve C: x = 8 cos t, y = 16 sin t…

A:

Q: Let f(x, y) = x sin ry. Find: fyy (x, y) fyz(x, y)

A: f(x,y)=xsinxyfx=sinxy+xycosxyfyx=xcosxy+x(cosxy-xysinxy)=2xcosxy-x2ysinxyfy=x2cosxyfyy=-x3sinxy

Q: Assume that x and y are both differentiable functions of t. Find dy/dt when x = 16 and dx/dt = 9 for…

A: Solution is given below:

Q: Assume that x=x(t) and y=y(t). Let y=x^2+2 and dx/dt=2 when x=5. Find dy/dt when x=5

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Q: Compute dy/dx using chain rule where y = (u^2+2u)/(u+1) and u = x(x + 1)

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Q: Assume w = x2y4, where x and y are functions of t. Find dw>dt when x = 3, dx>dt = 2, dy>dt…

A: Given:  w = x2y4, x and y are functions of t  To Find: dw/dt when x = 3, dx/dt = 2, dy/dt = 4, y = 1

Q: Find f. fy for f(x,y) = xcosy + %3D x5 siny3 ey

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Q: dy Sølve +y = 0, under the conditions that y= 1, dy 0. when t = 0. dt %3D

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Q: Find dydx? for f(x, y) = eXy

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Q: dz If z = e2x+ 3y and x = In t, y = t2, calculate from the total dt derivative formula and show that…

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Q: Xyz f(x.y.z) = function is given dt P(2,t,) in point fax + fy + f2z colculate.

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Q: Find dy or y' using implicit differentiation. r'y-y= Oy 1- 3r'y %3D Oy = 1 3z'y 3z'3y 1 %3D 1 3z'y…

A: The given equation is x3y3-y=x

Q: Find the gradient of the function f(x,y,z)=2xe^(y/3)sin(3z). grad f=

A: Given f(x,y,z)=2xe(y/3)sin(3z) We need to find gradient of f(x,y,z)

Q: Eval S fCx,y,z)dV Por f auel regien W M, fory2)= 2x W: x'+ y? < zs 4

A: The region W : x2+y2≤z≤4, and f(x,y,z)=2x, we have to evaluate the value of ∬∫Wf(x,y,z)dV.

Q: dy at z = 5 if y = 5x² + 4 and dt dz Find 3. dt dy dt

A: Given y=5x2+4 -1and dxdt=3

Q: Find the differential of f(x, y) = Vr2 + y³ at the point (2, 3). df = Then use the differential to…

A: given function is f(x,y) =x2+y3 we know, for the function z = f(x,y), the total differential of the…

Q: Find fr and fy- f (x, y) = 4etlo dt f#(x, y) f„(x, y) =|

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Q: Let f(x, y) = 3e" sin(5y) af af dy

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Q: Assume that x and y are differentiable functions of t. Find  dx/dt given that x= -3 , y= -2 , and…

A:

Q: Find y', where t (a) y = Jo dt VE2 +1 (b) y /t² +t+1 dt 1 • sin²3x (c) y = | 1 dt t2 +1 Cos x

A: The given function is: y=∫cos xsin2(3x) 1t2+1 dt

Q: If y is a function of t and z= 4y²+y• find dz/ dt when y=2 and dy/ dt = – 1 %3D

A: z=4y2+y, dy/dt=-1

Q: If f(x, y) = 3x? – by, then the total differential of f at x = 1 and y = -2 is : O df = 24 dx – 6 dy…

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Q: Find the gradient of the functions below. a)f (x, y, z) = sin(xyz)

A: Here we will find out the required value.

Q: Find a potential function for F(x, y, z) = (y² cos z, 2xy cos z, –ry² sin z). %3D COS

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Q: dz Suppose that z = f(x, y) is differentiable at the point (8, 16) with f,(8, 16) = 3 and f, (8, 16)…

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Q: Use the Chain Rule to find dQ where Q=√√3x² + 3y² + 4z², x= sint, y = cost, and z= cost. dt'

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Q: What is the integrating factor of (ry + y) dr + (r + 2y-1) dy %3D

A: Given xy+y2dx+(x+2y-1)dy=0Compare above differential equation with…

Q: dy whenx = 4 and dt dx = 17 for the equationy = Vx. Assume that x and y are both differentiable…

A: First find the derivative of y with respect to x at given point x = 4 then solve further.

Q: Find the first partlal derivatives with respect to x, y, and z. f(x, y, z) = 4x²y – 6xyz + 9yz? %3D…

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Q: Find fr and fy- f(x, y) = fetlo dt f.(r, y) = fy(x, y)

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Q: dz Suppose that z = f(x, y) is differentiable at the point (32, 8) withf,(32, 8) = 4 and f, (32, 8)…

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Q: Find fr and fy- f(x, y) = ets dt %3D f (x, y) = %3D fy(r, y) =

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Q: Please show your answer to at least 4 decimal places. Determine the gradient of f(r, y) = e sin(4y)…

A: The gradient is just the vector of partial derivatives.

Q: 2. Show that f(t) = 1 – sint defines a smooth flow on the manifold S = ( (cost, sint) ), but g(t) =t…

A: The given question can be solved as shown in step2