(Solved): 1. Values of a function \( f(x) \) are tabulated over a range of \( x \) as follows: (a) Use approp ...

1. Values of a function \( f(x) \) are tabulated over a range of \( x \) as follows: (a) Use appropriate values from the table to approximate \( f(0.7) \) using linear interpolation. [3 marks] (b) Estimate \( f^{\prime}(2.5) \) using the two-point formula and appropriate values from the table. [3 marks] (c) Use the tabulated values and a finite difference formula to estimate \( f^{\prime \prime}(2) \) using \( h=1 \). [3 marks] (d) Use the trapezium rule and the values given in the table to approximate \( \int_{1}^{3} f(x) d x \), using \( n=4 \) strips. [3 marks] (e) Use Simpson's rule and the values given in the table to approximate \( \int_{1}^{3} f(x) d x \), using \( n=4 \) strips. [3 marks] (f) If both trapezium rule and Simpson's rule are employed to approximate the integral \( \int_{-1}^{1}|x| d x \) with 50 strips, explain which method (possibly none or both) will give the exact integral \( \int_{-1}^{1}|x| d x=1 \). [5 marks] [End of Question 1; 20 marks total]