f(9(x))g'(x)dx = / f(t)dt. g(a)

Q: V1– 3x, if x -1 2x +1 Show that f is continuous at x = -1 is differentiable there by computing the…

A: See below the calculation

Q: Give examples of functions that do not have any fixed points, but do have these char- acteristics:…

A: We know that a function f:A→A is said to have a fixed point if there exists x∈A such that fx=x. In…

Q: Consider the set C = {(x, y) = R² | y² = x(1-x²)}. Is C a curve? If yes, why yes? If no, why not?

A: We shall plot the given curve y2=x6(1−x2).

Q: 1. Let f : Z6 → Z6 be such that f(a) = x2 + 3. (a) Is f a well defined function? (b) Is f…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub parts for…

Q: Using modifications of these functions, construct a functionf : R → R so that (a) Df = Zc. (b) Df =…

A: Consider the given all information.(a)Df=ZcThe domain of the given function is compliment of Z.…

Q: Let f be a real valued function defined on [a, b] and k ∈ ℝ . Suppose f is differentiable at a point…

A:

Q: Discuss, in detail, the continuity of the function f(x, y) for the whole domain making detailed…

A:

Q: Prove that if the function f : I → R has a bounded derivative on I, then f is uniformly continuous…

A:

Q: Let f: IR be a function. (a) Prove that if f is differentiable at some c I for which f(c) ‡ 0, then…

A:

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.