# You can use formulae from Appendix to the Textbook by G. James or from the Formula Sheet available on the Study

You can use formulae from Appendix to the Textbook by G. James or from the Formula Sheet available on the Study Desk

Question 1: [40 marks]

A surface is defined by the following equation: z(x,y)=x^2y/ (x^2+y^2).

a) Find the equation of the tangent plane to the surface at the point P(3, 5). Present your answers in the exact form (don’t use a calculator to convert your result to the floatingpoint format). [25 marks]

b) Find the gradient of function z(x, y) at the same point P. [5 marks]

c) Find the angle between the gradient and the x-axis. Present your answer in degrees up to one decimal place. [10 marks]

Rubric:

3 marks if z0 was correctly calculated;

5 marks for correctly calculated a partial derivative with respect to x;

3 marks if number p was correctly calculated;

5 marks for correctly calculated a partial derivative with respect to y;

3 marks if number q was correctly calculated;

5 marks if equation of the tangent plane was correctly presented;

4 marks if equation of the tangent plane was properly simplified and presented in the exact form;

5 marks if the gradient was correctly presented at the point P;

4 marks if the angle alpha was found correctly;

3 marks if the angle alpha was correctly presented in degrees up to one decimal place.

2 Question 2: [35 marks]

Find all critical points of the function z= x^2- xy + y^2 =3x – 2y + 1 and determine their character, that is whether there is a local maximum, local minimum, saddle point or none of these at each critical point. In each critical point find the function value in the exact form (don’t use a calculator to convert your result to the floating-point format).

Rubric:

3 marks for the correct calculation of the partial derivative with respect to x;

3 marks for the correct calculation of the partial derivative with respect to y;

5 marks if the set of equations to determine critical points is found correctly;

6 marks if the critical point is found correctly;

4 marks for the correct calculation of number A;

4 marks for the correct calculation of number B;

4 marks for the correct calculation of number C;

marks for the correct calculation of the discriminant D;

4 marks for the correct determination of the nature of the critical point.

Question 3: [40 marks]

Calculate the total charge Q of a thin plate with the charge density distribution rho(x, y) = x^2y mC/m2. The plate shape is restricted by the lines:

? = 2 – ?^2 and ? = 2? – 1,

where x and y are measured in metres.

a) Sketch the plate shape. [10 marks]

b) Present the total charge through the double integral. [4 marks]

c) Reduce the double integral to the repeated integrals and show limits of integration. [6 marks]

d) Calculate the integral and present your answer with five significant figures. [20 marks]

Rubric:

2 marks if the upper boundary of the domain of integration is plotted correctly;

2 marks if the lower boundary of the domain of integration is plotted correctly;

6 marks if the points of intersection are found correctly;

4 marks if the total charge correctly presented through the double integral;

6 marks if the double integral is reduced to the repeated integrals correctly;

2 marks if the inner integration was correct including limit substitution;

2 marks if the function of x was correctly simplified;

6 marks if the outer integration was correct;

4 marks if limit substitution was correct;

3 marks if the result was correctly presented with five significant figures;

3 marks if the result was correctly presented in mC.

Question 4: [40 marks]

Consider the double integral V= integral integralD 4r^2 tan phi dAover the region D enclosed between the lines:

0

a) Reduce the integral to the repeated integral and show limits of integration. [12 marks]

c) Calculate the integral and present your answer in the exact form. [28 marks]

Rubric: 12 marks if the double integral is reduced to the repeated integrals correctly;

2 marks if the inner integration was correct; 8 marks if substitution of limits was correct;

8 marks if function of phi was correctly simplified;

8 marks if the outer integration was correct;

1 mark if limit substitution was correct;

1 mark if the result was properly presented with two significant figures.

Question 5: [30 marks]

Determine which of the differentials (or both) is the total differential:

a) 2[exp(2x + 2y^2 ) + x ] dx + [4y exp(2x + 2y^2 ) + 1] dy; [15 marks]

b) 6 sin(y) e^(2x + y)dx + 3(sin y + cos y) e^(2x + y) –2 y sin(y^2 ) dy.

[15 marks] Find the potential function U(x, y) subject to the condition U(0, 0) = 5 in the case of the total differential.

Rubric: 3 marks if the total differential was determined correctly in case (a);

6 marks if the potential function was derived correctly in case (a);

4 marks if the constant of integration was found correctly in case (a);

2 marks if the potential function was presented explicitly in the final form in case (a);

3 marks if the total differential was determined correctly in case (b);

6 marks if the potential function was derived correctly in case (b);

4 marks if the constant of integration was found correctly in case (b);

2 marks if the potential function was presented explicitly in the final form in case (b).

Question 6: [35 marks]

The line mass density of wire is given by the formula

rho(x, y) = (5x^3 + 3y^2 + 8z^2 ) g/cm. Calculate the total mass M of the wire if its shape is described by the equations:

x (t) = sin^2 t, y (t) = cos^2 t, z(t) = (cos^2 t) /2, where 0 = t = pi/2 and length is measured in centimetres.

Present you answer in the exact form (don’t use a calculator) with the appropriate dimension.

Rubric: 7 marks if the arc element ds was calculated correctly;

8 marks if the formula for the total mass is presented in the correct form;

12 marks if the integration was correct;

3 marks if the substitution was correct;

3 marks if the result was correctly presented in the exact form;

2 marks if the mass dimension is given correctly in grams.

Question 7: [35 marks]

Calculate the line integral of the vector-function

F(x, y,z) = (y^2 + z^2 ) i – yzj + xk along the path

L: x = t, y = 2 cost, z = 2 sin t (0

Rubric: 6 marks if the differential dr has been calculated correctly;

8 marks if the vector-function has been calculated correctly in terms of t;

6 marks if the dot-product has been correctly calculated;

12 marks if the integration was correct;

3 marks if the result has been correctly presented in the exact form.

Question 8: [45 marks]

For the given matrix A= (-1 5 -1, 0 -3 0, -4 7 2)

(a) Find all eigenvalues and present them in the ascending order. [25 marks]

(b) Which of two given vectors v1 and v2 is the eigenvector of the matrix A, where v1^T = (1, 0, 4) and v2^T = (1, 0, –4)? What is the corresponding eigenvalue? [20 marks]

Rubric: 2 marks if the matrix A – lambda I has been calculated correctly;

5 marks if the determinant |A – lambda I | has been developed correctly;

2 marks if one of the roots has been guessed correctly;

8 marks if the quadratic polynomial has been found correctly by either method;

6 marks if other two roots have been found correctly;

2 marks if the eigenvalues are presented in the ascending order correctly.

Rubric: 5 marks if the product Av1 has been calculated correctly;

3 marks if it was correctly determined that vector v1 is not an eigenvector of matrix A;

5 marks if the product Av2 has been calculated correctly;

3 marks if it was correctly determined that vector v2 is the eigenvector of matrix A;

4 marks if the eigenvalue corresponding to vector v2 was determined correctly