9. Determine whether each of these functions from R to R is a bijection. а. f(x) %3D 2х + 4 b. f(x) = x² +1 c. f(x) = x³ d. f(x) = (x² + 2)/(x² + 3)

Q: If  x2 + y2 = z2, find dy/dt when x = 3, y = 4, dx/dt = 6, and dz/dt = 2.  (Enter your answers as a…

A: Solution: the objective is to find dy/dt

Q: 2- - Calculate the part a) using the given countour and assuming that R → o. For part b), use…

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Q: Q3: Prove that Σ(y₁ - y) = 0 AND Σfi(y₁ - y) = 0

A: The question is about proving given equation. To prove : ∑ ( yi - y¯ ) = 0∑ fi ( yi - y¯ ) = 0

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A: Sincewe can write every element of P2 in linear combination of f(x),g(x),h(x) so this functions…

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Q: Show that the two linearly independent solutions of xy" – y + xy = 0, x > 0 can %3D be obtained as…

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Q: Determine whether the following are linear trans- formations from R³ into R²: (а) L(x) 3 (х2, хз)"…

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Q: 2. Consider the following quadratic form; F(x,y) = ax? +2bxy + cy. Examine under which conditions on…

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Q: Show that o (z) = x² – x=1 is an implicit solution to z² = 2y on the interval (0, 0). %3D %3D

A: Here, the given equation is: x2d2ydx2=2y...(1) To show that ϕ(x)=x2-x-1 is the solution of (1) on…

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A: Consider the given equation.

Q: a²z If z = In(x² + y²)., find ax2 ay?

A: Here we have;  z=ln(x2+y2) ...................(1) partial derivative with respect to x

Q: 9. Show that the function (-, -) : IR² × R² → IR defined by setting = #1#2 + 2x12 + 2y1*2 + 8y1Y2 /2…

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Q: 1 Given the function g(a, b) = g(a, b) where suppose m %3D a²b+b a = cd b = c3 + 6d %3| дт The task…

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Q: Compute the Wronskian of the following equation. y" – 4y' + 4y = xe2x %3D

A: Given: y''-4y'+4y = xe2x

Q: Solve the following Cauchy-Euler D.E: x²y" – xy' +y = 2x2 + In x %3D

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Q: Let z = f(x, y) = 6(x + 3y - 2)². Find %3D 36x + 108y + 72 O18x + 54y O 36x + 108y - 72 18x + 54y -…

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Q: Save the following Sef of Lincar equations using eith 2=£1 over relaxation 10X, +X2 + X3 =12 X,+10X2…

A: Lower Triangular Component L* 10 0 0 1 10 0 1 1 10 Upper Triangular Component U 0…

Q: If after evaluating the Wronskian of the functions y1 and y2 the result is W=7x2+19 in the interval…

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Q: Draw the bifurcation diagram for à = (x + 1)(x² – 2.x +r).

A: We are given : Now, the bifurcation diagram for this is :

Q: 2/3 Determine the length of x=| for 0<ys2. Assume that y is positive.

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Q: () Let z = x + iy. Find Im and Im 2 in terms of x and y. Im Im z2 ||

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Q: Let D, be the linear transformation from C'(a, b] into C[a, b]. Find the preimage of the function.…

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Q: Compute the discriminant D(x, y) of ƒ(x, y) = x2y2.

A: fx, y=x2y2 Partially differentiating with respect to x, we get ∂f∂x=2xy2 Partially differentiating…

Q: Find the bounded อน əx solution of อน 2 + U if U(x, 0) = 6e-3x. at

A: We have ∂u∂x=2∂u∂t+u  The initial condition is ux,0=6e-3x Now, ∂u∂x=2∂u∂t+u can be written as…

Q: dz Find if yz – In z = x + y ax

A: To find the partial derivative of z of yz-ln z=x+y with respect x, differentiate yz-ln z=x+y, while…

Q: 2): Consider the DE y" – y" – 6y = 0 -

A: Follow the procedure given below.

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Q: 1. Let s, =2+j8 md -, = 4+j36. Show that a. -1+- = -+- 1 -1- -2 -1 -2 = -1-2 -1 : b. C d. =2 ||

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Q: 12. Show that F (x, y, z) = xy +xz + yz has the value 1 if and only if at least two of the variables…

A: This Question is From Boolean algebra where + sign denotes or Operation. So F(x,y,z)=xy + xz+yz if…

Q: (x) = 3x +12x+2, %3D ues of a, b and c such that

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Q: 3 ) If f:(1,00)→ [2, 00) is given by F(x)=Dx+, find f(x), (assume bijection).

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Q: Find the positive integer A such that t²x" +tx' + t²x = λx has a nontrivial solution satisfying x(0)…

A: Hint: Substitute t=ey , t2x''=D(D-1), tx'=D-1in the given equation and calculate to obtain the…

Q: . If 4=(x+2y+az, bx-3y-z,4x+cy+22) is irrotational. Find constant a,b,

A: Follow the procedure given below.

Q: Find the clomain for the following funclions:- -y=2X (x+11 b-ys a - ৮-9 (3-X) C. y= Jx +4

A: We find domain of following function

Q: 10 Consider A :3D and y:= Find all re R that minimize || Ar - y|l2.

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Q: Let f(x, y, z) = In(x² + z) + y²e#². Find fzyæ-

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Q: Find the Wronskian of a given set {yı, y2} of solutions of 6. y"3(x21)y- 2y 0, given that W(T) = 0.

A: Given the differential equation and the value of the wronskian at pi as:

Q: Find the Wronskian of a given set {y1, Y2} of solutions of (1 – a²)y/" – 2ry' + a(a+ 1)y = 0 when…

A: According to the given information it is required to find the Wronskian of solution set

Q: 4. Prove that (À × B) · (B x T) × (T x A) = [ABT)

A: The objective is to prove that

Q: (b) Let z = x + iy = re?". Find |z| and z² first in terms of x and y and then in terms of r and 0.…

A: Given:

Q: Prove that (X+Y) (X+Z) =X+YZ

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Q: If the functions f(x², y²) and g(x, y) are homogeneous of degree 2 and 4, f (x, y) g(x, y) 1…

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Q: In C[0, 1], are the functions {x2 , x3 , e2x} linearly independent?

A: Linearly dependent and independent functions: Two differential functions f(x) and g(x) are said to…

Q: Verify Young's Theorem z = In(2x + y) y? x2 +e(x+2y) y

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Q: 2. Use a theorem to find the ff. Zx if yz? + z = 5xy – 1.

A: Find out the zx when function is given as yz2+z=5xy-1 or yz2+z-5xy+1=0

Q: 2/ Compute fc E.dr and Where F Gxry2=(3+ 12x²,3xy²2 is a line segment from Co,01 to (4,2) %D Fdr =

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Q: D Reduce the Q.F 22,xz t 22,X3 2ズュズ3 weag ayf of oothogonal toans fermation by an Ans: y+92-295

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Q: Suppose that x2 = y2 = e and yz = zxy. Show that xy = yx.

A: Assume that x2 = y2 = e and yz = zxy.  Now from  yz = zxy we have  yz-1 = zxy-1z-1y-1 = y-1 x-1 z-1…