We know that if then
Provided that is small enough (but not infinitesimally small) , so
Example 4.23:
We can measure the volume of a sphere by measuring its radius and then use the formula, . Suppose we measure . Find the approximate error in .
Solution:
If then . The small error will cause an error in given by . Hence
Example 4.24:
We measure the height of a tower at a distance , by measuring and the angle with the horizontal. We then use the formula .
Solution:
Find error in due to an error in assuming to be known exactly. We solve for , , . Therefore
Example 4.25:
Given the relation between current, voltage and resistance, , with , , find the change in the current 1) if increases by , and 2) if increases by .
Solution:
We use the rule for small changes for partial derivatives,
We find
Using the numerical values, we find