We consider the subintervals

to evaluate L₄(f). By definition we obtain

For

we have

Hence, the area can be calculated with the formula

Hence the right answer is D.

Starting with the trial solution 

We obtain 

Hence the general solution is 

The initial conditions gives us

and by this equations we have 

Thus the solution of IVP is 

For an integrable function f on

the definite integral can be written as

Considering this we can write the question as

So;

So the correct answer is A.

Let us calculate the Gini index for the first function. Inserting the Lorenz function into the formula for the Gini index given above, we obtain

Correct answer is C.

A function F is called an antiderivative of f on an interval (a, b) if F' (x) = f (x) for all x in (a, b). Thus, an antiderivative of

is a function such that its derivative is 

We can take

since

By the linearity of the definite integral it is clear that

Hence the right answer is D.

Since this equation is an first order ODE with a constant coefficient, it is clear that y=ce^4x is the general solution.

We know the initial population P0=10000, and that at t=5 the population is P=10000∙1.10. Hence, from (13)

herefore, after 20 years,

Therefore the correct answer is the D option.

Setting u=x-1, ve get du=dx, then

Correct answer is E .

Let us use the change of variables method. Let u=x²+3x+7 which gives du=(2x+3)dx and 2du=(4x+6)dx. Substituting these values into the integral we get

where c is an arbitrary real number. 

The sought for area is expressed in terms of the definite integral as