Derive the Runge-Kutta 2nd Order formula for the differential equation
Consider the first-order ordinary differential equation dy/dx = f(x, y), with the initial condition y(x₀) = y₀. The aim of the Runge–Kutta second-order method is to obtain an approximation for y at xₙ₊₁ = xₙ + h that is accurate up to terms of order h². Expand the exact solution y(xₙ + h) in a Taylor series about xₙ: y(xₙ + h) = yₙ + h(dy/dx)ₙ + (h²/2)(d²y/dx²)ₙ + O(h³). Using _______ ____ ______ ______ ___ ________ ________.
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Using the linear congruential random number generator determine the first five random numbers given that: ,
,
, and
. Transform these random numbers to lie between 0 and
Explain why a polynomial is a useful choice for an interpolating function.
What would be the maximum possible period of this random number sequence?
Explain round-off error and truncation error, with one example of each.
Evaluate the integral using the 2-point Gaussian Quadrature Rule with the following values:
| Points | Weights | Arguments |
| 1 | 1.0 | 0.577 |
| 2 | 1.0 |
|
The series for ex can be written as:
Calculate the truncation error when the first four terms of the series are used to evaluate e1.5.
A population of single-celled organisms was grown in a Petri dish over a period of 24 hours. The number of organisms at a given time is recorded in the table below. Fit the data in the table below to an exponential model :
| x (time in hours) | 0 | 4 | 8 | 12 | 16 | 20 | 24 |
| y | 25 | 36 | 52 | 68 | 85 | 104 | 142 |