x - Balliol College - University of Oxford

SJ Roberts - July 2011 Revision 1/ page 6
4 Functions
You should be familiar with the properties of standard functions, such
polynomials, rational functions (where both numerator and denominator
are polynomials), exponential functions, logarithmic functions, and
trigonometric functions and their identities.
25. (i) For what value(s) of x is the function f x x / x 2 1
undefined Describe the behaviour of f as x approaches these
values from above and below.
(ii) Find the limits of f x
and dx
df / as x and x .
(iii) Does the function have stationary values If so, find the values
of x and f x
at them.
(iv) Now make a sketch of the function.
26. Sketch
y
t
e and
y
3t
e versus time t for 0 t 3. When a
quantity varies as
t
e
/
, is called the time constant. What are
the time constants of your two plots Add to your sketch two curves
showing the variation of a quantity with (i) a very short time
constant, (ii) a very long time constant.
27. A quantity varies as
y
100e
10t
e
t
/10
. Which part controls the
behaviour of y at short time scales (i.e. when t is just above zero),
and which at long time-scales