We know that: Theorem (*) If y₁, y₂ are linearly dependent on some interval, and their derivatives y'₁ and y', exist on that interval, then W[y₁, y₂] =0, there if W[y₁, y₂] =0, then: y₁, y₂ are linearly dependent on that interval? That is, the converse theorem of theorem(*) would be true. Study this situation in the following cases: Y₁ (x)=x² Y₂ (x)=x|x|, functions defined at -1

Q: Question 4 Find the gradient of the function at the given point. w = 2x - 4yz+z², (1, 1, −2) 21+…

A:

Q: Find the following statement is true? ' W[y1,y2] = 0 implies y1 and y2 are linearly dependent'

A:

Q: Explain why the function is differentiable at the given point. Then find the linearization L(x, y)…

A: Note that : Since you have posted multiple questions, we will provide the solution only to the first…

Q: Find the linear approximation of the function f (x, y) x² + y² at the point (2,-1, 1)

A: Given function:- fx,y=5x2+y2

Q: 3. Let f(x,y)=x²y/(x² + y²). (a) Show that f(x,y) goes to zero if (x,y) goes to (0,0) along any…

A: Solution: (a) If (x,y) approaches (0,0) along the straight line y = mx, then we can substitute mx…

Q: Prove that if two functions f(x, y) and g(x, y) have the same gradient, then they differ by at most…

A:

Q: Q2: Discuss the fundamental theorem for gradients using E = 5x2 i + 20 xy j+ 10 yz3 k and the…

A: Fundamental theorem of gradient states that a if  line integral through a  F gradient field depends…

Q: Let f(x, y, z) = x²y³ + zª and x = s², y = st², and z = (a) Calculate the primary derivatives - af…

A: fx, y, z=x2y3+z4  &  x=s2, y=st2,  &  z=s2t a). ∂f∂x=∂∂xx2y3+z4=y3·2x+0=2xy3…

Q: 3) A hot cup of tea has temperature T(1) = 198 °F, one minute after it has been sitting on the…

A: To calculate the local linearization of the temperature at t = 1.

Q: Find a vector parametric equation r(t) for the line through the points P = (4, -3, 4) and Q = (3,…

A:

Q: Minimizing the length of the residual vector r = b – Ax is equivalent of mini- mizing the quadratic…

A:

Q: If f(x, y, z) = x²y³ — 4x, find the directional derivative at (1, 1) (3,-4). in the direction given…

A:

Q: -3) The curve C has equation y = x³ - 11x + 1 a) Find the gradient at the point on C, where x = 3.…

A:

Q: 2. Show that e*, sin x, cos x are linearly independent using Wronskian Method.

A: As per the policy of bartleby the provisions is to solve only one question at a time  I am solving…

Q: z= f(x,y) ; x= r-s1 , y= r2s and zx(0,-1) = -24 ; If zx(0,-1)=-18, what is the value of the…

A: Given that z=f(x,y),     zx(0,-1)=-24 and  zy(0,-1)=-18x=r-s and y=r2s We have to find zr(0,-1).

Q: Determine whether the functions y, and y2 are linearly dependent on the interval (0,1). y, = e' 3t…

A: Take the Wronskian of these two functions function and if the value is not equal to zero then it is…

Q: f(x. y.z) = + xy + yz + Q8) Find the Linearization at (1,1.2)

A:

Q: please see image

A: Taking the cues from hint in the question:

Q: Let f(x, y) denote the elevation of boring national park, and suppose britbmw is on a slanted part…

A:

Q: aw for w = f(p, q); p = g(x, y), q = h(x, y) ax А) dw ax C) dw ax = = aw ax ах др aw др + dw да aw…

A:

Q: Suppose z is given implicitly by the equation x3y-yz2+z/x=4 in a neighborhood of the point P (1,…

A:

Q: 9. Find the direction of zero change for the following function at given point (Баллов: 2) xy + y,…

A:

Q: ? ? ? 1. f(t) = 4t² + 28t, g(t) = 4t² - 28t 2. f(0) cos(30), = g(0) = -8 cos³ (0) + 6 cos (0) 3.…

A: