Approximations Using Rates of Change
How can I use rates of change to approximate changes in value?
- Remember that a derivative in maths represents a rate of change
- e.g. if then
- When ,
- That's the rate of change of with respect to when
- As soon as changes away from 2, is no longer equal to 12
- But it will still be close to 12 so long as is still close to 2
- When ,
- We can use this to approximate the change in one variable based on a change in the other variable:
- That is, when changes by a small amount
- the change in the value of , ,
- will be approximately equal to times
- That is, when changes by a small amount
- This approximation is only valid when the change in is small
- The smaller the change in is,
- the more accurate the approximation will be
- The smaller the change in is,
- You may need to derive rates of change starting from standard geometric formulae
- e.g. the volume of a sphere is
- Take the derivative with respect to ,
- That's the rate of change of volume with respect to radius
- You may also need to use the relation
- e.g. you may need to answer a question
- Then
- e.g. you may need to answer a question
Exam Tip
- Look out for calculus questions asking you to 'estimate' or 'approximate' the change in a quantity
- The approximation is likely to be required
- Remember that's only valid when is small
Worked example
A sphere has a radius of .
The surface area of the sphere is increased by .
Using calculus, find an estimate for the increase in the radius of the sphere. Give your answer in , correct to 2 significant figures.
'Using calculus' and 'find an estimate for the increase' are hints that we should use
Here we know the change in the surface area,
We want to estimate the change in the radius,
Write down the formula for the surface area of a sphere from the exam formula sheet
Differentiate that with respect to to find
But we need for our approximation formula, so use
Substitute that into the approximation formula
We want to know the value of that when and
(Note that that is a 'small' change, , compared to the total surface area of when )
Round to 2 significant figures, as required