Polynomial Approximation 3

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BC3
Polynomial Approximations III
Name: ____________________
It's time to generalize the process of finding an approximating polynomial. Assume we have an
arbitrary function ƒ, and assume that we wish to find a polynomial p(x) that has the same value,
derivative, second derivative, … at x = 0. Let p( x)  a0  a1 x  a2 x 2   an x n be an arbitrary nth
degree polynomial. If we set each derivative of p(x) equal to that of f(x) at x = 0 (that is, set
p( k ) (0)  f ( k ) (0) for k = 0, 1, 2, ..., n) we can solve for each ak,
(1)
Set ƒ(0) = p(0) to find a0 in terms of ƒ(0).
(2)
Find ƒ'(x) and p'(x). Set ƒ' (0) = p'(0) to find a1 in terms of ƒ'(0).
(3)
Set ƒ''(0) = p''(0) to find a2.
(4)
Set ƒ'''(0) = p'''(0) to find a3.
(5)
Find p(n)(x) and set ƒ(n)(0) = p(n)(0) to find an in terms of ƒ(n)(0).
(6)
Write the polynomial p(x) in terms of the derivatives of ƒ evaluated at 0.
(This is called either the nth order Taylor polynomial at a = 0 or the nth order Maclaurin
polynomial for the function ƒ.)
IMSA
Poly Approx 3 p.1
S’05
What’s so special about matching all out derivatives at x = 0? What if we wanted to (or had to—
consider f(x) = ln x) match our derivatives at some other x? The thing that made x = 0 easy was
that when we plugged in 0 into a polynomial, all the terms except the constant go away. But if
we write p( x)  a0  a1 ( x  x0 )  a2 ( x  x0 ) 2   an ( x  x0 ) n we could plug in x = x0 and have
everything cancel out.
Definition: Suppose that the first n derivatives of f(x) all exist at x = x0. The nth-order Taylor
polynomial at x = x0 is defined as
f ( x0 )
f ( x0 )
f ( n ) ( x0 )
2
3

pn ( x)  f ( x0 )  f ( x0 )( x  x0 ) 
( x  x0 ) 
( x  x0 )  
( x  x0 ) n
2
6
n!
(7)
Verify that pn(x), its first four derivatives, and its nth derivative match those of f(x) when
x = x0.
(8)
Use the definition to find the following Taylor polynomials:
(a) f(x) = cos(x), x0 = 0, n = 4 (you’d better get the same thing as in the previous packet!)
(b) f(x) = sin(x), x0 = 0, n = 4
(c) f(x) = ex, x0 = 0, n = 4
(d) f(x) = ln x, x0 = 1, n = 4
IMSA
Poly Approx 3 p.2
S’05
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