Comparison & related change

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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.

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.

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