Calculus – GATE CS Premium Module (High-Accuracy Topic)

Calculus contributes 2–3 guaranteed marks in GATE CS every year.

Questions are predictable: limits, continuity, basic differentiation, and 1–2 quick integrals.

This chapter becomes an easy scoring zone once you learn:

How limits behave near special points
How to differentiate functions using shortcuts
When to use L’Hôpital’s rule
How to identify maxima/minima quickly

Let’s build the entire topic the way GATE expects.

1. Limits & Continuity (Most Tested Area)

Limits tell how a function behaves just before reaching a point.

1.1 Important Standard Limits (Must Memorise)

(displaystyle lim_{x o 0} rac{sin x}{x} = 1)
(displaystyle lim_{x o 0} rac{1 - cos x}{x} = 0)
(displaystyle lim_{x o 0} rac{e^x - 1}{x} = 1)
(displaystyle lim_{x o infty} left(1 + rac{1}{x} ight)^x = e)

GATE frequently asks indirect forms of these.

GATE Trick:

If the expression looks messy, try rewriting it so it matches one of the standard limits above.

1.2 L’Hôpital’s Rule (Only for 0/0 or ∞/∞)

(displaystyle lim_{x o a} rac{f(x)}{g(x)})

gives 0/0 or ∞/∞, then

[

lim_{x o a} rac{f(x)}{g(x)} = lim_{x o a} rac{f'(x)}{g'(x)}

]

This rule can save 60–70 seconds on many problems.

Example (Exam-Type):

[

lim_{x o 0} rac{e^{2x} - 1}{x}

]

Direct substitution → (0/0). Apply L’Hôpital:

[

lim_{x o 0} rac{2e^{2x}}{1} = 2e^{0} = 2

]

1.3 Continuity

A function (f(x)) is continuous at (x = a) if:

(f(a)) is defined
(displaystyle lim_{x o a} f(x)) exists
(displaystyle lim_{x o a} f(x) = f(a))

Types of Discontinuity:

Removable: Limit exists but (

eq f(a))

Jump: Left-hand limit (

eq) right-hand limit

Infinite: Function goes to (pm infty)

GATE Pattern: Continuity questions usually check:

Whether left/right limits match
Whether piecewise functions join smoothly at boundaries

2. Differentiation (Fastest Scoring Area)

Differentiation gives the rate of change of a function.

2.1 Essential Rules

Power Rule

[

rac{d}{dx}(x^n) = n x^{n-1}

]

Product Rule

[

(fg)' = f'g + fg'

]

Quotient Rule

[

left(rac{f}{g} ight)' = rac{f'g - fg'}{g^2}

]

Chain Rule (very important in GATE)

[

rac{d}{dx}[f(g(x))] = f'(g(x)) cdot g'(x)

]

2.2 Common Derivatives

(dfrac{d}{dx}(e^x) = e^x)
(dfrac{d}{dx}(ln x) = dfrac{1}{x})
(dfrac{d}{dx}(sin x) = cos x)
(dfrac{d}{dx}(cos x) = -sin x)
(dfrac{d}{dx}( an x) = sec^2 x)

These alone solve most differentiation questions.

2.3 Applications in Computer Science

Gradient Descent (Machine Learning)

[

heta_{ ext{new}} = heta_{ ext{old}} - alpha

abla J( heta)

]

Derivatives help compute gradients to minimise the loss function.

Algorithm Analysis & Optimisation

Understanding growth rates
Finding minima of continuous approximations of cost functions

3. Maxima & Minima (Direct 1‑Mark Questions)

A critical point occurs where:

(f'(x) = 0), or
(f'(x)) is undefined

To classify the critical point, use the second derivative test:

(f''(x) > 0) → local minimum
(f''(x) < 0) → local maximum
(f''(x) = 0) → inconclusive

GATE Hack (Quadratic Functions):

If (f(x) = ax^2 + bx + c):

(a > 0) → parabola opens up → minimum
(a < 0) → parabola opens down → maximum

No need to differentiate in many MCQs.

