Polynomials And Interpolation

a) From the following data, estimate the value of $\int_{1}^{5} \log x \, dx$ using Simpson's 1/3 rule. Also obtain the value of $h$, so that the value of the integral will be accurate up to five decimal places.

(8 marks)

(b) Determine the maximum step size that can be used in the tabulation of $f(x) = e^x$ in $[0, 1]$, so that the error in the linear interpolation will be less than $5 \times 10^{-4}$. Find also the step size if quadratic interpolation is used.

(8 Marks)

c) Evaluate the integral $I = \int_{0}^{1} \frac{x}{1+x^2} \, dx$ using trapezoidal rule with a subdivision of $0.25$. Estimate the maximum error involved.

(4 marks)

a) Define the following terms as used in computation

i) Relative Error
ii) Reformulation
iii) Interpolation [3 Marks]
b) i) Use three digit rounding arithmetic to perform the calculation below:

$$
\left( \frac{13}{14} - \frac{6}{14} \right) \div (2e - 5.4)
$$

ii) Compute the percentage error with the exact value determined to five digits rounding. [5 Marks]
c) Two qualities are found to be related as below

Assuming (y = f(x)) is continuous find

$$
\int_{1.0}^{1.2} f(x)dx, \text{ numerically using both trapezoidal and Simpsons rule.}
$$

[4 marks]

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