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8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
Activity 8.22.
We have just seen that the nth order Taylor polynomial centered at a = 0 for the
exponential function e x is
n
k=0
x k
k!
.
In this activity, we determine small order Taylor polynomials for several other familiar functions, and look for general patterns that will help us find the Taylor series
expansions a bit later.
(a) Let f (x) =
1
1−x .
(i) Calculate the first four derivatives of f (x) at x = 0. Then find the fourth
order Taylor polynomial P 4 (x) for
1
1−x centered at 0.
(ii) Based on your results from part (i), determine a general formula for f (k) (0).
(b) Let f (x) = cos(x).
(i) Calculate the first four derivatives of f (x) at x = 0. Then find the fourth
order Taylor polynomial P 4 (x) for cos(x) centered at 0.
(ii) Based on your results from part (i), find a general formula for f (k) (0).
(Think about how k being even or odd affects the value of the kth derivative.)
(c) Let f (x) = sin(x).
(i) Calculate the first four derivatives of f (x) at x = 0. Then find the fourth
order Taylor polynomial P 4 (x) for sin(x) centered at 0.
(ii) Based on your results from part (i), find a general formula for f (k) (0).
(Think about how k being even or odd affects the value of the kth derivative.)
⊳
It is possible that an nth order Taylor polynomial is not a polynomial of degree n;
that is, the order of the approximation can be different from the degree of the polynomial.
For example, in Activity 8.22 we found that the second order Taylor polynomial P 2 (x)
centered at 0 for sin(x) is P 2 (x) = x. In this case, the second order Taylor polynomial is a
degree 1 polynomial.
Taylor Series
In Activity 8.22 we saw that the fourth order Taylor polynomial P 4 (x) for sin(x) centered
at 0 is
P 4 (x) = x −
x 3
3!
.
8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
Activity 8.22.
We have just seen that the nth order Taylor polynomial centered at a = 0 for the
exponential function e x is
n
k=0
x k
k!
.
In this activity, we determine small order Taylor polynomials for several other familiar functions, and look for general patterns that will help us find the Taylor series
expansions a bit later.
(a) Let f (x) =
1
1−x .
(i) Calculate the first four derivatives of f (x) at x = 0. Then find the fourth
order Taylor polynomial P 4 (x) for
1
1−x centered at 0.
(ii) Based on your results from part (i), determine a general formula for f (k) (0).
(b) Let f (x) = cos(x).
(i) Calculate the first four derivatives of f (x) at x = 0. Then find the fourth
order Taylor polynomial P 4 (x) for cos(x) centered at 0.
(ii) Based on your results from part (i), find a general formula for f (k) (0).
(Think about how k being even or odd affects the value of the kth derivative.)
(c) Let f (x) = sin(x).
(i) Calculate the first four derivatives of f (x) at x = 0. Then find the fourth
order Taylor polynomial P 4 (x) for sin(x) centered at 0.
(ii) Based on your results from part (i), find a general formula for f (k) (0).
(Think about how k being even or odd affects the value of the kth derivative.)
⊳
It is possible that an nth order Taylor polynomial is not a polynomial of degree n;
that is, the order of the approximation can be different from the degree of the polynomial.
For example, in Activity 8.22 we found that the second order Taylor polynomial P 2 (x)
centered at 0 for sin(x) is P 2 (x) = x. In this case, the second order Taylor polynomial is a
degree 1 polynomial.
Taylor Series
In Activity 8.22 we saw that the fourth order Taylor polynomial P 4 (x) for sin(x) centered
at 0 is
P 4 (x) = x −
x 3
3!
.