Note 4.1 - Related Rates and Differentials
1 Introduction
We apply chain rules/implicit differentiations and linear approximations to some practical problems.
2 Related Rates
In many situations, there are quantities, related by equation(s), depend upon each other:
F (x, y) = 0.
It can be mathematically unclear to tell which one is independent variable or dependent variable (i.e. the function). They can also be functions of another variable (the parameter) as well:
F (x(t), y(t)) = 0.
In any case, we are interested to see how a certain variable varies with respect to one of the other variables or parameters. This is done by implicit differentiation.
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3 Differentials
We have discussed how differentiations are interpreted as linear approximations.
Now we use them to approximate some numerical values of functions:
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The tangent lines are also quick tools to estimate the change of function value:
∆f := f (x + ∆x) − f (x)
These are often used to estimate the errors of some quantities:
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