Use the function of two independent variables x andy

Question I Use the function of two independent variables x andy, f(xY)=f+4rY-&+4Y-9 (a) Find all the first and second orderpartial derivatives. (b) Find the stationary poin(s) of the function. (c) Classi& the stationarypoint(s) of the function. (4 marks) (4 marks) (4 marks) Question 2 Consider the differential equation il -' -d*-'' *Ji, where />0, with the initial conditiony :0.427. when x: 0.3. (a) Use Euler mid-point method with step size 0.01 to calculate an approximate value fory(0.31) accurate to 6 decimal places, wherefix) is the solution of the equation satisffing the given initial condition. (5 marks) O) Using a computer program to calculate an approximation I(0.4), for y(0.4) by the Euler-trapezoidal method, the following values for I(0.4) were obtained for two different step sizes. Steo size r0.4) 0.05 0.s5r536 0.02 0.551521 Estimate the upper bound of the step size which would give an approximation to y{0.4) accurate to 6 decimal places, when used in the Euler-trapezoidal method. (7 marks) ?. Question 3 (a) Sketch the region in the r.y-plane bounded bf th1 ltnes x = l, ! = , *O r4:Jall; ' ." tt: ' (b) Evaluate the area integral f" (r' -y\dA , where S is the region in the rry-plane bounded by the lines x = L,!: 1 and the curvey:v * l. (9 marks) Question 4 (a) Considerthethreedimensionalsurfacegivenby, 2nt -3ry-4x=7 Determine a unit vector normal to the surface at the point (1,-1,2). (4 marks) O) Consider the scalar fietd given by, .f (x,y,4=*yz+ +x* Determine the directional derivative of this scalar field/at the point (1, -2, -l), and in the direction 2i_i_2b-(5 marks) (c) A vector field is given by, B(x, y, z) = 3xy* i + 2xf i -ly, k Determine the divergence of F at the point (1, -1, 1). (4 marks) Question 5 The function/is defined onttre interval [0, a] by f (t)= lsin (r)l is to be represented by a Fourier series ofthe form r (t) = Ao * 2n, -{T)* ir,'*(ry) (a) Describe the definitions of the odd and even extensions of/that have period 2n and sketch...