TRY QUESTIONS
1.
Suppose Gold Fields Ltd. models its expenditure at the end of a financial year. The model
considered two departments (maintenance and production). In the maintenance department, it costs
the company ₵ 1 per maintenance of a faulty machine plus a fixed cost of ₵ 5 for other materials.
It then costs the company an additional ₵ 3 per repair to automate the machine after maintenance.
In the production department, it costs ₵ 2 to produce an ounce of gold plus a fixed cost of ₵ 1 for
other materials. It then costs additional ₵ 7 per production of gold in the form of insurance package
for the production team plus ₵ 12 for insurance processing.
If the later cost function is a composition in the former in each respective department.
(a) Write a separate composition cost function for each department if the respective
composition function for the maintenance department and the production department is
raised to power four and three respectively.
(b) If the company’s total yearly cost function is an interactive cost function which is a product
of the two formulated cost functions in (a)
i. Quote and verify the Leibniz formula
ii. Using the Leibniz formula, find the actual total yearly cost, if the Leibniz formula for
the cost function is truncated at the third derivative at
3x=
.
(C) Suppose the company later realised that their actual total yearly cost function using the Leibniz
formula was to be truncated at the fourth order at
3x=
instead of the third order at
3x=
.
Did Gold Fields over spend and why?
NB: Let
( ), ( ), ( )C x M x P x
be the cost, maintenance and the production functions respectively.
2.
a. Evaluate
2
2
4 26 6
2 9 4
xx
dx
xx
++
++
b. Evaluate
24
os sin sin sin 3 cos 5
x
c x xdx e xdx x xdx
−+
3.
a. Suppose
2
w x y y xy= + +
where
cosx
=
,
siny
=
and
2
z
=
. Find
dw
d
and evaluate
at
3
=
.
b. Use the reduction formula to determine the
65
sin 2 cos 2xdx xdx−
c. Use Taylor’s theorem to expand
sin 6h
+
in ascending powers of h as far as the term in
4
h
.
