Using the points

let us divide the interval

into four subintervals

and

Since the function f is an increasing function, it takes its smallest values at the left end of each interval. The length of each subinterval is

and

therefore

the correct answer is C.

We need to calculate the future value of a continuous income stream, hence

We assume that

and it then follows that m must be a root of the characteristic equation

Thus the possible values of

the general solution of the differential equation is

Since

we clearly see that

the correct answer is E.

It is obvious that

which means the right answer is C.

where c=c₁+c₂+c₃.  After these calculations we obtain the integral as

the correct answer is A.

Using the initial condition

hence “+” sign chosen

The correct answer is B.

If

then the characteristic equation is

and its roots are

Therefore the general solution of the differential equation is

Applying the initial conditions, we obtain the following two equations for

The solution of these equations is

and the solution of the initial value problem is

The differential equation can be written as

Integrating the left side with respect to

and the right side with respect to

gives

For this question, we only need to apply formula for the future value of an income. Since the time period T=3 and the interest rate r=0,2 is given, we have,

The average value is calculated using the definite integral as follows:

Consumer surplus is derived whenever the price a consumer actually pays is less than they are prepared to pay. A demand curve indicates what price consumers are prepared to pay for a hypothetical quantity of a good, based on their expectation of private benefit.

The total revenue function may be found by integrating the marginal revenue, so that we have

Let us use the change of variables method. Let  u=x³+6   which gives 

Substituting these values into the integral we get 

where c is an arbitrary real number.