\(\blacktriangleright \) A Peek ahead to Calculus
In Calculus, we will see all of the rules for obtaining the derivative of any funciton
for ourselves.
Right now we know three rules:
- if \(f(x) = a \, x^2 + b \, x + c\), then \(f'(x) = 2a \, x + b\)
- if \(f(x) = a \, x + b\), then \(f'(x) = a\)
- if \(f(x) = a\), then \(f'(x) = 0\)
When we get to Calculus, we’ll be able to “differentiate” any function we want.
Right now, you would be given the derivative of a function, which you could then use
to analyze the function.
When analyzing \(g(t) = t - \ln (t)\), you might be given the derivative.
We could then use this derivative to identify critical numbers and decide where the function \(g\) is increasing and decreasing.
Critical Numbers
Note:. The domain of \(g\) is \((0, \infty )\).
To find critical numbers, we need to solve \(g'(t) = 0\).
\(\blacktriangleright \) \(1 = \frac {1}{t}\), when \(t = 1\).
\(g\) has \(1\) as its only critical number.
Behavior: Increasing and Decreasing
The sign of the derivative will tell us where \(g\) is increasing and decreasing.
\(\blacktriangleright \) Where is \(1 = \frac {1}{t}\) positive and negative?
Since \(g\) is continuous, this tells us that there is a local (possibly global) minimum at \(1\).
We know logarithmic functions as an inverse to exponential functions.
\(\log _A(B)\) is the thing you raise \(A\) to, to get \(B\).
In particular, \(\ln (x)\) is the inverse to \(e^x\), meaning their composition gives the identiy function.
If we were in Calculus, the chain rule will tell us how their derivatives relate to each other.
We need some more experience with limits to see why this is true. Experience which
Calculus will give us.
For now, you would need to be given such derivatives. You don’t know how to get these derivative yet.
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more examples can be found by following this link
More Examples of Rate of Change