MATH256 Problem Sheet 1
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH256 Problem Sheet 1
1.1 Use repeated differentiation to obtain all terms in the Taylor series for the
function
p(x) = 2x3 + 4x2 - 7x + 5
about the point x = 3.
1.2 (a) (i) Find a formula for the nth derivative of f (x), in the case where
f (x) = 1 lxl < 1.
(ii) Hence obtain the full Taylor series for the function f (x) about the point x = 0.
(iii) Can you see an easy way to confirm that your result is correct?
Hint: think about the types of infinite series that you know how to sum.
(b) Use the result of part (a) to deduce the full Taylor series about x = 0 for
(i) g(x) = 1 (ii) h(x) = arctan(x), lxl < 1.
Hint: what is the derivative of arctan(x)?
1.3 Use the Taylor series for f (c + h) and f (c - h) to obtain a centred difference formula for f\\ (c) with an O (h2 ) error. Make sure you retain enough terms to clearly establish the order of the error.
1.4 Suppose that the function g(t) has the convergent Taylor series expansion g(t) = j g(j)(0) .
(a) Show that
-h(h) g(t) dt = 2 j h2j+1 .
Show that
g(h) - 2g(0) + g(-h) h2
= g\\ (0) + O(h2 ).
Make sure you include enough terms in the Taylor series to clearly
establish the order of the error.
(c) Using the results of parts (a) and (b), show that
-h(h) g(t) dt = ←g(-h) + 4g(0) + g(h)] + O(h5 ).
1.5 Suppose the series
o
j=1
is such that
lAj+1l s klAj l, for j > N, where 0 < k < 1. (a) Find an upper bound for the magnitude of the tail
T = Aj ,
j =N+1
in terms of lAN+1l.
What does this tell you about the error in the approximation
N
j=1
2023-05-05