Download Engineering Mathematics | CHEN30101 problem sheet 1 1. The

Engineering Mathematics | CHEN30101
problem sheet 1
1. The equation 2x−1 = cos x has a root between x = 0 and x = 1. Compute
the root to an accuracy of two digits. (Check your answer by trying both
values as starting points.)
2. A tank is fed at a constant rate and also drains a second tank. In turn,
this tank drains into a third tank. The volume of fluid in the tank varies
like V = 10 − 9e−t − e−3t . Compute an accurate estimate (correct to 5
digits) of the time when the volume of fluid in the tank is equal to 8.
3. The coupled equations
2x + y + e−
5x − 2y −
1
10
xy
5
− 11 = 0
sin(x + y) − 14 = 0
have a solution near the point x = 4, y = 3. Construct a Newton iteration
to solve these two equations∗ and use this strategy to compute the solution
to an accuracy of 5 digits.
4. The proportion of two kinds of impurities in a liquid vary with height and
temperature as follows:
T
A = e− 100 /100 + T × sin(πh)/11000 + h/10
T
B =e− 60 /80 + T × sin(πh)/5000 + (1 − h)/5.
Find a temperature near T = 600K and a position near h = 0.5 where
A = 0.1 and B = 0.2.∗ Record your final answer to an accuracy of 3 digits.
5. Consider the coupled equation system
x2 − 2xy + y 2 − 5 = 0
x2 + 4xy + 4y 2 − 7 = 0
Compute (on paper) the first Newton iteration for solving the system
starting from from x0 = 1, y0 = 1. What happens on a computer?
Repeat the above experiment, this time starting from the point x0 = 1,
y0 = 3. Does the iteration converge?
6. Compute a solution to the system of equations
x2 + y 2 + z − 40 = 0
x2 − y + 3z 2 − 22 = 0
x + y 2 − 2z 2 − 20 = 0
that is accurate to two decimal places by constructing a Newton iteration†
and starting from the point x = 5, y = 4, z = 0.
∗
An efficient way of doing this is to modify the function fdef.m that is associated with
Computational Exercise I.
†
An efficient way of doing this is to modify the functions in the M-file newtonit.m that are
presented in Computational Exercise I.