GATE DA & AI Engineering Calculus and Optimization Flashcards

24 sample cards from the Calculus and Optimization deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the Maclaurin series for $\ln(1 + x)$ and its interval of validity.

   $\ln(1 + x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^{n}}{n} = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \cdots$, valid for $-1 < x \leq 1$.

2. State the Maclaurin (geometric) series for $\frac{1}{1 - x}$ and its interval of validity.

   $\frac{1}{1 - x} = \sum_{n=0}^{\infty} x^{n} = 1 + x + x^{2} + x^{3} + \cdots$, valid for $|x| < 1$.

3. State the binomial series for $(1 + x)^{k}$.

   $(1 + x)^{k} = \sum_{n=0}^{\infty} \binom{k}{n} x^{n} = 1 + kx + \frac{k(k-1)}{2!}x^{2} + \cdots$, valid for $|x| < 1$.

4. What is the first-order (linear) Taylor approximation of $f$ near $a$?

   $f(x) \approx f(a) + f'(a)(x - a)$, the tangent-line linearization of $f$ at $a$.

5. Define a critical (stationary) point of a single-variable function.

   A point $x = c$ in the domain where $f'(c) = 0$ or $f'(c)$ does not exist. Local extrema can only occur at critical points (Fermat's theorem).

6. State the First Derivative Test for classifying a critical point $c$.

   If $f'$ changes from $+$ to $-$ at $c$, $f$ has a local maximum; from $-$ to $+$, a local minimum; if $f'$ does not change sign, $c$ is neither (e.g. an inflection).

7. State the Second Derivative Test for a critical point where $f'(c) = 0$.

   If $f''(c) > 0$, $f$ has a local minimum at $c$; if $f''(c) < 0$, a local maximum; if $f''(c) = 0$, the test is inconclusive.

8. What is the difference between a local (relative) extremum and a global (absolute) extremum?

   A local extremum is the largest/smallest value in some neighborhood of the point; a global extremum is the largest/smallest value over the entire domain.

9. State the Extreme Value Theorem.

   If $f$ is continuous on a closed bounded interval $[a, b]$, then $f$ attains both an absolute maximum and an absolute minimum on $[a, b]$.

10. What is the closed-interval method for finding absolute extrema of $f$ on $[a, b]$?

    Evaluate $f$ at all critical points in $(a, b)$ and at the endpoints $a$ and $b$; the largest value is the absolute maximum and the smallest is the absolute minimum.

11. Define concavity in terms of the second derivative.

    $f$ is concave up (convex) where $f''(x) > 0$ and concave down (concave) where $f''(x) < 0$.

12. What is a point of inflection and what condition characterizes it?

    A point where concavity changes sign. A necessary condition is $f''(x) = 0$ or $f''$ undefined there, and $f''$ must actually change sign across the point.

13. For a higher-order test, how do you classify a stationary point where $f'(c) = f''(c) = \cdots = f^{(n-1)}(c) = 0$ but $f^{(n)}(c) \neq 0$?

    If $n$ is even: local minimum when $f^{(n)}(c) > 0$, local maximum when $f^{(n)}(c) < 0$. If $n$ is odd: it is a point of inflection (neither max nor min).

14. What is the necessary first-order condition for an interior optimum in single-variable optimization?

    The stationarity condition $f'(x^{*}) = 0$ at an interior optimum $x^{*}$ of a differentiable objective.

15. What is the second-order sufficient condition for an interior point $x^{*}$ with $f'(x^{*}) = 0$ to be a strict local minimum?

    $f''(x^{*}) > 0$ (strictly positive second derivative).

16. In a single-variable minimization, why must boundary points of the feasible interval be checked separately?

    Because the optimum may occur at a boundary where $f' \neq 0$; the stationarity condition only locates interior optima, so endpoints must be evaluated directly.

17. For a convex function on an interval, what can be said about a stationary point?

    If $f$ is convex ($f''(x) \geq 0$ throughout) and $f'(x^{*}) = 0$, then $x^{*}$ is a global minimum; for convex functions any local minimum is global.

18. What is the general process to solve an applied single-variable optimization (max/min) problem?

    (1) Express the quantity to optimize as a function of one variable using constraints; (2) determine the feasible domain; (3) find critical points via $f'(x) = 0$; (4) test critical points and endpoints; (5) select the optimal value.

19. Define a convex function on an interval using the inequality definition.

    $f$ is convex if for all $x_{1}, x_{2}$ and $\lambda \in [0,1]$: $f(\lambda x_{1} + (1-\lambda) x_{2}) \leq \lambda f(x_{1}) + (1-\lambda) f(x_{2})$. The reverse inequality defines concave.

20. Evaluate $\lim_{x \to \infty} \frac{a_{m} x^{m} + \cdots}{b_{n} x^{n} + \cdots}$ for a rational function (degrees $m$ numerator, $n$ denominator).

    If $m < n$, the limit is $0$; if $m = n$, it is $\frac{a_{m}}{b_{n}}$; if $m > n$, the limit is $\pm\infty$ (diverges).

21. State Taylor's theorem with the second-order expansion used to derive the second derivative test.

    $f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^{2}$; at a stationary point ($f'(a)=0$) the sign of $f''(a)$ determines whether $f$ increases or decreases on both sides.

22. What is the radius of convergence of a power series $\sum a_{n}(x-a)^{n}$ and how is it found by the ratio test?

    The radius $R$ is the distance from $a$ within which the series converges: $R = \lim_{n \to \infty} \left|\frac{a_{n}}{a_{n+1}}\right|$ when this limit exists; the series converges for $|x - a| < R$.

23. Compare necessary versus sufficient conditions for a local minimum at an interior differentiable point.

    Necessary (first-order): $f'(x^{*}) = 0$. Sufficient (second-order): $f'(x^{*}) = 0$ and $f''(x^{*}) > 0$. Necessary conditions can hold without an extremum (e.g. inflection), but sufficient conditions guarantee one.

24. Using the linear approximation, estimate $\sqrt{4.1}$ via $f(x) = \sqrt{x}$ at $a = 4$.

    $f'(x) = \frac{1}{2\sqrt{x}}$, so $\sqrt{4.1} \approx \sqrt{4} + \frac{1}{2\sqrt{4}}(0.1) = 2 + \frac{0.1}{4} = 2.025$.

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