Step-by-step explanation:


Here, the Taylor approximation for a square root was applied, and O(x) stands for all negligible terms of Taylor's sum with respect to variable x.
So, 
b. For an increase of 2%, that is:


The simplest path from (0, 0, 0) to (1, 1, 1) is a straight line, denoted
, which we can parameterize by the vector-valued function,

for
, which has differential

Then with
, we have



Complete the square in the quadratic term of the integrand:
, then in the integral we substitute
:


Make another substitution of
:

Integrate by parts, taking




So, we have by the fundamental theorem of calculus that



C
the sum to infinity of a geometric series = 
where a is the first term and r the common ratio
=
= 81 × 3 = 243
The recipe calls for 736 milliliters of liquid in total.