4. Integration (Less Frequent but Easy Marks)

Integration is the reverse of differentiation.

4.1 Basic Rules

Power Rule

[

int x^n , dx = rac{x^{n+1}}{n+1} + C quad (n

eq -1)

]

Substitution Rule

If (u = g(x)), then:

[

int f(g(x)) g'(x) , dx = int f(u) , du

]

Integration by Parts

[

int u , dv = uv - int v , du

]

Used for: (x e^x), (x sin x), (ln x), etc.

4.2 Common Integrals

(displaystyle int e^x dx = e^x + C)
(displaystyle int sin x , dx = -cos x + C)
(displaystyle int cos x , dx = sin x + C)
(displaystyle int rac{1}{x} , dx = ln|x| + C)

4.3 Definite Integrals

[

int_a^b f(x),dx = F(b) - F(a)

]

where (F(x)) is an antiderivative of (f(x)).

5. GATE PYQ‑Style Solved Questions

Q1. Limit (PYQ)

Compute (displaystyle lim_{x o 0} rac{sin 3x}{x}).

Rewrite:

[

rac{sin 3x}{x}

= rac{sin 3x}{3x} cdot 3

o 1 cdot 3 = 3 ext{ as } x o 0

]

Answer: 3

Q2. Differentiation (PYQ)

Let (f(x) = ln(x^2 + 1)). Find (f'(x)).

[

f'(x) = rac{1}{x^2 + 1} cdot 2x = rac{2x}{x^2 + 1}

]

Q3. Maxima–Minima (PYQ)

Let (f(x) = x^2 - 6x + 5). Find the minimum.

[

f'(x) = 2x - 6 = 0 Rightarrow x = 3

]

[

f''(x) = 2 > 0 Rightarrow ext{local minimum at } x = 3

]

Minimum value:

[

f(3) = 9 - 18 + 5 = -4

]

Q4. Integration (PYQ)

Evaluate (displaystyle int_0^1 (3x^2 + 2x),dx).

[

int_0^1 (3x^2 + 2x),dx

= left[ x^3 + x^2 ight]_0^1

= (1 + 1) - 0 = 2

]

6. Common Mistakes (Avoid These)

Using L’Hôpital when the limit is not 0/0 or ∞/∞
Forgetting the chain rule inside composite functions
Dropping minus signs in trigonometric derivatives
Wrong substitution in integrals
Treating discontinuous functions as differentiable

These errors easily cost 1–2 marks if you’re not careful.

7. Fast Revision Sheet (Night Before Exam)

Limits

(displaystyle rac{sin x}{x} o 1) as (x o 0)
(displaystyle rac{e^x - 1}{x} o 1) as (x o 0)

Derivatives

((e^x)' = e^x)
((ln x)' = 1/x)
((sin x)' = cos x)
((cos x)' = -sin x)

Max/Min

Critical point: (f'(x) = 0), then apply (f''(x)) test.

Integrals

(displaystyle int e^x dx = e^x + C)
(displaystyle int sin x dx = -cos x + C)
(displaystyle int rac{1}{x} dx = ln|x| + C)

8. Practice Set (Exam Simulation)

You can turn these into practice questions in Schoolabe:

Evaluate (displaystyle lim_{x o 0} rac{e^{5x} - 1}{x}).
Differentiate (f(x) = sin(x^2)).
Find the max/min of (f(x) = -2x^2 + 4x + 1).
Compute (displaystyle int (2x^3 - 3x),dx).
Evaluate (displaystyle int_1^2 rac{1}{x},dx).

Conclusion

Calculus in GATE CS is built on predictable patterns.

If you master limits, continuity checks, basic differentiation, and simple integrals, you can reliably secure the full 2–3 marks from this chapter.

Calculus