Show that the function is differentiable by finding values of ε1 and ε2 as designated in the definition of differentiability f(x, y) = x2y , and verify that both ε1 and ε2 approach 0 as (∆x, ∆y)→(0, 0).

Q: Assume that x and y are both differentiable functions of t and are related by the equation…

A: topic - derivative of a function

Q: Assume that x and y are both differentiable functions of t and are related by the equation…

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

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Q: Suppose that z is a differentiable function of x and y with dz (2, 5) = 108 and- dx dz (2, 5) = 9.…

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Q: Let x and y be differentiable functions of t and let s = 2x2 + y2 be the distance between the points…

A: Differentiate both sides with respect to t.

Q: Suppose that z = xy² and x and y are differentiable functions of t with x = 1, y = −1, dx/dt = −2,…

A: Given  z=xy2 Also at t=-1 x=1y=-1dxdt=-2dydt=3

Q: Let A E R"Xn, let b e R", and let n n f (x) = xT Ar + 6Tx = Aikr;xk +> b;x;. i,k=1 i=1 Show that f :…

A: Definition: A function f:ℝn→ℝm is differentiable at x0 with derivative M if M:ℝn→ℝm is linear and…

Q: Assuming that z is a differentiable function of x and y, Let F(x.y.2) = In(x³y) – e" -tan-z. Find F,…

A: We can find the answer as below.

Q: Let f(x) = e2sinz 1. Use the local linear approximation of f at x = 0 to approximate e2 sin(0.01

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Q: Find and by implicit differentiation, and confirm that the - dy results obtained agree with those…

A: The solution is as follows

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

A: Given that, s=x2+y2

Q: Differentiate f(x) with respect to g(x) of yx=xsiny

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Q: 4/(A)Find the derivative of f(x, y) = xe + cosxy at the point (2,0) in the direction. дz əz (B)If Z…

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

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Q: Find the linear approximation L (x) to ƒ (x) = xcosx at a = T.

A: Given that, f(x) = x cosx and we have to find linear approximation L(x) at a = π

Q: Suppose f(x, y) =                             ((xy2) / (x2 + y4)), for (x, y) ≠ (0, 0)…

A: The given function is,

Q: Suppose F(x,y)=sin²(xy +e"), where x=u² +v and y = In(u² +v). Find the . ƏF ¸ ÔF value of when u =1…

A: The given function is: F(x,y)=sin2xy+eywhere:x=u2+v         and         y=lnu2+v

Q: If x, y, and z are lengths of the edges of a rectangularbox, then the common length of the box’s…

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Q: Assume that x and y are both differentiable functions of t and find the required values of dy/dt and…

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Q: ) Find the linearization of f(x) = sinx at x = 0.(b) Compute dy if y = x2 +x and x changes from x =…

A: Since you have submitted two questions, we'll answer the first question. For the second question…

Q: Find the linearization L(x)of the function at a .

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Q: of Let f (x,y)=tanh-1(x-y) with x=e" and y= usinh (t). Then the value of evaluated at the point dt…

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Q: tion z2 – (x2 + y2)2 and let p = (0, 0, 0). ermine the set Tp(Zf ) of all tangent prs to Zf at p. Is…

A: Sol

Q: Assume that x and y are both differentiable functions of t and find the required values of dy/dt and…

A:

Q: Find the (absolute) maximum and minimum values of f(x, y) = sin x cos y on the square R = {(x, y)|0…

A: Given,            f(x, y)= sinxcosy                    (1) on the sqaure             R=x, y| 0≤x≤2π,…

Q: Let z = In(x² + y²) be a differentiable function of x and y, where x = sin t, y = cos t. dz when t =…

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Q: 3.Find the linear approximation to f(x) = cos x at x, = ÷ and use the linear approximation to…

A: We will use formula for linear approximation.

Q: Assume that x and y are both differentiable functions of t and find the required values of dy/dt and…

A: Given : Equation x2 + y2 = 25 and x and y both differentiable functions of t.   To find : (a)…

Q: Calculate of of and for the function f(x, y) defined as : ax' dy sin (xy) 1 + y f(x, y) =

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Q: Assume that x and y are both differentiable functions of t and are related by the equation…

A: Given, y = cos2x We have to find dy/dt when x=π/4, given dx/dt=−4 when x=π/4.

Q: Let f (x, y, z) = ln(x2 + z) + y2 exz. Find fzyx

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Q: Assume that x and y are both differentiable functions of t and find the required values of dy/dt and…

A: partial differentiation

Q: prove that xy^2/x^2+y^2 is not differentiable at 0,0 when  xy^2/x^2+y^2 if (x,y) not equal to (0,0)…

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Q: (i) Find the linear approximation of the function f(x, y) = 1 - xy cos πy at (1, 1) and use it to…

A: Here, the given function is: f(x,y)=1-xycos πy To find the linear approximation of the function. For…

Q: Express in form y=f(x) by eleminating the parameter x = ln t , y = 2 − t

A:

Q: Suppose that z is a differentiable function of x and y with dz (1, 1) = 162 and(1, 1) = 6. dz ду dx…

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Q: he tangent plane to the function f (x, y) = cosh x + cos(ny) + e*+1)y at the point (0, 1, f(0, 1))…

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Q: b) f(x): 1 + sec2

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Q: Suppose that z = f(x, y) is differentiable at the point (27, 81) withf,(27, 81) = 2 and f, (27, 81)…

A: Given that:                            z=f(x, y)fx(27, 81)=2fy(27, 81)=-6x=t3y=t4

Q: The force exerted by our home planet is GMx if x < R, R3 h(x) = GM if x 2 R, measured on a unit mass…

A: In the question we have to identify if the function is continuous.

Q: xY 2. (a) Prove that the function f(x, y) is differentiable everywhere in its do- %3D x2 + y? main.

A: Solution is given below:

Q: Find the antiderivative F for f(x) = | sin z|, -T <x <0 satisfying F(0) = 1. (1) F(x) = - cos x – 1…

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Q: 4. Given f(x, y) = arctan(x + 2y). Find the local linear approximation of f at (1,0) and use it to…

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Q: Assume that x and y are both differentiable functions of t and find the required values of dy/dt and…

A:

Q: Suppose that z is a differentiable function of x and y with dz (3, 6) дх dz = 36 and (3, 6) = 2. ду…

A: This question is related to differentiation, we will solve it using given information.

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: (a) Show that y = x* is concave upwards at the origin and y= e* is everywhere concave upwards.

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Q: Suppose that z is a differentiable function of x and y with dz (8,7) = 80 and dx dz (8, 7) = 2. ду…

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Q: Assume that x and y are both differentiable functions of t and find the required values of dy/dt and…

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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