By taking the Laplace transform Y (x, s) of y(a, t), solve the boundary value problem for a >0 andt 2 0. given the boundary conditions: a) y(x, 0) = e- for all a > 0. dy b) at \t=0 0 for all x > 0. c) lim Y(x, s) = 0 for all a> 0. 800 d) y(0, t) = 0 for all t > 0.

Q: (1 point) Evaluate the line integral y dx + x dy where C is the parameterized path x 1 x = t², y =…

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Q: * Find (fzyx) for the function (f = sin(xy)* .cos(yz)* cot(xz) ) B) zycos(yz)-sin(yz) O A)…

A: Given: f=sinxycosyzcotxz To find: fzyx

Q: 2. f(x, y) = (x² + y²)² — 2(x² - y²) + 1. Ans.: global minimum 0 at (±1,0), saddle at (0, 0).

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Q: Determine whether the function xy²/³ + xy ³√x⁹ + 2y³¹ 1 √3' if (x, y) = (0,0) if (x, y) = (0,0) is…

A: Let us consider a function f then f is said to be continuous at c if f(c) is defined limx→cf(x)…

Q: Isin , it (z, y) # (0, 0) 2. Determine if a function f : R → R defined by f (r, y) = 0, if (z, y) =…

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Q: 9. Use the chain rule for multivariable functions to find where f(u, v) = u + v u(x, v, z) = x+y,…

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Q: 5. f(x, y) = (y — x²)(y — 2x²). - Ans.: no extrema, saddle at (0, 0)..

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Q: Let h(x, y): = x5 + y sin²x x² + y² 0, if (x, y) = (0,0) if (x, y) = (0,0) (a) Show that h is…

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Q: (x* + y° ) In (x² + y* ). (x,y)= (0,0) f(x, y) = k +1, (x, y) = (0,0)

A: If f is continuous then lim (x, y) tends to (0, 0) =k+1 And using l hospital rule.

Q: integrate f over the given curve f(x,y)= x3/y     , C: y=x2/2 ,    0<= x <= 5

A: NOTE: Refresh your page if you can't see any equations. . the given curve is  substitute x=t . the…

Q: Find the limit of the function f(x, y) = 4e10z–3y (x, y) → (0,0). as lim (x,y)→(0,0) 4e10x-3y

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Q: f(x, y) = xy(3-a-y) OA. (1, В. О в. (0, 1) O C. (0,0) O P. (-3,3)

A: Given function is f(x,y)=xy(3-x-y) At first we differentiate it partially w.r.t. x and y and setting…

Q: Let h(x, y): = x² + y sin² x x² + y² 0, if (x, y) = (0,0) if (x, y) = (0,0) (a) Show that h is…

A: Let us consider the function f(x.y). The limit is essential discontinuity if the limit of the…

Q: Let f(x, y) = sin (x - y) [x] + [y] ' 0, |x | + (x, y)

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Q: 2. Let u(x,y) = e* cos(y) for any (x,y) = R². Define B, ((0,4)) to be the ball whose center is (0,)…

A: a) Let ,where

Q: (d) S (1, y) = sin (2), (x, y) # (0,0) 0, (x, y) = (0,0) (e) , r + y # 0 7+y x² + y = 0 0, f (x, y)…

A: Limit of a function: A function f is said to tend to a limit l as a point x, y tends to the point…

Q: Let y: (0, 3] → R?, y(t) = (t,?V³) and I := ran y. Sketch T and compute L(T) :!!

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Q: Given a function f(x, y, z) = ryz° +z°yln (z) – z'y? + 4, find: of (1,2, 1) = (1, 2, 1) = (1,2, 1) =…

A: Given problem based on partial derivative.

Q: Let h(x, y) = x5 + y sin²x x² + y² 0, { 25 if (x, y) = (0,0) if (x, y) = (0,0) (a) Show that h is…

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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: Let f be a differentiable function such that f(h) f(x + y) = f(x) + f(y) + 2xy and lim = 15. Find…

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Q: A. What is the domain of h(x) (that is, on which x values is the functi B. Find h'(). C. At what x…

A: h(X) is defined by the integral of f(X). We need to find the domain, derivative and the relative…

Q: Let a differentiable function z = f(h²(x² + y3), 2łn(x +y)) satisfying the condition f(4,0) 1 be…

A: We need to find the partial derivative of z with respect to x for the given function.

Q: I grad(f)•dr where C is graph of y = x² + 3x from (0,0) to (2, 14) and f (x,y) = ev/lx*•1}

A: The function is fx, y=eyx+1. Find: ∫Cgradf·dr…

Q: Consider the following graph of y = f(t) which is composed of line segments and semicircles: y =…

A: Here in the question, the graph of y=fx is given which is composed to the line segments and…

Q: For F(x, y,z)=(6xsin y+10xz' )i+(3x cos y-4ye"-2y ze" ) j+(-2y'e* +15x²z² )k , find | a function f…

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Q: 1. (y-1) x²+4 Cy-1)² (x,y)# (0,1) f(ox,y)= %3D (x,y) = (0,1)

A: Explained below...

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…