Engineering Mathematics

Calculus

Derivatives, definite integrals, maxima–minima and area between curves — the rate-of-change and accumulation engine of GATE numericals.

PART 1

Topic Breakdown & Traps

The Engineering Principle

Calculus answers two complementary questions. The derivative gives the *instantaneous rate of change* — the slope of a curve — and at a maximum or minimum that slope is zero. The definite integral gives *accumulation* — the signed area under a curve between two limits. The Fundamental Theorem of Calculus ties them together: integrating a function and then differentiating returns the original, so a definite integral is evaluated by finding an antiderivative and taking the difference at the limits.

The Core Formula Matrix

Power rule:

\frac{d}{d x} x^{n} = n x^{n - 1}

, and

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)

.

Fundamental theorem:

\int_{a}^{b} f (x) d x = F (b) - F (a)

, where

F^{'} = f

.

Stationary points: solve

f^{'} (x) = 0

; a maximum has

f^{''} < 0

, a minimum has

f^{''} > 0

.

Area between curves

y = g (x)

(upper) and

y = h (x)

(lower):

A = \int_{a}^{b} [g (x) - h (x)] d x

Area between the line y = 2x (upper) and the parabola y = x² (lower) from x = 0 to x = 2 — the shaded region equals ∫₀²(2x − x²)dx.

The ‘IIT Traps’

⚠
Don't forget the lower limit. A definite integral is $F (b) - F (a)$ ; evaluating only at $b$ drops the $- F (a)$ term.
⚠
Upper minus lower curve. For area between curves, subtract the *lower* function from the *upper* one over the interval where they don't cross; reversing them gives a negative (wrong-sign) area.
⚠
** $n = - 1$ exception.** $\int x^{- 1} d x = ln ∣ x ∣$ , not $x^{0} /0$ . The power rule breaks at $n = - 1$ .
⚠
Slope zero ≠ always a maximum. $f^{'} = 0$ marks a stationary point; you must check $f^{''}$ (or sign change) to classify it.

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · MCQ

f (x) = 3 x^{3} - 2 x^{2} + 5

, then

f^{'} (1)

equals:

Q2MEDIUM2 Marks · NAT

The value of the definite integral

\int_{0}^{2} (3 x^{2} + 2 x) d x

is ______. (Round off to two decimal places.)

Q3HARD2 Marks · NAT

The area enclosed between the curves

y = 2 x

and

y = x^{2}

(see figure) is ______ square units. (Round off to two decimal places.)

y = 2x and y = x² intersect at x = 0 and x = 2; the line is the upper boundary in between.