GATE DA & AI Engineering Calculus and Optimization Syllabus

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

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

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

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

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 $x$.

State the Maclaurin series for $e^{x}$.

$e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$, valid for all $x$.

State the Maclaurin series for $\sin x$.

$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n+1}}{(2n+1)!} = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots$

State the Maclaurin series for $\cos x$.

$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots$

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 $-1 < x \leq 1$.

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 $|x| < 1$.

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 $|x| < 1$.

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 $a$.

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).

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).

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.

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.

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 $[a, b]$.

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.

Define concavity in terms of the second derivative.

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

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.

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).

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.

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).

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.

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.

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.

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.

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).

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.

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 $|x - a| < R$.

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.

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

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