\(\blacktriangleright \) 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 \(f(x) = (3x - 2) e^{4x + 5}\), you might be given the derivative.
We could then use this derivative to identify critical numbers and decide where the function \(f\) is increasing and decreasing.
Critical Numbers
To find critical numbers, we need to solve \(f'(x) = 0\).
By the zero product property, one of the factors must equal \(0\).
\(\blacktriangleright \) \(e^{4x + 5}\) is never \(0\), since it is an exponential function.
\(\blacktriangleright \) \((12x - 5) = 0\), when \(x = \frac {5}{12}\).
\(f\) has \(\frac {5}{12}\) as its only critical number.
Behavior: Increasing and Decreasing
The sign of the derivative will tell us where \(f\) is increasing and decreasing.
\(\blacktriangleright \) Where is \(e^{4x + 5} (12x - 5)\) positive and negative?
This is a product, so we need the signs of the factors.
\(\blacktriangleright \) \(e^{4x + 5}\) is always positive, since it is an exponential function with a positive leading
coefficient.
\(\blacktriangleright \). \((12x - 5)\) is a linear function with a positive leading coefficient, which means the linear
function is increasing, which means it is negative to the left and positive to the right
of its zero, which is \(\frac {5}{12}\).
Since \(f(x)\) is continuous, this tells us that there is a local minimum (possibly global) at \(\frac {5}{12}\).
We know exponential functions as a description of constant percentage growth.
- If you change the same amount in the domain, then the function changes by the same percentage.
- If you change the same amount in the domain, then the function value is multiplied by the same constant.
This description compares changes in the function with changes in the domain. That
sounds a lot like a derivative.
Exponential functions can be described through characteristics of a derivative.
Exponential functions, \(exp(x) = A \, r^x\) are the only functions who are their own derivatives.
We need some more experience with limits to see why this is true. Expereince 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