+1 862 207 3288 Assignment, Semester 2 2015
Instructions:
1. Submit a hand-written or typed assignment.
2. Organise your answers in the order in which the questions are asked below.
3. Provide full working/justi cation for your answers.
4. Submit a hard-copy of your assignment into the assignment box at reception in building 23.
You must attach a university cover sheet.
Part 1 Due: 5pm Friday 28th of August
A particle travels along the curver(t) =t
2
i+
t
3
3
t
j, 1< t <1.
a) Graph the trajectory of the particle using Matlab for 3t3. Be sure to label your axes
(xandy), and indicate the direction in which the particle is moving along the trajectory.
[3 Marks]
b)From the graph you will note that the particle traverses a loop. Calculate the arclength of
the loop, expressing your answer exactly.
[7 Marks]
c) Calculate the velocity and acceleration vectors for this curve.
[2 Marks]
d)Determine the unit tangent and unit normal to the curve, and write the acceleration vector
as the sum of a vector parallel to the unit tangent and a vector parallel to the unit normal
vector. Fully simplify your answers.
[10 Marks]
Part 2 Due: 5pm Friday 11th of September
Letf(x;y) = sin(x+ 2 cos(y)).
a) State the domain and range off.
[2 Marks]
b)Use Matlab to graph the cross-section off for x= 0 and 2y2. Label the axes.
[2 Marks]
1
c) Use Matlab to produce a labeled contour plot off for 2x2and 2y2.
Label the axes.
[2 Marks]
d)Use Matlab to plot the graph off for 2x2and 2y2. Label the axes.
[2 Marks]
e) Determine all rst and second partial derivatives off.
[6 Marks]
f) Determine the equation of the tangent plane to the surface at the point
4
;
2
.
[4 Marks]
Part 3 Due: 5pm Friday 9th of October
a) Continuing from Part 2.
i. Determine all critical points off and show that
2
2;0
and
2
2;0
are critical
points. Express your answer exactly.
[5 Marks]
ii. For the critical points
2
2;0
and
2
2;0
determine the equation of the second-order Taylor polynomial off for (x;y) near the critical point. Use Matlab to plot the
surface of each of the second-order Taylor polynomials and use the graph to decide if
the corresponding critical point is the location of a local maximum, local minimum or
saddle point. Plot the second-order Taylor polynomial about
2
2;0
on the domain
(x;y)2[ 1;0][ 1;1], and the second-order Taylor polynomial about
2
2;0
on
the domain (x;y)2[ 4; 3][ 1;1]. Label the axes.
[12 Marks]
iii. Can you use the second derivative test to con rm your answers in ii? Justify your answer.
[2 Marks]
b)Consider a circular plate of radius 2 given byx
2
+y
2
4. The temperatureTat any point
(x;y) on the plate is given byT(x;y) = 2y
2
+x
2
x+ 20. Find the hottest and coldest points
on the plate. Use the method of Langrange multipliers to check for optimal solutions on the
boundary of the plate. Express your answer exactly.
[13 Marks]
Part 4 Due: 5pm Friday 16th of October
a) For the integral given below, sketch the region of integration, reverse the order of integration
and evaluate the integral:
Z
8
0
Z
2
3
p
x
dydx
y
4
+ 1
Express your answer exactly.
[8 Marks]
2
b)In spherical polar coordinates (;; ) the surface of an apple may be modelled by
= 1 cos( )
where denotes the angle with thez-axis.
i. Determine the volume of the apple. Express your answer exactly.
[7 Marks]
ii. Assuming the apple has a constant density, determine the coordinates of the centre of
mass of the apple.
[8 Marks]
Part 5 Due: 5pm Friday 30th of October
Suppose thatCis the line segment from the point (0;0) to the point (4;12) andF(x;y) =xyi+xj.
a) Use Matlab to draw the vector eld at a representative assortment of points onC. Use the
`axis equal’ command to ensure that angles can be accurately observed.
[2 Marks]
b)Is
R
C
F drgreater than, less than or equal to 0? Give a geometric explanation using your
plot from a).
[2 Marks]
c) A parameterisation ofCis (x(t);y(t)) = (t;3t), 0t4. Use this to compute
R
C
Fdr.
[3 Marks]
d)Is the vector eld Fconservative? Justify your answer.
[3 Marks]
e) Now letC1
be the semicircle that starts at the point (4;12) and nishes at (0;0) in an anti-clockwise direction. Consider the closed curveC[C1
. Calculate the circulation of Faround
C[C1
. Express your answer exactly.
[10 Marks]
3
