5. Find the inverse of each of the following bijections. (а) и: Q — Q, u(x) = 3x – 2. (b) v: Q – {1} → Q – {2}, v(x) = . I-1 (c) w: Z → Z, w(n) = n+ 3.

Q: Find R(t) if R′(t)=⟨1+2t,2e2t⟩ and R(0)=⟨1,3⟩.

A:

Q: 4. Let aj = 9. and b = 63 , a2 = 45 (a) Write b as a linear combination of a1 and a2. b = aj + a2.…

A:

Q: I| (2 + x²y)dA, R = [0, 1] × [–1, 1]

A:

Q: The domain of the function f (x, y) x2+5x+1 is x2 -1 A R? - { (1,y)} B R2 – { (1,y), (-1,y)} C R2 -…

A: Let f:D→E be a function then domain of f is D. Moreover, domain of f is where the function f is well…

Q: For 3 and 4given the translation rule: T(x,y) (x+2,y-3)

A: Find the image of (5,-1) and preimage of (-6,7)

Q: ? 4 1. F(x, y) : = x (7) x? +y° ° x2 + y? ? ; 2. F(x, y) = ? ਦ ਵਿੱਚ) 4 3. F(x, y) = (x, y) y A…

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Q: Evaluate f (x + y - z)² dV for the box B with dimension: [0, 8] × [0, 7] × [0, 3]. B (Use symbolic…

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Q: A = {x € Rx ≥ 5 and x 8 and r < 12}

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Q: V(12r), v°(-8r)

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Q: If f(z) = z² − (z)² + zz, calculate the following: a) u(x, y) and v (x, y) b) u(-10,-5) and…

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Q: F.T. 3) If X(t) + y(t) = x (2t - 1) =it W+1 Ans: Yow) = [4+jCW+1)]² X(w) and X (w.) = -j(w+1) e 2…

A:

Q: Which of the following functions V(y₁, Y2) is a Lyapunov function for the dynamical system with…

A:

Q: a(k) = eak coshw.k coshw, k ewok te-wok 2 eak coshw.k eak ewokteake-wok 2 X(Z) 3D글 z(enk e.k) + 글…

A:

Q: b) (i) Show that dz is a constant independent of z, and R. { Hint : z= Z, + Re"} 1 +- 2z-4 2 -z 3…

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Q: Find curl (3z i + 4x j + 6y k). curl (3z i + 4x j + 6y k) = ( i )i + ( i )j + ( i )k

A:

Q: Given f(x, y) 90x + 80y - 2 – 4y? – xy, and evaluate f(8, 5). f(8, 5) = %3D

A: Given : fx , y = 90x + 80y - 2x2 - 4y2 - xy  To fine : f8 , 5 Solution : For value of fx , y , put x…

Q: [x +1 0 2 1 2 (3)if A = 3 1 and B = 3 y -5 4 -1 -5 4 z - 4. Find x,у аnd z.

A: If two matrix A and B are equal then Each matrix of A and B has the same number of rows. Each matrix…

Q: Find f f(x,y) dA, where f(x, y) = x and R = [5, 10] × [-5, –4]. SSR f(x, y) dA=

A: The given function is f(x,y)=x, and the domain is x,y∈5,10×-5,-4. To Evaluate: ∬Rf(x,y)dA.

Q: If f(x,y) = In(x?y² +3x) find- дудх O a. 12x?y (x²y² + 3x)? O b. 6x?y (x2у2 + 3x) Ob. 6x²y (x²y² +…

A:

Q: 30z2 – 23z + 5 Evaluate J (2z – 1)²(3z – 1) dz, C : |2| = 1 (counterclockwise). |

A:

Q: Write f(z) = Im(z - f(z) = u(x, y) + iv(x, y). 3z) zRe(z²) - 5z in the form ○ a. (4y+x³ − xy²) +…

A: Step 1Step 2Step 3 : Step 4

Q: 2 + v - , 2+z+ 11. 0. di

A:

Q: d d Calculate[ri(t) r₂(t)] and[r₁(t) × r₂(t)] first by differentiating dt the product directly and…

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Q: 1-t2 -i+ t+2 j+ In(t + 2)k. Find the domain of r(t) t2+1

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Q: If (x+iy)³ = u + iv, then show that

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Q: Q4 If z = x In(xy) a3 z a4z Find and əy²əx əx²əy²

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Q: d d Calculate[r₁(t) r₂(t)] and [r1(t) × r₂(t)] first by differentiating dt dt the product directly…

A: give r1(t)=9ti+2t2j+6t3k and r2(t)=t4k So  r'1(t) =9i+4tj+18t2k and r'2(t)=4t3k

Q: Find det (T) for T:R- R, where T(x,,×2,*3) = (x,- x2, X2 - X3, X3-x).

A: We will find the matrix [T] from the image of the standard basis vectors of R³ under the…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.