GATE Statistics Calculus Flashcards

24 sample cards from the Calculus deck

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

1. State the error bound for an alternating series satisfying Leibniz's conditions.

   If $S$ is the sum and $S_N$ the $N$-th partial sum, then $|S - S_N| \leq b_{N+1}$: the truncation error is bounded by the first omitted term.

2. Define absolute convergence of a series.

   $\sum a_n$ converges absolutely if $\sum |a_n|$ converges. Absolute convergence implies convergence of $\sum a_n$.

3. Define conditional convergence and give a standard example.

   $\sum a_n$ converges conditionally if $\sum a_n$ converges but $\sum |a_n|$ diverges. Example: $\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n}$ converges (to $\ln 2$) but $\sum \frac{1}{n}$ diverges.

4. What does Riemann's rearrangement theorem say about conditionally convergent series?

   A conditionally convergent series can be rearranged to converge to any real number, or to diverge. Absolutely convergent series have the same sum under any rearrangement.

5. Give the $\varepsilon$-$\delta$ definition of $\lim_{x \to a} f(x) = L$.

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

6. State the relationship between one-sided limits and the existence of a limit.

   $\lim_{x \to a} f(x)$ exists and equals $L$ iff both one-sided limits exist and are equal: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$.

7. Give the $\varepsilon$-$\delta$ definition of continuity of $f$ at $a$.

   $f$ is continuous at $a$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|x - a| < \delta$ implies $|f(x) - f(a)| < \varepsilon$; equivalently $\lim_{x\to a} f(x) = f(a)$.

8. State the Intermediate Value Theorem.

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

9. State the Extreme Value Theorem (boundedness theorem).

   A function continuous on a closed bounded interval $[a,b]$ is bounded and attains its maximum and minimum values on $[a,b]$.

10. Define a monotone (monotonic) function on an interval.

    $f$ is increasing if $x_1 < x_2 \Rightarrow f(x_1) \leq f(x_2)$ and decreasing if $x_1 < x_2 \Rightarrow f(x_1) \geq f(x_2)$. Strict versions use strict inequalities on the values.

11. What kind of discontinuities can a monotone function on an interval have?

    A monotone function can have at most countably many discontinuities, and they are all jump (first-kind) discontinuities; it has no removable or essential discontinuities.

12. Define uniform continuity of $f$ on a set $A$.

    $f$ is uniformly continuous on $A$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that for all $x, y \in A$, $|x - y| < \delta \Rightarrow |f(x) - f(y)| < \varepsilon$ ($\delta$ independent of the point).

13. State the theorem linking continuity on a closed interval to uniform continuity (Heine–Cantor).

    A function continuous on a closed and bounded interval $[a,b]$ is uniformly continuous on $[a,b]$.

14. Give an example showing continuity does not imply uniform continuity, and explain.

    $f(x) = \frac{1}{x}$ on $(0,1)$ and $f(x) = x^{2}$ on $\mathbb{R}$ are continuous but not uniformly continuous, because the required $\delta$ shrinks to $0$ as $x$ approaches $0$ (resp. grows large).

15. Define differentiability of $f$ at $a$.

    $f$ is differentiable at $a$ if $f'(a) = \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$ exists (finitely).

16. What is the relationship between differentiability and continuity?

    Differentiability at $a$ implies continuity at $a$, but not conversely. Example: $f(x) = |x|$ is continuous at $0$ but not differentiable there.

17. State Rolle's Theorem with all hypotheses.

    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 $f'(c) = 0$.

18. State Lagrange's Mean Value Theorem.

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

19. State Cauchy's Mean Value Theorem (generalized MVT).

    If $f, g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'(x) \neq 0$, then there exists $c \in (a,b)$ with $\dfrac{f'(c)}{g'(c)} = \dfrac{f(b) - f(a)}{g(b) - g(a)}$.

20. What does the sign of $f'$ tell you about monotonicity (a corollary of the MVT)?

    If $f'(x) > 0$ on an interval then $f$ is strictly increasing there; if $f'(x) < 0$ then strictly decreasing; if $f'(x) = 0$ throughout then $f$ is constant.

21. State Taylor's Theorem with the Lagrange form of the remainder.

    If $f$ is $n+1$ times differentiable, then $f(x) = \sum_{k=0}^{n} \dfrac{f^{(k)}(a)}{k!}(x-a)^{k} + R_n$, where $R_n = \dfrac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}$ for some $\xi$ between $a$ and $x$.

22. Write the Maclaurin series (Taylor at $0$) for $e^{x}$.

    $$e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$$

23. State L'Hôpital's Rule for the $\frac{0}{0}$ indeterminate form.

    If $\lim_{x\to a} f(x) = \lim_{x\to a} g(x) = 0$, $f,g$ differentiable near $a$ with $g'(x) \neq 0$, and $\lim_{x\to a}\frac{f'(x)}{g'(x)}$ exists, then $\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$.

24. To which indeterminate forms does L'Hôpital's Rule directly apply, and how are others handled?

    It applies directly to $\frac{0}{0}$ and $\frac{\infty}{\infty}$. Forms $0\cdot\infty$ and $\infty-\infty$ are converted to one of these by algebra; $0^{0}$, $1^{\infty}$, $\infty^{0}$ are handled by taking logarithms first.

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