The heat released by a reaction whose power rises as kW

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Integration: error expansions, Gauss rules and adaptivity

Problem

The heat released by a reaction whose power rises as

q(t) = e^{t}

kW, with

t

in hours, over the first hour is

I = \int_0^1e^t\,\mathrm{d}t

kWh, and its exact value is

e - 1 = 1.7182818

kWh. Let

Q_1

be Simpson's rule on the whole interval

[0,1]

(three points) and

Q_2

the sum of Simpson's rule on

[0,\tfrac12]

and on

[\tfrac12,1]

(five points). ((a)) Compute

Q_1

and

Q_2

. ((b)) Form the estimate

(Q_2 - Q_1)/15

I - Q_2

from Eq. (5.26) and compare it with the true value of

I - Q_2

. Give the corrected value

Q_2 + (Q_2 - Q_1)/15

and its error, as value minus exact value. ((c)) An adaptive code uses the test

|Q_2 - Q_1| < 15\epsilon_{\rm tol}

on this interval. Does it accept with

\epsilon_{\rm tol} = 10^{-4}

kWh? With

\epsilon_{\rm tol} = 10^{-5}

kWh? ((d)) State the one assumption behind the estimate, say whether it holds here, and name a feature of an integrand that would make the estimate misleading.

Browse the problem library