Example (2a): Suppose x and y are positive real numbers and
z = x/y.
If the value of y changes by α percentage and x stays the same, then what would be the percentage change β in the value of z?
Answer:
Let the initial values be x, y and z = x/y.
After the change,
x2 = x, y2 = y(1 +α/100), z2 = x2/y2 = x/[y(1 + α/100)]
Then, percent change in z is:
When the percent change α in y approaches to zero, what would happen to the percent change of z? When α → 0, 100 + α → 100. Then,
When percent change in y (which is α) is very small, then percent change β in z can be approximated as – α. Formula [C] can also be derived using differential calculus.
That is,
(Percent change in z) = - (percent change in y) or β = - α.
Sample calculations:
Let x = 1 for simplicity. We do some sample calculations to compare the above three formulas for percent change in z compared to percent change in y.
Percent change in 1/y compared to percent change in y by three formulas
It can be observed that for large changes in y-values, formula [C] deviates from true values significantly. For small changes in y-value, formula [C] is close to true values provided by the formula [A] and {B]. For all inverse relationships such as z = x/y with x being a fixed number, change in z and change in y are in the opposite directions.
Example (2b): Comparing the change in y2 and change in y.
Let
When the change in y-value is very small, y2/y → 1. In this case,
(Percent change in z) = 2 (percent change in y).
This can also be derived using differential calculus.
That is,
(Percent change in z) = 2 (percent change in y).
Extension: In physics, there are inverse square laws such as gravitational field force and electrostatic field force. A simplified formula for such laws can be written as:
Interested persons can practice developing the relationship between (percent change in z) and (percent change in y) in three methods.
Method 1: using function composition. For all real numbers t > 0,
Combine the formulas from the above examples.
Method 2: Derive the relationship directly from z = 1/y2 for y > 0.
Method 3: Use differential calculus to calculate dz/z in terms of dy/y, where z = 1/y2 for y > 0.
Compare and confirm the formulas. Then find an application in estimating the change of gravitational field force in terms of change in the position.