David Thibodeaux
Calculus 1 – H03
MATLAB Assignment #2
11/4/2010
Introduction:
In this assignment, we will use the concept of the Taylor series to
approximate the trends seen in complex functions.
Using theory, we will first derive the
formula used to approximate these curves, and then apply these approximations to
various graphs.
1. Find a secondorder (quadratic) approximation P
2
(x) = Ax
2
+ Bx + C to the function
f(x), stipulating the following for the best approximation:
(a) P
2
(a) = f(a)
(b) P
2
′
(a) = f
′
(a)
(c) P
2
′′
(a) = f
′′
(a)
Using the Definition of the Taylor Series:
P
2
(x) = f(a) + f’(a)(x – a) + ½f’’(a)(x – a)
2
Substituting a for all x…
P
2
(a) = f(a) + f’(a)(a – a)
+ ½f’’(a)(a – a)
2
P
2
(a) = f(a)
For P
2
’(a), we can say that f(a) is zero, since f(a) is a constant…
P
2
’(x) = 0 + f’(a)(1) + (x – a)(0) + ½f’’(a) x 2(x – a)(1)
Now substitute a for all x…
P
2
’(a) = 0 + f’(a) + (a  a)(0)
+ f’’(a)(a – a)(1)
P
2
’(a) = f’(a).
We can also say that for P
2
’’(a) of f’(a) is zero, since f’(a) is a constant.
P
2
’’(a) = 0 + (0)(x – a) + ½f’’(a)(x – a)
2
.
Substituting all a for x, and using the chain rule for the term
½f’’(a)(x – a)
2
…
P
2
’’(a) = 0 + (0)(a – a)
+ ½f’’(a)(2)(1)
P
2
’’(a) = f’’(a).
We have satisfied all conditions.
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2.
Now that we have determined the formula for an approximation, we will now test this
approximation on the function f(x) =
e
x
at a = 0.
Using the Taylor approx. formula, we
derive the following approximations:
P
1
(x) = x + 1
P
2
(x) = 1 + x + ½x
2
.
Graph of
f(x),
P
1
(x)
,
and
P
2
(x)
Description: the linear approximation is only truly accurate at x = 0 due to the fact that it
is linear and the function is exponential.
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 Derivative, Taylor Series, Natural logarithm, Logarithm

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