Problem 2. Now consider the relation D' on the domain z+ where rD'y if and only if r|y. Use the properties of functions and the relation between them to answer the following questions and to reach a conclusion about (Z+, D'). 2(a) Prove that D' is reflexive. 2(b) Prove that D' is anti-symmetric. 2(c) Prove that D' is transitive.

Q: Show that the directiona f(x, y, z) = z²x + y ² at ( X (1/15" + ) ; 255. them

A: fx,y,z=z2x+y3 P1,1,2v→=15i+25j We know directional derivative Dv→fP=∇fP.v→v→

Q: 3. Suppose that the following information about a function f is known. It has continuous partial…

A: Answer well explained above

Q: ,2 For function f (x, y) x2+1 Evaluate the gradient at the point (1,2)

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Q: 9. The graph of f “(x) is given below. When is the function f(x) concave down? 4 3 1 5 -1 -2 -3

A: To find the correct option

Q: 10. A function f(x) has domain {x ɛ R} and range (y ɛ R, y 2 2}. What would be the domain and range…

A: The domain of f is the range of f-1 and the range of f is the domain of f-1. Here domain of f(x) is…

Q: 32.A point of symmetry of the function f where f(x) = a) (3,-4) b) (3,5) c) (-3,-4) d) (-3,4) 1 - +…

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Q: Graph these parametric equations on the TI and include screenshots x = 2t – 2 sin t, y = 2 – 2 cos t…

A: In the Cartesian coordinate system, the parametric equations use a third variable(usually "t") to…

Q: - Given f(x) is graphed below, sketch f(x) on the coordinate plane with f(x). -1,2) -5- (1, 1)

A: Consider the points on the graph of f(x) are (-1,2) and (4,1). The objective is to sketch the…

Q: . Eliminate the arbitrary constants c1 and c2 from the relation y = Cje-2« + Cze&e а. у" -бу - у -…

A: Solution : We need to find out the arbitrary constants c1 and c2 from the relation y = C1e-2x +…

Q: Let h = fog where ƒ : R² → R and g: R² R² is defined by g(x, y) = (3x²y, x – 2y). Find the…

A: Letandfind the derivative

Q: 9. What is the domain of the function f(x) = Vx- 3? | A. (3, 0) B. (-00, 3] С. [3, о) D. (-00, 3)

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Q: 10. Let F(x, y) = 1 + v4 – y². (a) Evaluate F(3, 1). (b) Find and sketch the domain of F. (c) Find…

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Q: 3. Prove properties (3) and (4) analytically, by making x = -t in the part of the Integral from -a…

A: Concept:

Q: 5. Determine algebraically whether the function f(x) = -3x5(4-9x2) ¹2 is even, odd or neither.

A: "According to the guidelines issued by the company We are supposed to answer only one question.…

Q: 7. Which of the following sets of functions is linearly dependent on R0,2? (a) {6, z, sinz} (b) {1,…

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Q: 4. Let f-(x) = g(x) = x-5, k-1(10) + 7 = 10 and h(x) = then find: b. Domain of h(x). c. (hof)(2). (Z…

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Q: Suppose f(x) is a function with the following properties: • ƒ"(1) = ƒ"(2) = ƒ”(4) = 0, f"(x) > 0 for…

A: Let's solve for inflection points at which points.

Q: A function is defined as f(x,y) = 25xy + 3x³ - 7x² sin(y) aj ay a2f (ii) Find and 1325 3²x (iii)…

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Q: u+iv is analytic function, under what condition (if any) v+iu

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Q: 12. Let f(z,y) = < then f(1,2) = (b) 글 (d) 금

A: Given :fx, y=x-yx+y-1

Q: 4. Let f(x) = g(x) = x -5, k-(10) +7 = 10 and h (x) = find: ther b. Domain of h(x). c. (hof)(2). (Z…

A: (NOTE:   As nothing is mentioned, so all functions are taken as real valued functions.)

Q: 6. The domain of y = v1 – x? is

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Q: Suppose (-8, 6) is a critical point of a function f with continuous second derivatives. If fra (-8,…

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Q: 4. Make a table for each function in Question 3. a) у3 3х- 1 b) y=-0.5x + 2 c) y= 2x y y y 1 1 1 2

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Q: The inverse of function f(x) = x3 + 2 is а) f-1 (y) %3 (у - 2) 1/2 b) f-1 (y) = (y – 2) 1/3 c) f-1…

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Q: (6) If f(z) is analytic function show that: [v] + [0] =

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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.