Consider the following polynomial function, where x ≥ 0 and n is a positive integer.?
Consider the following polynomial function, where x ≥ 0 and n is a positive integer.
p(x)=1+x+(x^2/2!)+(x^3/3!)+....(x^n/n!)
a) Evaluate p(1) . Let n go as high as 10 (you have graphing calculators, after all).
b) Evaluate this function for x = 2, 3, 4, and 5. What simpler function is this polynomial
function similar to?
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That is the Maclaurin series, Σ(n = 0,∞) xⁿ/n! = e^x, you have!
a) By calculator, we have...
Σ(n = 0,10) xⁿ/n! = x^10/3628800+x^9/362880+x^8/40320+x^7/5040+x^6/720+x^5/120+x^4/24+x^3/6+x^2/2+x+1
If x = 1, then we obtain around 2.71828, which is nearly e^1!
b) p(2) = e²
p(3) = e³
p(4) = e^4
p(5) = e^5
It's best to think of that series as e^x since p'(x) = p(x)!
Good luck!
p(x)=1+x+(x^2/2!)+(x^3/3!)+....(x^n/n!
p(x) = e^x