1. A natural cubic spline S.on [0,2] is defined by (S.(x) S(x) = {x)=1+ 2x-x s,(x) = a+ b(x- 1) + c(x - 1) + d(x- 1) if 1sxS2 if Find a, b, c, and d.

Q: 3. Find the maximum and minimum values of a 1 āz - on |z| ≤ 1 (where a with la| < 1 is fixed)

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Q: For u(x, t) defined on the domain of 0 <x< 2n and t 0, solve the PDE, a?u a?u a*u +t + cos(2x) +…

A: Given: To solve: ∂2u∂t2 = ∂2u∂x2+ ∂4u∂x4 + t + cos2x +sinxcost

Q: 5. A clamped cubic spline S on [0, 2] is defined by So(x) – 1+ Ar + Bx - 23, 0<x < 1, S(x) – S1(x) –…

A: Using the conditions of cubic spline,

Q: 3) Find the general solution by using reduction of order method to ty" – (t + 1)y' + y = 0 given…

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: 2) Use Green's Theorem to compute f F df where F= (3x +4y)i + 2xyj and C is the triangular path from…

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Q: (b) State (without proof) the dimension formula for the linear transforma- tion L:V → W where both…

A: Dimension theorem

Q: (1.-13) Show that the line integral (2xy+3)dx +(x² - 4z) dy-4 ydz is independent of path and (0.0,0)…

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Q: In a cubic spline approximation assume that we know that S,(r) = 5 4r– 2² +3x³ S+1(x) = a + br + cr²…

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Q: 15.4.59 Assume that F(x.y,z(x,y))3D 0 implicitly defines z as a for xyz + 8x + - 4z = 0. dz

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Q: 6. Let u (r, t) be the solution of the initial-boundary value problem ut = 9uzz 0 0, %3D uz (0, t)…

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Q: 1. A function f(x, y) is homogeneous of degree 7 in x and y if and only if , 2. The function f(x,…

A: Topic:- calculus

Q: 1. A function f(x, y) is homogeneous of degree 7 in x and y if and only if, 2. The function f(x, y)=…

A: We have to find degree of homogeneous function

Q: 4.Determine the correct Wronskian setup for v, and v, for the equation y" – 3y + 2y = 10sin(x) e 2x…

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Q: -8 6. Evaluate the integral +y² dydx

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Q: (b) Solve the following IE u(x) = f* (z + 1²) dz by "Adomian Decomposition" method.

A: Solution:

Q: Determine Taylor's formula of order 1 and center a = (0, 0) of f(x)=ex1sin(x2).

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Q: What is the minimum order of the Taylor polynomial centered at 0 for cos x required to 10-42

A: Given,        cos(-0.35) with an absolute error no greater than 10-4.

Q: 2) Use Green's Theorem to compute f F. df where F=(3x +4y) i + 2xyj and C is the triangular path…

A: Use green's theorem

Q: 2) Use Green's Theorem to compute f F df where F=(3x + 4y)i + 2xyj and C is the triangular path from…

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Q: Please solve

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Q: (1. 1. 1) Evaluate the integral (5xy + 22°) dx + 5xf dy + (12z + 1) dz. (0.0.0) (Hint. Try the path…

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: Q3: Find the Wronskian for the following solutions. Y, = 1, Y =In

A: Wronskian= 112ln1+x1-xd1dxddx12ln1+x1-x= 112ln1+x1-x01x+11-x=1x+11-x

Q: 1. Find the maximum of the function: f (x) = 4sinx (1+ cosx ) a. Using Golden Section Search with…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: 3. For Z = x²secy,prove that Zxy is equal toZyx

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Q: Let C be the shortest path that connects a to ni and I = . (sinhz + 1)dz %3D O I-sinh rt+ T(i-1) O…

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Q: 8.Z. Show that the set S = { (x, y) € R² : y = sin (1/x), x # 0} U (0, y) : –1 <yS 1}, is connected…

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Q: 1. The loop of y² = x * (4 - x) 2. у %3D 6х — х?, y = x² – 2x - 3. у %3D 2х y2 + 4x – 8 = 0 - 4. An…

A: Question 4 : Arch of y = (1/3)sin(x/3) First find out intersection points :  y = 0 (1/3)sin(x/3) = 0…

Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: H.W/ using convolution theorem to find L-1

A: We have to find the Laplace inverse by using convolution theorem

Q: Let C be the shortest path that connects t to ni and I = J. (coshz + 1)dz. Then %3D O I=-1-cosh t+…

A: Since you are asking multiple questions. We are answering only the first questions. Let C be the…

Q: 5. Compute the 1st and 2nd degree Taylor polynomials L(x, y) and Q(x, y) for f(x, y) = x /ỹ at…

A: Given function is fx,y=xy Taylor polynomial at point a,b is…

Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

A: We have to use the Euler method

Q: Chapter 7, Section 7.7, Question 012 Use the inequalities JEM| = ƒ (x) dx 24n2 (b - a) K2 |Er| = f…

A: Given that the integral is ∫0414x+1dx. a) It is required to find the upper bounds on error obtained…

Q: 11 ComputePydz - zy'dy, nhere Dis descnbed by z² + y²s1. -1/2

A: Given an integral : ∫∂D x2y dx - xy2 dy  where D is : x2+y2≤1 We need to compute the integral.

Q: Expand xy² + 2x – 3y in powers of (x + 2) and (y – 1) upto third degree | terms.

A: Given ,  f(x,y) = xy2 + 2x - 3y  By Taylor's theorem  f(x,y) = f(a,b) + {(x-a)fx(a,b) +…

Q: 15 5) Find the area bounded by y= 2+ 7x+ 12 y = 0 x=1,and x = 2 Use partial fractions.

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: 2. If the Wronskian of y1 and y2 is e2" (x² + 5) and y1 = e2", find y2. %3D

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Q: 8. Use the Chain Rule to find az/as and az/at. z = (y-2x)³, x = s³t², y=s+t²

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Q: 7] The Wronskian of the functions 1, x, and e -2x is (а) е' -2x (2x+1) (b) 4e-2x (c) -e(2x +1) (d)…

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Q: 4. Expand f (x) = 1 around a = 0, to get linear, quadratic and - 1-x cubic approximations.

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Q: 1. Find the optium path for f=xy³ from (2,1,2)

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

A: Here we have to use Euler's method to obtain a four-decimal approximation of the indicated value.…

Q: Let f(2)= = csc(TZ) 22 Determine all isolated singularities off and find the residue at each…

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Q: A clamped cubic spline for a function f on 0,3 is given by [1+ax² -2x', S(x) = . [1-2(x-1)+b(x-1)² +…

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Q: Let C be the shortest path that connects a to zi and =J. (coshz - 1)dz. Then I=0 |=-sinh n+ n(1-i)…

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Q: 16.1. Use an antiderivative to show that for every extending from a point 21 to a point 22 2" dz =…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.