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

We know the initial population P0=12000, and that at t=12 the population is P=12000•1.20. Hence

Therefore, after 20 years,

The average value is evaluated as follows:

Hence the right answer is C.

Let find

Then producer surplus by the integral formula

The characteristic equation

The general solution

Correct answer is  A.

Using the formula for the present value of money we write

Therefore the correct answer is the D option.

In order to calculate this integral, we use the change of variable method. Let

then we get

Returning back to the original variable x, we find that the antiderivative of

Thus the definite integral is

The correct answer is B.

If the average value of a function defined and continuous over the interval

is denoted by

then the average value is calculated by the definite integral

For

integral becomes

Therefore, the correct answer is A.

Observe that for F(x)=xe^x the derivative is F'(x)=e^x+xe^x=(x+1)e^x. Therefore F(x)=xe^x is an antiderivative of (x+1)e^x. 

The antiderivative of the integrand is

The correct answer is A.

Applying method of separation of variables, we obtain for y>0

Letx-8=y then dy=dx and x=y+8. So we have

We need to use the method of integration by parts to find the first integral above. Let u=y and dv=e^ydy, then we get du=dy and v=e^y. Hence it is obvious that

Therefore, we find

We first calculate the quantity when the price is 8TL. Using the supply function, we find

, which gives q = 3. Inserting the given values in the formula

we obtain,

Therefore the correct answer is the B option.

If we make a change of variables as u=x-1, dx=du, we get

Return back to the original variable x, we find that the antiderivative of f is

Thus, the definite integral is 

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

Therefore the correct answer is the B option.

Let y=4x-6. Then we obtain 

It means that

In other words,

f(a)=5a-12=13.

5a=25

a=5.