a. Use the Chain Rule to find the indicated partial derivatives. z = x4 + x2y, x = s + 2t − u, y = stu2; ∂z ∂s ∂z ∂t ∂z ∂u when s = 1, t = 2, u = 3 b. Use the Chain Rule to find the indicated partial derivatives. w = xy + yz + zx, x = r cos(θ), y = r sin(θ), z = rθ; ∂w ∂r ∂w ∂θ when r = 8, θ = pi/2 c. Use the...
Du (b). Use the chain rule to find and y = scost, z = s sint when s = 1 and t=0 Ow at where w= = 22 + y2 + 22, r = st,
(b). Use the chain rule to find aw and as y = 8 cost, z = s sint when s= 1 and t=0 aw at where w = = 22 + y2 + z2, x = st,
Exercise 3. Chain rule (15 pts) Let f(x,y,z) = xy +z-5, x = r + 2s, y = 2r - sec(s),z=s af Then is: ar a. r - sec(s) b. sec(s) c. r+s+sec(s) d. 4r + 4s - sec(s) a. b. d.
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
3. (a) Find the partial derivatives (with respect to r and s) using the chain rule:[express the final answer in r and s only ,y= r2 +In(s) and z-2r wx2y +z2 ; where x (b) Find dt if f (x, y) = xy + z; where x cos t ,y = sint and z = 3t2
aw 4. Find when (r, s) = (1, -1) if w = (2+y+z)?, r=r-s, y = cos(r +s), z = sin(r +s). ar 5. Find the directional derivative of f(x, y, z) = 3x² + yz + 2yz? at P(1,1,1) in a direction normal to the surface x2 – y + z2 = 1.
2. Given R(x,y, z, w, k, t). There are two keys: (x,y) and z. Given the following functional dependency: F = { {x,y} {z,w,k,t}, z {x,y,w,k,t }, yt}. Is R in 2nd normal form? Justify your answer. 3. Given R(x,y, z, w, k, t). There are two keys: (x,y) and z. Given the following functional dependency: F = { fd1:{x,y} {z,w,k,t}, fd2: z {x,y,w,k,t }, fd3:k x}. Is R in 3rd normal form? Justify your answer....
ANSWER BOTH PARTS » Suppose that u depends on z and w, while z and w depend on x and y, where both x and y depend on t. Draw a tree diagram and write the chain rule for computing อน ot ·Explain why the direction of maximal ascent coincides with the direction of the gradient » Suppose that u depends on z and w, while z and w depend on x and y, where both x and y depend...
4.let U= {q,r,s,t,u,v,w,x,y,z}; A= {q,s,u,w,y};and C={v,w,x,y,z,}; list the members of the indicated set , using set braces A'u B A.{Q,R,S,T,V,X,Y,Z} B.{S,U,W} C.{R,S,T,U,V,W,X,Z} D.{Q,S,T,U,V,W,X,Y}