Q1Advanced Engineering Mathematics I

To find the number of students obtaining marks below or above a certain value, we must first form a Cumulative Frequency Table.

| Marks Less Than (x) | No. of Students (y) | |---|---| | 40 | 31 | | 50 | 31 + 42 = 73 | | 60 | 73 + 51 = 124 | | 70 | 124 + 35 = 159 | | 80 | 159 + 31 = 190 |

We then construct a forward difference table for $y$. $\Delta y_0=42,{\Delta }^2{y}_0=51-42=9,{\Delta }^3{y}_0=-16-9=-25$, etc.

a) Less than 45 marks: Use Newton's Forward Interpolation formula to find $y$ at $x=45$. Here $x_0=40,h=10,p=(45-40)/10=0.5$. Apply formula $y=y_0+p\Delta y_0+\frac{p(p-1)}{2}{\Delta }^2{y}_0+\dots$ to get the cumulative frequency at 45.

b) More than 45 marks: Subtract the result of (a) from the total number of students (190).