5. Let g(t) be a smooth function and u(x, t) be a solution of the following problem: Uų = k?u 0 < x < L,t > 0, u(0, t) = g(t), t > 0, u(L, t) = 0, t > 0, u(r, 0) = f(x), 0 < x < L. Prove u(r, t) decays to 0 if g(t) → 0 as t→ o.

Q: Find f'(2), where f(t) f'(2) = = u(t) · v(t), u(2) = (2, 1, −2), u'(2) = (7, 1, 8), and v(t) = (t,…

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Q: (x, y) = (b) 1 1 (d) 1/2 Does not exist. (x, y) = Ty³ x² + y6 = Find limf(x,y) along the path x =…

A: We will find the required.

Q: IT ) Verify that ,0) is a critical point of f(x, y) = y² - 2y cos x. Then use the second derivative…

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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: Q10

A: Given F(u,v)be a function of two variables. let Fu(u,v)=G(u,v) and Fv(u,v)=H(u,v) Here f(x)=F(u,v)…

Q: 5. Find the linear appoximation for the funciton f(x, y) = (ro, Yo) = (3,0). x-y xy+1 at the point

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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: (-4,4) 1 -5 1 -2 (-1,0) -3 -1 -2 2 3,5

A: Note that we have common data only for (-4.4). For others, we don't have it. So please provide other…

Q: 2. Suppose that {f(t)} = F(s) with F a differentiable function and that {f(t)} exists. Then L{-te*…

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

A: We will solve both the parts using the basics of integral calculus, in particular double integral,…

Q: Find the linear approximation of the function f (x, y,z)= x' Vy +z° at the point (2, 3, 4) and use…

A: Given:-               Find the linear approximation function f (x,y,z)= x 3y2+z2 at the point…

Q: In problem (A), find a linear approximation f(x, y) at the indicated point. (x + y)´| at (-1,1) A)…

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Q: 4. Suppose that the following information about a function f is known. It has continuous partial…

A: To find the gradient of the function  f  at the origin given the directional derivatives, we need to…

Q: 1. Find the critical points of f (x,y) = 8y³ + x³ – 6xy and use the second derivative test to…

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Q: The function f(x) = |ax - b| + c|x| : x in (-∞, ol where Find the

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Q: f(r, y, z) = sec='(9x + 3yz) f. = fy = f; =

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Q: 26. Evaluate the gradint of f (x,y,₁z) = log (= ² + y² +₂²) at (1₁91)-11

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Q: Find the saddle point/s of the function f(x,y) = x + y² +x²y+4 O a. (E ,-1) & (- 2 -1) O b.(4,0) &…

A: Let's find.

Q: Valero's 5k Fun Run was held yesterday in the downtown district. The top runner 4 runners (one of…

A: Given 1. Ralph finished 1 minute after the competitor in the white shirt. 2. The competitor in the…

Q: (ii) Suppose z = f(x, y), where x = g(s, t), y = h(s, t), g(1,2) = 3, gs (1, 2) = −1, gt(1, 2) = 4,…

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Q: Let f(x, y, z) = x + yz and let C be the line segment from P = (0, 0, 0) to (4, 5, 5). (a) Calculate…

A: Given function is  and let  be the line segment from . We have to calculate  and for , .Since,…