In what way can polynomials be used to approximate functions?
Functions you may not have encountered before
Consider the function \[c(t)=\frac{t}{t^2+4}.\]
Without a graphing tool, it may be hard to visualize what this function looks like.
Functions you may not have encountered before
Consider the function \[c(t)=\frac{t}{t^2+4}.\]
With a computer, it becomes trivial to even graph this function.
Functions you may not have encountered before
Consider the function \[c(t)=\frac{t}{t^2+4}.\]
So why should we care about this function if a computer is available?
Functions you may not have encountered before
Consider the function \[c(t)=\frac{t}{t^2+4}.\]
To gain insight in real-world cases!
- Most functions are not as “simple” as \(c(t)\).
- At some point we may need functions which depend on more than one input.
Functions you may not have encountered before
Consider the function \[c(t)=\frac{t}{t^2+4}.\]
How do we gain insight into the behavior of this function?
- Use functions that you know very well.
- In particular, focus on quadratic functions.
Quadratic functions
These are polynomials of the form \[P(t)=a_0+a_1 t +a_2t^2.\]
How could these polynomials be used to gain insight about \[c(t)=\frac{t}{t^2+4}?\]
They seem too different on the surface!
Quadratic functions
These are polynomials of the form \[P(t)=a_0+a_1 t +a_2t^2.\]
You can make them look similar by being a bit more modest:
- Work within a small range.
- Choose \(a_0\), \(a_1\), and \(a_2\) properly.
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How did I get these polynomials?
- In general, you pick a point where you want to approximate the function. Call this \(t_0\).
- Next, create the polynomial \[P(t)=\underbrace{c(t_0)}_{a_0}+\underbrace{c^\prime(t_0)}_{a_1}(t-t_0)+\underbrace{\frac{c^{\prime\prime}(t_0)}{2}}_{a_2}(t-t_0)^2 \]
New objects!
- You need objects which are called derivatives: specifically \(c^{\prime}(t_0)\) and \(c^{\prime\prime}(t_0)\).
- Why not look into polynomials of degree greater than 2?
- I am hoping you will join us to try to answer these questions and more in our online course on differential calculus for economic analysis!