Q1) Given that

Find at .

A1)

Here the first derivative is to be calculated at the beginning of the table, therefore forward difference formula will be used

Forward difference table is given below:

By Newton’s forward differentiation formula for differentiation

Here 

Q2) Find the first and second derivatives of the function given below at the point :

A2)

Here the point of the calculation is at the beginning of the table,

Forward difference table is given by:

By Newton’s forward differentiation formula for differentiation

Here  , 0.

Again 

At

Q3) Given that

Find ?

A3)

Backward difference table:

Newton’s Backward formula for differentiation

Here

Q4) State the trapezoidal rule for finding an approximate area under the given curve. A curve is given by the points (x, y) given below:

(0, 23), (0.5, 19), (1.0, 14), (1.5, 11), (2.0, 12.5), (2.5, 16), (3.0, 19), (3.5, 20), (4.0, 20).

Estimate the area bounded by the curve, the x axis and the extreme ordinates.

A4)

We construct the data table:

Here length of interval h =0.5, initial value a = 0 and final value b = 4

By Trapezoidal method

Area of curve bounded on x axis =

Q5) Evaluate using trapezoidal rule with five ordinates

Here

A5)

We construct the data table:

Q6) Evaluate

Using Simpson’s 1/3 rule with .

A6)

For , we construct the data table:

By Simpson’s Rule

Q7) Evaluate

A7)

By Simpson’s 3/8 rule.

Let us divide the range of the interval [4, 5.2] into six equal parts.

For h=0.2, we construct the data table:

By Simpson’s 3/8 rule 

= 1.8278475

Q8) Evaluate 

A8)

Let

Here the interval of x and y are and .

Let 

Consider the following table:

Or        

By Trapezoidal Rule

.

Q9) Evaluate

A9)

Let 

And

Or

By Trapezoidal Rule

Q10) Evaluate 

A10)

Let

Here

Let the number of intervals be .

Or

By Simpson’s 1/3 Rule

Q11) Evaluate

A11)

Let 

And

Or

By Simpson’s 1/3 Rule