GATE DA & AI Engineering Calculus and Optimization Syllabus

Calculus and Optimization syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Calculus and Optimization in GATE DA & AI Engineering, not a summary of it.

1. 1

   Functions of a Single Variable

   5 topics

   • Limit
   • Continuity and Differentiability
   • Taylor Series
   • Maxima and Minima
   • Optimization Involving a Single Variable

Calculus and Optimization flashcards for GATE DA & AI Engineering

23 of 50 cards from the Calculus and Optimization deck — real questions with worked answers.

1. What is the formal (epsilon-delta) definition of $\lim_{x \to a} f(x) = L$?

   For every $\varepsilon > 0$ there exists $\delta > 0$ such that $0 < |x - a| < \delta$ implies $|f(x) - L| < \varepsilon$.

2. State the condition relating one-sided limits to the existence of a two-sided limit.

   $\lim_{x \to a} f(x)$ exists and equals $L$ if and only if $\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = L$.

3. What is the value of the standard trigonometric limit $\lim_{x \to 0} \frac{\sin x}{x}$?

   $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (with $x$ in radians).

4. Evaluate $\lim_{x \to 0} \frac{1 - \cos x}{x^{2}}$.

   $\lim_{x \to 0} \frac{1 - \cos x}{x^{2}} = \frac{1}{2}$.

5. What is the limit definition of Euler's number $e$?

   $e = \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x} = \lim_{x \to 0} (1 + x)^{1/x}$.

6. Evaluate $\lim_{x \to 0} \frac{e^{x} - 1}{x}$.

   $\lim_{x \to 0} \frac{e^{x} - 1}{x} = 1$.

7. Evaluate $\lim_{x \to 0} \frac{\ln(1 + x)}{x}$.

   $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$.

8. State L'Hopital's rule for the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

   If $\lim \frac{f(x)}{g(x)}$ is of form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ and $g'(x) \neq 0$ near $a$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the latter limit exists.

9. List the seven standard indeterminate forms encountered when evaluating limits.

   $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $0^{0}$, $\infty^{0}$, and $1^{\infty}$.

10. State the Squeeze (Sandwich) Theorem for limits.

    If $g(x) \leq f(x) \leq h(x)$ near $a$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

11. What three conditions must hold for a function $f$ to be continuous at a point $x = a$?

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

12. Classify the types of discontinuities a function can have.

    Removable (limit exists but $\neq f(a)$ or $f(a)$ undefined), jump (one-sided limits exist but differ), and infinite/essential (a one-sided limit is infinite or does not exist).

13. What is the definition of the derivative $f'(a)$ as a limit?

    $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$, equivalently $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$.

14. State the relationship between differentiability and continuity.

    If $f$ is differentiable at $a$, then $f$ is continuous at $a$. The converse is false (e.g. $f(x) = |x|$ at $x = 0$ is continuous but not differentiable).

15. Give the condition for a function to be differentiable at a point in terms of one-sided derivatives.

    $f$ is differentiable at $a$ iff the left-hand derivative $\lim_{h \to 0^{-}} \frac{f(a+h)-f(a)}{h}$ equals the right-hand derivative $\lim_{h \to 0^{+}} \frac{f(a+h)-f(a)}{h}$.

16. State the product rule and quotient rule of differentiation.

    Product: $(uv)' = u'v + uv'$. Quotient: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}$.

17. State the chain rule for $\frac{d}{dx} f(g(x))$.

    $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$.

18. What is the Intermediate Value Theorem (IVT)?

    If $f$ is continuous on $[a, b]$ and $N$ lies between $f(a)$ and $f(b)$, then there exists $c \in (a, b)$ with $f(c) = N$.

19. State Rolle's Theorem.

    If $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists $c \in (a, b)$ such that $f'(c) = 0$.

20. State the Mean Value Theorem (Lagrange's MVT).

    If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

21. What is the general Taylor series of $f(x)$ about $x = a$?

    $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^{n} = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^{2} + \cdots$

22. What is a Maclaurin series, and how does it relate to a Taylor series?

    A Maclaurin series is a Taylor series expanded about $a = 0$: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}$.

23. Write the Lagrange form of the remainder $R_{n}(x)$ in Taylor's theorem.

    $R_{n}(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x - a)^{n+1}$ for some $\xi$ between $a$ and $x$.

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