6. Let 8(t) be the Dirac-delta function. The position u(x, t) of a vibrating string which satisfies Up = 8(t – 1) + 8(t – 2), ) < x < L,t > t, Utt - u(0, t) = u(L, t) = 0, t > 0, u(x,0) = u,(x, 0) = 0, 0 < x < L. (a) Find the series solution. (b) Does the solution decay in time? Explain the physical interpretation of your result.

Q: If (x, y) moves from (-2, –1) towards (1, 5) for a distance 0.02, then which of the following is…

A: fx,y=exyPoints from−2,−1 towards 1,5 for a distance 0.02

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: In Exercises 61-67,firstfind f' and then find f.

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Q: *) Use the Wronskian to determine whether the functions y₁ = sin(3x) and y₂ = cos(4x) cos(4x) are…

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Q: The Domain of f (x) = V1 – x is 1. [-1, 0) 2. (-∞, 1] 3. (-0, 1) 4. [1, c0)

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Q: Suppose that y is a function of u, u is a function of x, and u = -1 when x = 2. If du/dx|x=2 = 3 and…

A: Solution

Q: Which of the following most accurately de- scribes the behaviour of f. 1. local maximum at (2, 0) 2.…

A:

Q: What is the B of f(x, y) = tan (x3³+ 2xy)? [sec ² (x²y³+ 2xy] (3x²y² +2x) [sec² (x²y³ + 2xy ] (+²+*+…

A:

Q: = 0. c) y(0) = 1, y'(0) = 0, y"(0) = -1. Is the set of functions linearly independent or linearly…

A:

Q: -3t k Compute the first, second, and third derivatives of r(t) = 3ti + 5 In(t)j + 5e

A: r(t)=(3t)i+(5lnt)j+(5e-3t)k   We are required to findFirst derivative Second derivativeThird…

Q: The graph of the function below intersects at t axis on the interval [0, 4rt] at points f(t) = cos t…

A: We know that f(t) = cos(t) It cuts the t axis at the points π2,0 , 3π2 , 0 , 5π2 , 0 , ...

Q: f (x, y) ~ bo + b1,0(x − 1)+b0,1(y−2)+ b2,0 (x - 1)² + b₁,1(x − 1)(y-2) + bo,2(y − 2)² +…...

A: We find the Taylor series of the given function

Q: Q2// B :- If u+v3e"cos y and u- v=e' sin y find the Jacobain of the function alu,v at dx.yl (x.y)…

A:

Q: (x + 3)³ 12 16 11 f(x) Consider the function 4(х — 1)2 4 1 (x – 1)2 4. Note that - (x – 9)(x + 3)²…

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Q: 1 Find the partial derivative with respect to y of the function (2 x – 3) (3 y + 1)i el-5 z+5). -5…

A: To find the partial derivative of the given function with respect to y (2x-3)1/4(3y+1)1/4e(-5z+5)

Q: f(x,y) = –x² + 2xy + 3y² – 6x – 2y – 4 | - Serialize the function around the point (-2,1). a) f(x,y)…

A:

Q: Let a(t), 2 < t< 8 be a positive function. Also, let the function 8(t) be defined as a'(t)/a(t) on…

A: We will find out the required expression.

Q: Let ƒ (x, y) = x³ − x² − 8x + 2xy − - - 12y². (a) Compute all first order and second order partial…

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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: Which conditions make (a, b) a critical point (stationary point) of f(x,y)? What does the sign of…

A: What is Critical Point: A critical point is a point lying in the domain of a function where the…

Q: 0, 1) Knowing that u(t-a): |1, t<a =H¸ (t)=H(t-a)=u(t-a)=u̟ sketch the following tza functions. 1-a)…

A:

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: Calculate the divergence of cach of the following functions (a) r²î + y²§ + 2²k. (b) yzî + xzj+ xyk.…

A: As per the instruction, we will answer only the first 3 subparts. For (a), For a given vector fx, y,…

Q: (2) Consider the function f(x, y) = -x* - ²- 3 + ry³ + 5. (a) Find the critical points of f. (b)…

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Q: Locate and identify the critical points of f(x, y) = (x²-1)²-(x²y-x-1)²

A: The given function is fx,y=-x2-12-x2y-x-12.   The critical points can be found by solving the system…

Q: Consider the function F whose criterion is = / dt. 1+ t2 F(x) sena When determining dF/dx the result…

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Q: 3. Find f. (g) f'(x) = 2x – e

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Q: 5- Verify that the given function u is harmonic in an appropriate domain D. Find v(x,y), the…

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Q: Consider the following graph. Let u-(t) = u(t – 7) be the unit step function shifted 7 units to the…

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Q: Transform the function y=(1)/(2)x right 6 and vertical shrink of (1)/(4)

A: The given function is y=12x. To transform the function rightwards by 6 units and to shrink…

Q: 1.16. (a) If f and g are differentiable scalar functions, show that V(fg)=fVg+gVf.

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Q: z} 0,1} definition for f: X→ A that is a well-de- (0, w), (-1, w), (-2, w)} , (0, w), (-1, w), (-1,…

A: We have to choose the correct option from the given options according to the given information

Q: Show all work! Find the maximum, minimum, and saddle point of f(x,y) = cos(x)+ (1/y^2 + 1/x)+1

A: We will use the basic knowledge of 2-variable calculus to answer this question.

Q: Let f(z) = r? – 9z + 5 with domain 0 <r < 4, 5. Calculate (f ')(-3). Note: Round to two decimal…

A: