GATE CS - ENGINEERING MATHEMATICS:Calculus

Calculus for GATE CS

Calculus usually contributes about 2–3 marks: standard limits, continuity checks, a differentiation (often chain rule), and a short definite/indefinite integral. Keep the standard limits automatic; rewrite messy expressions until they match one.

Topics that recur:

• Limits near 0 / ∞ (including forms that need L’Hôpital for 0/0 or ∞/∞)
• Basic derivatives and maxima/minima of polynomials
• Simple integrals

1. Limits & Continuity (Most Tested Area)

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

1.1 Important standard limits

• $\displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1$
• $\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} = 0$
• $\displaystyle \lim_{x \to 0} \frac{e^x - 1}{x} = 1$
• $\displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

GATE frequently asks indirect forms of these.

Tip:

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 ∞/∞)

If

$\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)}$

gives 0/0 or ∞/∞, then

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

Example (Exam-Type):

Direct substitution → $0/0$. Apply L’Hôpital:

1.3 Continuity

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

1. $f(a)$ is defined
2. $\displaystyle \lim_{x \to a} f(x)$ exists
3. $\displaystyle \lim_{x \to a} f(x) = f(a)$

Types of Discontinuity:

• Removable: Limit exists but $\neq f(a)$
• Jump: Left-hand limit $\neq$ 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

Product Rule

Quotient Rule

Chain Rule (very important in GATE)

2.2 Common Derivatives

• $\dfrac{d}{dx}(e^x) = e^x$
• $\dfrac{d}{dx}(\ln x) = \dfrac{1}{x}$
• $\dfrac{d}{dx}(\sin x) = \cos x$
• $\dfrac{d}{dx}(\cos x) = -\sin x$
• $\dfrac{d}{dx}(\tan x) = \sec^2 x$

These alone solve most differentiation questions.

2.3 Applications in Computer Science

Gradient Descent (Machine Learning)

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

Substitution Rule

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

Integration by Parts

Used for: $x e^x$, $x \sin x$, $\ln x$, etc.

4.2 Common Integrals

• $\displaystyle \int e^x dx = e^x + C$
• $\displaystyle \int \sin x \, dx = -\cos x + C$
• $\displaystyle \int \cos x \, dx = \sin x + C$
• $\displaystyle \int \frac{1}{x} \, dx = \ln|x| + C$

4.3 Definite Integrals

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

5. GATE PYQ‑Style Solved Questions

Q1. Limit (PYQ)

Compute $\displaystyle \lim_{x \to 0} \frac{\sin 3x}{x}$.

Rewrite:

Answer: 3

Q2. Differentiation (PYQ)

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

Q3. Maxima–Minima (PYQ)

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

Minimum value:

Q4. Integration (PYQ)

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

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

• $\displaystyle \frac{\sin x}{x} \to 1$ as $x \to 0$
• $\displaystyle \frac{e^x - 1}{x} \to 1$ as $x \to 0$

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

• $\displaystyle \int e^x dx = e^x + C$
• $\displaystyle \int \sin x dx = -\cos x + C$
• $\displaystyle \int \frac{1}{x} dx = \ln|x| + C$

8. Practice set

You can turn these into practice questions in Schoolabe:

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

Timed drills: GATE hub.