Mathematics 2

Yanıt Açıklaması: Setting in the Euler equation we obtain the general solution of the given equation is

Yanıt Açıklaması: The solution is quite straightforward. The given equation is a separable differential equation so The correct answer is E.

Yanıt Açıklaması: By the formula we have Let us calculate and from integration by parts we have Therefore, we obtain The right answer is C.

Yanıt Açıklaması: Employing the properties of the integral, Using the information supplied in the question, we obtain

Yanıt Açıklaması:

Yanıt Açıklaması: We know that if the original mass was M 0 , then after 15 days we have Hence Thus we obtain

Yanıt Açıklaması: The general solution of the ODE is Substituting we obtain This equation is quadratic in y and we obtain Only satisfy the given initial condition.

Yanıt Açıklaması: The Gini index is calculated using the formula Where is the Lorenz curve, Inserting given in Correct answer is  D .

Yanıt Açıklaması: Since and

Yanıt Açıklaması: The area under the graph of the supply function corresponds to the expected income. So we have The right answer is E.

Yanıt Açıklaması: Let us use the change of variables method. Let which gives and for upper bound and for lower bound Substituting these values into the integral we get The correct answer is A.

Yanıt Açıklaması: Let u=x 3 +x that gives du=(3x 2 +1)dx and we substitute and in the integral we have

Yanıt Açıklaması: We first find the intersection points of the given curves. Let us equate them We find the intersection points For the interval we have so the area A is evaluated as The correct answer is C.

Yanıt Açıklaması: To use the integration by parts method, let u=ln3x and dv=6x 2 dx.  Then we have and v=2x 3 Inserting above equalities into ∫udv=uv-∫vdu we get; where c is an arbitrary real number.

Yanıt Açıklaması:

Yanıt Açıklaması: the correct answer is D.

Yanıt Açıklaması: Correct answer is A.

Yanıt Açıklaması: Since the function in question is a linear one, the integral may be calculated by using the areas of triangles and squares. For –3≤x≤0, let A1 denote the area of the triangle above the x-axis and under the graph. In the interval [0, 2], the area is denoted by A2, and for 2≤x≤4, the area under the graph is denoted by A3. Then the definite integral is found as Therefore the correct answer is the D option.

Yanıt Açıklaması: The integrating factor is Multiplying the equation by this factor leads to the equation By integrating both sides of above equation, using integration by parts on the second term on the right side, we obtain where c is an arbitrary constant. Thus