Due date: 5 October
2002

1-
Consider the
function _{}. Estimate
the value of the function at x = 0 from at its value at x = 1 using Taylor
Series. Find the minimum order of Taylor series that gives a true relative
error є_{t} < 1%. (Hint: the limit of the value of
the function as x -> 0 is 1.)

2-
Consider the
function Ln x

a-
Estimate the
first derivative of the given function at x = 1 using the backward, forward and
centered finite difference with h = 0.2.

b-
Repeat part
(a) for the second derivative.

3-
The velocity
distribution of a fluid near a flat surface is given by the following table,
where v is the velocity and x is the distance from the surface:

X(m) |
v(m/sec) |

0.000 |
0.00000 |

0.001 |
0.00723 |

. |
. |

. |
. |

. |
. |

If α is the viscosity, the Newton’s law for the
shear stress is given by

_{}

Use the forward finite
difference to calculate the shear stress at the surface (x = 0). Assume
that α = 0.0011 N sec/m^{2}.

4-
Use *Mathematica*
to generate Taylor series for the function f= x-cos(x) with x_{0 }= 0
and n=5. Use the series to approximate f(1) and calculate the percent relative
errors.