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(MATH100)Sample Exercises for the Final Examination.pdf
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Sample Exercises for the Final Examination
2 2(. x2 . y2)/2
1. Given the surface z = (x . y )e ,
(a)
write the equation of the tangent plane at an arbitrary point (x,y,z) on the surface.

(b)
find the maxima, minima and the saddle points of the surface and sketch the surface.

(c)
using a double integral, compute the volume under the surface and over the square formed by the points (-2,-2), (2,-2), (2,2) and (-2,2).


2. (a) Compute using cylindrical or spherical coordinates, the volume of the

22 22
solid enclosed by the cones z = x + y and z = 1 . 2 x + y .
Hint: divide the solid in two regions.

(b) Use the divergence theorem to find the outward flux of the vector field F=x2i+y2j+z2k across the pyramid with basis the square formed by the points (0,0,0), (2,0,0), (2,2,0) and (0,2,0) and vertex at (1,1,2).
3. Show that the line integral
(1, )
sin ydx + x cos ydy

( .1,0)
is independent of the path and evaluate the integral by:
(a)
finding a potential function for the integrand (this means finding such that sinyi+xcosyj=.).

(b)
integrating along any convenient path.


4. (a) Calculate the surface integral of F=xj+2zk over the surface S defined by z=xy2 over the triangle bounded by x=0, y=0, and y=x-1.
(b) Let =2xeyi+(x2ey+z3)j+3z2yk.
Use Stokes theorem to compute (r)dr, where C is the boundary of the semi-


C
ellipsoid x2+4y2+9z2=25, z0.
5. Find the aria between the curves y=x3 and y =
x
(a)
directly by using a double integral

(b)
using a line integral by applying Greens formula.


6. Let the surface S be given by z=x2+y2.
(a)
Find the tangent plane to the surface at each point on the surface.

(b)
Find the outward oriented normal vector to the surface at each point on the surface.

(c)
Compute the flux of the vector field F=zk across the surface of the solid formed by S and the plane z=1.


7. Let the surface S:RR3 be given by x=ucosv, y=usinv, z=u2+v2, where R=[0,0][0,2].
(a)
Find the tangent plane to the surface at each point on the surface.

(b)
Find the outward normal vector to the surface at each point on the surface.

(c)
Compute the flux of the vector field F=xi+yj across S.