**Prerequisites:** Higher order derivatives

**Goals:** To become familiar with the concept of polynomials converging to a function. To conjecture the MacLaurin series for *e ^{x}* and

- Let's imagine that e
^{x}can be written as a polynomial (actually, it cannot, but that doesn't stop us from imagining that it can). Then for some appropriate integer n≥0. For the moment, let's pretend that n=0, so e

e^{x}=c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+...+c_{n}x^{n}

^{x}=c_{0}. Focusing on what is happening to e^{x}at x=0, we get c_{0}=___. Try graphing both (that is, e^{x}and the value you got for c_{0}) in Maple by using the code below.*You will need to replace the blank with the value you got for c*_{0}.restart:with(plots): f:=x->___: expPlot1:=plot(exp(x),x=-3..3,y=-1..7,color=red): expPlot2:=plot(f(x),x=-3..3,y=-1..7,color=blue): display(expPlot1,expPlot2);

- They are clearly not the same functions, so let's move next to n=1. Then since e
^{x}=c_{0}+c_{1}x, we expect that their derivatives are also equal. Differentiate both sides and plug in x=0 to find c_{1}=___. Again try graphing both (that is, e^{x}and c_{0}+c_{1}x) by substituting your value of c_{0}+c_{1}x into the blank in the code (above) that you pasted into Maple. What do you notice? - Again, they are clearly not the same functions, so let's try n=2. Then e
^{x}=c_{0}+c_{1}x+c_{2}x^{2}. We already know that their zeroth and first derivatives are equal, so set their second derivatives equal and find c_{2}=___. Again graph both with Maple. What do you notice? - Repeat the procedure for n=3 to find c
_{3}=___, and describe the relationship of the two graphs. - Repeat the procedure for n=4 to find c
_{4}=___, and describe the relationship of the two graphs. - Find a general formula for c
_{n}=___. What do you expect to happen to the graphs if we take larger and larger values of n? - Repeat the procedure to find a polynomial of degree 0, 1, 2, 3, and 4 to estimate cos(x), graphing your polynomials with the function cos(x) at each step. The above Maple code will need some adjustment to work properly. For example, you will need to change the function in the first plot from exp(x) to cos(x), and should probably adjust the domain and range of each plot, too.
- Explain why this procedure won't work (at least not in exactly the same way) for ln(x).
- Let's imagine that ln(x) can written as a polynomial in x-1, that is

ln(x)=c_{0}+c_{1}(x-1)+c_{2}(x-1)^{2}+c_{3}(x-1)^{3}+...+c_{n}(x-1)^{n}

restart:with(plots): h:=x->___: logPlot1:=plot(ln(x),x=-1..3,y=-3..2,color=red): logPlot2:=plot(h(x),x=-1..3,y=-3..2,color=blue): display(logPlot1,logPlot2);

- Find a general formula for c
_{n}=___ in problem 9. What do you think happens to the graphs if we take larger and larger values of n?