Engineering Mathematics

Numerical Methods

Newton–Raphson root finding and numerical integration (trapezoidal & Simpson's rules) — approximating answers when closed forms fail.

PART 1

Topic Breakdown & Traps

The Engineering Principle

When an equation can't be solved or an integral can't be evaluated by hand, we approximate. Newton–Raphson finds a root by repeatedly following the tangent line down to the axis — fast (quadratic) convergence near a good guess. Numerical integration estimates area by replacing the curve with simple shapes: the trapezoidal rule uses straight-line segments, while Simpson's 1/3 rule fits parabolas through pairs of intervals, giving far higher accuracy for the same number of points.

The Core Formula Matrix

Newton–Raphson iteration:

x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}

.

Trapezoidal rule (

n

strips, width

h

\int_{a}^{b} f d x \approx \frac{h}{2} [f_{0} + 2 (f_{1} + \dots + f_{n - 1}) + f_{n}]

.

Simpson's 1/3 rule (even number of strips):

\int_{a}^{b} f d x \approx \frac{h}{3} [f_{0} + 4 (f_{1} + f_{3} + \dots) + 2 (f_{2} + f_{4} + \dots) + f_{n}]

.

Here

h = \frac{b - a}{n}

and

f_{i} = f (a + ih)

The ‘IIT Traps’

⚠
Newton–Raphson sign. It's $x_{n} - f / f^{'}$ (minus). Using a plus sign sends the iteration the wrong way.
⚠
Trapezoidal weights are 1, 2, 2, …, 2, 1. Only the end ordinates get weight 1; the interior ones are doubled. Simpson alternates 4 and 2.
⚠
Simpson needs an even number of strips. With an odd $n$ the 1/3 rule can't be applied directly.
⚠
Coarse grids over-/under-estimate. The trapezoidal rule over-estimates a convex (cup-up) curve; refine $h$ or use Simpson for accuracy.

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · MCQ

Applying one Newton–Raphson iteration to

f (x) = x^{2} - 2

starting from

x_{0} = 1

gives

x_{1} =

Q2MEDIUM2 Marks · NAT

Estimate

\int_{0}^{2} x^{2} d x

using the trapezoidal rule with

n = 2

strips. The estimate is ______. (Round off to two decimal places.)

Q3HARD2 Marks · NAT

Estimate

\int_{0}^{2} x^{2} d x

using Simpson's 1/3 rule with

n = 2

strips. The estimate is ______. (Round off to two decimal places.)