CSEIT – GATE 1996/ Subject – Engineering Mathematics/ Topic – Calulus The formula used to compu.

Lui Kreyszig

1 week ago

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For comprehensive preparation in Calculus for the GATE 1996 Engineering Mathematics section, it’s beneficial to review previous years’ questions and their solutions. Here are some resources that can aid your study:

ExamSIDE: This platform offers a collection of GATE Mechanical Engineering (ME) previous year questions on Calculus, organized subject-wise and chapter-wise, complete with detailed solutions. GeeksforGeeks+2ExamSIDE+2ExamSIDE+2
GeeksforGeeks: They provide a compilation of GATE Computer Science and Engineering (CSE) previous year questions on Calculus, which can be instrumental in understanding the types of questions asked and the methodologies to solve them. GeeksforGeeks
GATE Overflow: This resource contains a comprehensive collection of previous year questions and solutions for GATE CSE, including those on Calculus. It serves as a valuable tool for in-depth understanding and practice. Gate Overflow

By thoroughly practicing these previous year questions and understanding the solutions, you can enhance your proficiency in Calculus and improve your performance in the GATE examination.

CSEIT – GATE 1996/ Subject – Engineering Mathematics/ Topic – Calulus The formula used to compu.

Mathematics and engineering in computer science

SK Mondal’s – GATE Mathematics

Here’s a concise and exam-focused explanation of the GATE 1996 CSE/IT – Engineering Mathematics topic related to Calculus, particularly focusing on the formula used to compute limits, derivatives, and integrals — often asked in various forms in GATE.

Topic: Calculus – GATE 1996 CSE/IT

1. Limits – Formulae Used

Limits are used to understand the behavior of a function as it approaches a specific point.

Standard Limit Results:

$lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1$ $lim⁡x→01−cos⁡xx2=12\lim_{x \to 0} \frac{1 – \cos x}{x^2} = \frac{1}{2}$ $lim⁡x→0ex−1x=1\lim_{x \to 0} \frac{e^x – 1}{x} = 1$ $lim⁡x→∞(1+1x)x=e\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

2. Derivatives – Important Rules

Definition of Derivative:

$\lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$

Standard Derivatives:

$ddx(xn)=nxn−1ddx(sin⁡x)=cos⁡xddx(cos⁡x)=−sin⁡xddx(ex)=exddx(ln⁡x)=1x\frac{d}{dx} (x^n) = nx^{n-1} \quad \frac{d}{dx} (\sin x) = \cos x \quad \frac{d}{dx} (\cos x) = -\sin x \quad \frac{d}{dx} (e^x) = e^x \quad \frac{d}{dx} (\ln x) = \frac{1}{x}$

Product Rule:

$ddx[u⋅v]=u′v+uv′\frac{d}{dx}[u \cdot v] = u’v + uv’$

Quotient Rule:

$ddx(uv)=u′v−uv′v2\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u’v – uv’}{v^2}$

Chain Rule:

$ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

3. Integration – Standard Formulae

Basic Formulas:

$∫xndx=xn+1n+1+C(n≠−1)\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \ne -1)$ $∫1xdx=ln⁡∣x∣+C∫exdx=ex+C\int \frac{1}{x} dx = \ln |x| + C \quad \int e^x dx = e^x + C$ $∫sin⁡xdx=−cos⁡x+C∫cos⁡xdx=sin⁡x+C\int \sin x dx = -\cos x + C \quad \int \cos x dx = \sin x + C$

Integration by Parts:

$dx)dx\int u \cdot v \, dx = u \int v \, dx – \int \left( \frac{du}{dx} \cdot \int v \, dx \right) dx$

Definite Integration:

$f(x))\int_a^b f(x) dx = F(b) – F(a) \quad \text{(where $ F(x) $ is antiderivative of $ f(x) $)}$

GATE 1996 Example Question (Conceptual)

Q:
Evaluate:

$lim⁡x→0ex−sin⁡x−xx3\lim_{x \to 0} \frac{e^x – \sin x – x}{x^3}$

Solution Outline:

Use Taylor Series Expansion:

$ex=1+x+x22!+x33!+⋯e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ $sin⁡x=x−x33!+⋯\sin x = x – \frac{x^3}{3!} + \cdots$

So:

$)−x=1+x22+x33e^x – \sin x – x = (1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \dots) – (x – \frac{x^3}{6} + \dots) – x = 1 + \frac{x^2}{2} + \frac{x^3}{3}$

Now divide by $x^3$ :

$1+x22+x33x3=1×3+12x+13→∞\frac{1 + \frac{x^2}{2} + \frac{x^3}{3}}{x^3} = \frac{1}{x^3} + \frac{1}{2x} + \frac{1}{3} \to \infty$

But since numerator starts with constant term, the limit does not exist as finite.

(However, if the problem had been $lim⁡x→0ex−sin⁡x−xx2\lim_{x \to 0} \frac{e^x – \sin x – x}{x^2}$ , then the approach would give a finite value.)

Summary Table

Concept	Formula Used
Limit	$sin⁡xx=1\frac{\sin x}{x} = 1$ , Series forms
Derivative	Chain Rule, Product Rule, Standard forms
Integration	Power Rule, By Parts, Definite Integrals

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CSEIT – GATE 1996/ Subject – Engineering Mathematics/ Topic – Calulus The formula used to compu.

CSEIT – GATE 1996/ Subject – Engineering Mathematics/ Topic – Calulus The formula used to compu.