Let f(x,y,z)=x^3y^2+z^2 and x=s^2t,y=s^3t^2, and z=st^3. (a) Calculate the primary derivatives ∂f∂x=  ∂f∂y= ∂f∂z= (b) Calculate ∂x∂s= ∂y∂s= ∂z∂s= (c) Use the Chain Rule to compute ∂f∂s=  In (c) express your answer in terms of the independent variables t,s

Q: find the gradient of the function and the maximum value of th directional derivative at the given…

A:

Q: What is the rate of increase of the scalar function f(x,y,z) = sin (πx/3) sin (πy/2) e^-2z from the…

A: What is the rate of increase of the scalar function f(x,y,z) = sin (πx/3) sin (πy/2) e^-2z from the…

Q: Find work done by the gradient of f(x, y, z) = zexy- 2y2/z  along the path from (1, 2, 4) to (-1, 0,…

A: By Fundamental theorem line integral,we have Where C is curve joining to

Q: Integrate f(x, y, z) = x2 +y – z over the line segments from (0, 0, 0) to (1, 1, 0) and then to the…

A: According to the given information, it is required to integrate the function over the line segment.…

Q: ag Use the Chain Rule to evaluate the partial derivative at the point (r, 0) = (2/2, 7) where g(x,…

A:

Q: Find the gradient of the function and the maximum value of the directional derivative at the point…

A: The given function is g(x,y)=lnx2+y23 We have to find the gradient of the function and maximum value…

Q: Find the gradient of the function f(x, y, z) = xeyz, and the maximum value of the directional…

A:

Q: Use the limit definition of partial derivatives to compute the partial derivative of the function…

A:

Q: 10

A: Given:

Q: .    Find work done by the gradient of f(x, y, z) = zexy- 2y2/z  along the path from (1, 2, 4) to…

A:

Q: a) Find the directional derivative of f(x, y, z) = x²yz + 4xz² at (1, -2, 1) in the direction 2i - j…

A:

Q: Find the maximal value of the directional derivative of w = xy°z at the point (2, 1, 1)

A:

Q: How is the linearization of f (x, y) at (a, b) defined?

A: given,          any function fx,y at a,b

Q: Explain why the function is differentiable at the given point. 4x f(x, y) = y + sin (0, 6) The…

A:

Q: Integrate f(x, y, z) = x2 -y + z over the line segments from (0, 0, 0) to (1, 1, 0)  and then to the…

A: According to the given information, it is required to integrate the function over the line segment.…

Q: 9(2, u, 2) sin(y) gx, y, z

A: The given function is: g(x,y,z)=x sin(y)z2

Q: 8. Find the directional derivative off at the p in the direction of v = () of f(x, y, z)=x²exx, p =…

A:

Q: How is the linearization of f(x, y) centered at (a, b) defined?

A: To define linearization of a function f(x,y) at point (a,b).

Q: Prove that the function ((z*)2 Iz| > 0 f (z) = z 0, z = 0 is not analytic at z = 0 by calculating…

A: Given function is f(z)=z*2z, z>00,        z=0 we have to prove, the above function is not…

Q: Find the gradient of the function w = xy2z2,  and the maximum value of the directional derivative at…

A: Given function:    w=xy2z2

Q: Answer the following using your knowledge of derivatives and Mathematica: a. Compute the first- and…

A: Solution a: Given function is f(x,y)=xexy Then first order partial derivative with respect to x is…

Q: Suppose that a duck is swimr - perature is given by the formı v²sin x. Find dT/dt in two chain rule…

A: Given query is to find the derivative of function T.

Q: ih fcuv, w) and its partial derivatives v and w with respecttou and and are Continows that u-x-y, V=…

A:

Q: Choose the gradient of the function f defined by f(x, y, z)= In[1 + (17xy + xz +9yz)²].…

A:

Q: Prove that div curl v = 0 where the components of v have continuous second partial derivatives

A:

Q: Compute the partial derivative of the function f(x, y, z) = e'-z cos(y) +ze-/(1+y²) with respect to…

A:

Q: Use chain rule to find given x² + 2y² + z² and x os (st), y %3D Cos

A: Solve the following

Q: Let z = z(x, y) be a continuous function of x and y defined implicitly by In(3æ* + 4y* + zyz*) = =…

A:

Q: Calculate the directional derivative of f(x,y)= In( Vx +y ) at point (1,1) in the %3D direction of v…

A:

Q: A function f has continuous second partial derivatives on an open region containing the critical…

A: Given: A function f has continuous second partial derivatives on an open region containing the…

Q: f(x,y)=ln(5x+4y). Find the directional derivative of f at (1,0) in the direction of the line…

A:

Q: The coordinates of a particle in the metric xy-plane are differentiable functions of time t with…

A: The distance of a point (x,y) and the origin is calculated by d=x2+y2 Given rate of change is,…

Q: Find all the second order partial derivatives of g(x.y) = x°y +5 sin (y) + 4y cos (x).

A:

Q: ih fcuv,w) and its partial derivatives with respect tou and v and w and are Continous that usx-, V=…

A:

Q: a) Describe the curve obtained when we make y = 2 and z = √2 . af b) Represent on this curve the…

A:

Q: what local linearizatoin of the function f(x,y) sin(xy) + cos(x/y) at (pie/4, 1)  (show all steps…

A:

Q: Show that following function xY (x, y) + (0, 0), f(x, y) = x² + y² (x, y) = (0,0) is continuous,…

A: To check the continuity of given function at origin firstly finding the limit of the function at…

Q: Suppose that f(z, y) = z*y°. The directional derivative of f(1, y) in the directional ( – 1, – 2)…

A: We can solve this as follows:

Q: 2. Let closed curve C consist of the line segment from (0, 0) to (2, 1), followed by the line…

A: We will solve both parts of this question using basic knowledge of line integrals and definite…

Q: Suppose that u = x²y, where x = r cos 0 and y = r sin 0. du Compute the partial derivatives du and…

A: Given that u=x2y  ….(1) where x=rcosθ  and y=rsinθ Differentiating (1) partially with respect to 'r'…

Q: Find the gradient of the function and the maximum value of the directional derivative at the point…

A: The solution is given as

Q: dg Use the Chain Rule to evaluate the partial derivative at the point (r, 0) = (2/2, 7 where g(x, y)…

A:

Q: Let f(x, y) be a function whose partial derivatives exist and are continuous at a point (a, b).…

A:

Q: What is the rate of increase of the scalar function f(x,y,z) = sin (πx/3) sin (πy/2) e^-2z from the…

A: See the attachment

Q: Find the gradient of the function f(x, y) = (x + y)/(y + 1),and the maximum value of the directional…

A: Given: fx,y=x+yy+1 The gradient of fx,y can be calculated as, ∇fx,y=i^∂∂xfx,y+j^∂∂yfx,y Here,…

Q: Let x and y be differentiable functions of t, and lets = √(x2 + y2) be the distance between the…

A: Given equation is s=x2+y2. Compute the derivative of s with respect to t.…

Q: Let F(r, y, z) = (y² + 2z, 2ry + 4, 2a), and let the curve C be described by x = t, y = t, z = t3…

A:

Q: Find the first partial derivatives of 2х — 4у - f(x, y) = at the point (x, y) = (2,2). 2х + 4y af…

A: