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 cos(x) and the Taylor series for ln(x) centered at 1.

1. 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
   e^x=c₀ +c₁x+c₂x²+c₃x³+...+c_nxⁿ
   
   for some appropriate integer n≥0. For the moment, let's pretend that n=0, so e^x=c₀. Focusing on what is happening to e^x at x=0, we get c₀=___. Try graphing both (that is, e^x and the value you got for c₀) in Maple by using the code below. You will need to replace the blank with the value you got for c₀.

   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);

2. They are clearly not the same functions, so let's move next to n=1. Then since e^x=c₀+c₁x, we expect that their derivatives are also equal. Differentiate both sides and plug in x=0 to find c₁=___. Again try graphing both (that is, e^x and c₀+c₁x) by substituting your value of c₀+c₁x into the blank in the code (above) that you pasted into Maple. What do you notice?
3. Again, they are clearly not the same functions, so let's try n=2. Then e^x=c₀ +c₁x+c₂x². We already know that their zeroth and first derivatives are equal, so set their second derivatives equal and find c₂=___. Again graph both with Maple. What do you notice?
4. Repeat the procedure for n=3 to find c₃=___, and describe the relationship of the two graphs.
5. Repeat the procedure for n=4 to find c₄=___, and describe the relationship of the two graphs.
6. 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?

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7. 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.

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8. Explain why this procedure won't work (at least not in exactly the same way) for ln(x).

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9. Let's imagine that ln(x) can written as a polynomial in x-1, that is
   ln(x)=c₀ +c₁(x-1)+c₂(x-1)²+c₃(x-1)³+...+c_n(x-1)ⁿ
   
   for some appropriate integer n≥0. Repeat the previous procedure for n=0, 1, 2, 3, and 4 to estimate ln(x), but focus on what is happening at x=1 instead of x=0. Here the following code may be helpful.

   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);

10. 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?