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Exponential function/Cubic Taylor-approximation/Zero/Exercise
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We consider the polynomial
P
=
1
+
X
+
1
2
X
2
+
1
6
X
3
.
{\displaystyle {}P=1+X+{\frac {1}{2}}X^{2}+{\frac {1}{6}}X^{3}\,.}
Compute the value of
P
{\displaystyle {}P}
for the points
−
2
,
−
1
,
0
,
1
,
2
{\displaystyle {}-2,-1,0,1,2}
.
Sketch the graph of
P
{\displaystyle {}P}
on the interval
[
−
2
,
2
]
{\displaystyle {}[-2,2]}
. Does there exist a relation to the exponential function
e
x
{\displaystyle {}e^{x}}
?
Determine a zero of
P
{\displaystyle {}P}
within
[
−
2
,
2
]
{\displaystyle {}[-2,2]}
, with an error of at most
1
4
{\displaystyle {}{\frac {1}{4}}}
.
Create a solution
for the points 
.
on the interval ![{\displaystyle {}[-2,2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43a81bc64cff8cbbca442d24d8a8275ff3776b53)
. Does there exist a relation to the exponential function 
?
within , with an error of at most 
.
