(c) The electric potential V at the point P(x, y, z) in a rectangular coordinate system is given by V = x2 + 4y² + 9z. Calculate the rate of change of V at P(2,–1,3) in the direction from P to the origin.

Q: Find the gradient of the function z = cos(x2 + y2), at the given point (3, −4)

A: Given=z=cos x2+y2 at point x,y=3,-4Gradient∇x,y=fxx,yi+fyx,yjwhere fx,y=z=cos x2+y2

Q: A particle moving along R(t) = r cos tî + r sin tj, r > 0, has an instantaneous speed that varies…

A:

Q: Find the derivative of the function h (x, y, z) = cos xy+ e+ In zx %3D at P. (1, 0, 1/2), in the…

A: Here we have to find directional derivative of h(x,y,z) in the direction of u.

Q: b. Determine the gradient of the following function at Po h(x, y, z) = cos xy + e + In zx, Po(1, 0,…

A: We need to find gradient.

Q: Use the total differential dz to approximate the change in z= 6-x² – y² as (x, y)moves from the…

A:

Q: (a) Find the derivative of f(x, y, z) =x²- xy²- z at Po(5, 1, 0) in the direction of v=2i-3j+ 24k.…

A: Given: f(x, y, z) = x2 - xy2 - z, P0(5, 1, 0) and v=2i - 3j + 24k.

Q: 3. find the derivative of the function at Po in the direction of u. f (x, y,z) = 3e* cos yz,…

A:

Q: find the directional derivative of f(x,y,z)=3e^xcos(yz) at P(0,0,0) in the direction of A=2i+j-2k

A: First find out the partial differentiation of f(x,y,z) and then put the values (0,0,0) and then for…

Q: Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,…

A:

Q: 2) Find the maximum rate of change of f at the given point P and the direction in which it occurs.…

A:

Q: Find a gradient of the equation: F(x,y) = v2x + :- y sin a + 4 VF =?

A: Here, the given vector equation is: F(x,y)=2x+xy-y2sin x+4

Q: b) Find the directional derivative of f (x, y) = xy + e*°vy + 1 at (-1,1) in the direction of the…

A: f=xy+ex3y+1 take the partial derivative with respect to x   fx=∂∂ xxy+ex3y+1 fx=∂ ∂ xxy+∂ ∂ xex3y+∂…

Q: Find the direction at which the function f(x, y, z) = x In(e - cz) increases fastest at (-3, C O (0,…

A: To Find: The direction at which the function  fx,y,z=x ln(ey -xz) increases fastest at ( -3, 0 ,0)…

Q: A particle was following the trajectory r(t) = (e', sin (2t), t), but it starts flying tangentially…

A: SolutionGivenA particle was following the trajectory r(t)= (et,sin(2t),t) , but it starts flying…

Q: Use the differential dz to approximate the change in z = √4 − x2 − y2 as (x, y) moves from the point…

A:

Q: Write the differential dw for the function w = ƒ(x, y, z).

A: Consider the given function w=fx,y,z

Q: A particle moves in a plane according to -t, x = (t+1)e; y=2tet -t Determine the simplified…

A:

Q: (a) Find the derivative of ƒ(x, y, z) = x3 - xy2 - z at P0(1, 1, 0) in the direction of v = 2i - 3j…

A: We have given function fx,y,z=x3-xy2-z, point P01,1,0 and vector v=2i^-3j^+6k^

Q: Find the total differential of the function f(r, y) = (3 y²+6 2):

A: Given,           The function is fx,y=3y2+6x13 .

Q: Find the directional derivative of the function f(x, Y, z) = 3e" sin(xy) in the direction of v =…

A:

Q: a particle moves along a curve x = t^3+1, y=t^2, z=2t+5 , where t represents the time . its velocity…

A: Velocity is obtained by differentiating the position function

Q: Find the derivative of the function at Po in the direction of A. f(x,y,z)=3 ex cos (yz), Po(0,0,0),…

A: Solution: f(x,y,z) =3exCos(yz) P0=(0,0,0)

Q: 30. Find an equation for the tangent plane to z = f(x, y) at (3, -2) if the differential at (3, -2)…

A:

Q: Consider the function g defined by 1 g(x, y) = cos (Ta y) + %3D log3(x – y) Do as indicated.…

A: please comment if you need any clarification. If you find my answer useful please put thumbs…

Q: Let f(x, y, z) = x² + 2y+ e³=2« + 2°. Find the rate at which f is changing at (1,2, –1) in the…

A: Use the formula of the directional derivative and simplify.

Q: Find the derivative of the function at Po in the direction of vector A h(x, y, z) = cos(xy) + e>% +…

A:

Q: (2) Evaluate the rate of change of i with respect to t when i 15t sin 3t and t=0.1. 2e" sin 2t when…

A: Calculation of rate of change of z with respect to t is given.

Q: Find the direction at which the function f(x, y, z) = x In(e – xz) increases fastest at (-2, 0, 0)

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: 2. Find (a) the gradient of f at P and (b) the rate of change of the function value in the direction…

A:

Q: 1. Find the gradient of f(x, y, z) = e2* (sin a – cos y).

A: Gradient of functions

Q: Consider the function defined by 1 g(x, y) = cos (Tx Vy) + log3 (x – y) Calculate the instantaneous…

A:

Q: The gravitational attraction F between two objects of masses m1 and m2 is jointly proportional to…

A: see the attachment for answer

Q: Find the maximum rate of change of f(z, y, z) = tan(6z + 2y + 3z) at the point ( – 1, 3, – 2).

A: We need to find maximum rate of change.

Q: Write the differential dw in terms of the differentials of the independent variables. w =…

A:

Q: - Consider the function g defined by 1 g(x, y) = sin 2 log3 (x – y) 1. Calculate the instantaneous…

A:

Q: 3. Evaluate the line integral sin(y) dx + (x cos(y) – sin(y)) dy, where C is the straight line from…

A:

Q: Find the gradient Vf for the function f(x,y)= e* sin(zy) and the directional derivative at the point…

A: to find the gradient take the partial derivatives of the function then substitute the x and y values…

Q: Find the directional derivative of f(x, y, z) = xy² + yz% at P(1, – 1, 1) in the direction of a = i…

A:

Q: Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,…

A:

Q: Find the derivative of the function at Po in the direction of A. f(x,y,z) = 3 ex cos (yz),…

A:

Q: Find the maximum rate of change in f (x, y, z) = tan(r + 2y + 3z) at P = (-5,1,1).

A:

Q: Find the maximum rate of change of f(x, y, z) = tan(7x + 3y + 6z) at the point (4, – 5, 2).

A:

Q: (b) Calculate the rate of change of the function f(r, y, 2) = ryz along the path r(t) = (sin(t), e“…

A:

Q: 4. Find (a) the gradient of f at P and (b) the rate of change of the function value in the direction…

A:

Q: 1. Find the directional derivative of the function f(x, y, z) = 3e" sin(ry) in the direction of v =…

A:

Q: Verify the Pierre-Simon Laplace equation ∂2f/∂x2+∂2f/∂y2=0 with f(x,y)=ln∣x2+y2∣

A: