GATE Textile Engineering ENGINEERING MATHEMATICS Flashcards

24 sample cards from the ENGINEERING MATHEMATICS deck

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

1. Give the linear (tangent-line) approximation of $f$ near $x = a$.

   $f(x) \approx f(a) + f'(a)(x - a)$.

2. State the Fundamental Theorem of Calculus (evaluation form) for $\int_{a}^{b} f(x)\,dx$.

   If $F'(x) = f(x)$, then $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$.

3. How is an improper integral $\int_{a}^{\infty} f(x)\,dx$ defined and when does it converge?

   $\int_{a}^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_{a}^{b} f(x)\,dx$; it converges if this limit exists and is finite.

4. For which values of $p$ does $\int_{1}^{\infty} \dfrac{dx}{x^{p}}$ converge?

   It converges if $p > 1$ and diverges if $p \leq 1$.

5. Define the Gamma function and state its factorial property.

   $\Gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x}\,dx$, with $\Gamma(n+1) = n\,\Gamma(n)$ and $\Gamma(n+1) = n!$ for non-negative integers.

6. What does it mean for a sequence $\{a_{n}\}$ to converge to a limit $L$?

   For every $\varepsilon > 0$ there is $N$ such that $n > N$ implies $|a_{n} - L| < \varepsilon$; i.e. $\lim_{n \to \infty} a_{n} = L$.

7. State the $n$-th term test (divergence test) for the series $\sum a_{n}$.

   If $\lim_{n \to \infty} a_{n} \neq 0$, the series diverges. (If the limit is $0$, the test is inconclusive.)

8. State the ratio test for convergence of $\sum a_{n}$.

   Let $L = \lim_{n \to \infty} \left|\dfrac{a_{n+1}}{a_{n}}\right|$. Converges if $L < 1$, diverges if $L > 1$, inconclusive if $L = 1$.

9. State the root test for convergence of $\sum a_{n}$.

   Let $L = \lim_{n \to \infty} \sqrt[n]{|a_{n}|}$. Converges if $L < 1$, diverges if $L > 1$, inconclusive if $L = 1$.

10. State the $p$-series test for $\sum_{n=1}^{\infty} \dfrac{1}{n^{p}}$.

    Converges if $p > 1$, diverges if $p \leq 1$.

11. State the comparison test for positive-term series.

    If $0 \le a_{n} \le b_{n}$: convergence of $\sum b_{n}$ implies convergence of $\sum a_{n}$; divergence of $\sum a_{n}$ implies divergence of $\sum b_{n}$.

12. How is the radius of convergence $R$ of a power series $\sum a_{n} x^{n}$ found via the ratio test?

    $R = \lim_{n \to \infty} \left|\dfrac{a_{n}}{a_{n+1}}\right|$; the series converges for $|x| < R$.

13. Write the Taylor series expansion of $f(x)$ about $x = a$.

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

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

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

15. State the Maclaurin series for $\sin x$.

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

16. Write the general first-order linear ODE and its integrating factor.

    $\dfrac{dy}{dx} + P(x)y = Q(x)$; integrating factor $\mu = e^{\int P(x)\,dx}$, giving $y\,\mu = \int Q\,\mu\,dx + C$.

17. What is the condition for the first-order ODE $M\,dx + N\,dy = 0$ to be exact?

    $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$.

18. What substitution reduces a Bernoulli equation $\dfrac{dy}{dx} + P y = Q y^{n}$ to linear form?

    Substitute $v = y^{1-n}$, which converts it into a first-order linear ODE in $v$.

19. For the higher-order linear ODE with constant coefficients, how are the complementary roots used when the auxiliary equation has real distinct roots $m_{1}, m_{2}$?

    The complementary function is $y_{c} = C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}$.

20. For a constant-coefficient linear ODE, give the complementary function when the auxiliary equation has complex roots $m = \alpha \pm i\beta$.

    $y_{c} = e^{\alpha x}\left(C_{1}\cos\beta x + C_{2}\sin\beta x\right)$.

21. State the standard form of the Euler-Cauchy (equidimensional) equation and the substitution that solves it.

    $x^{2}\dfrac{d^{2}y}{dx^{2}} + a x \dfrac{dy}{dx} + b y = 0$; substitute $x = e^{t}$ (or try $y = x^{m}$) to obtain a constant-coefficient ODE.

22. How is a second-order linear PDE $A u_{xx} + B u_{xy} + C u_{yy} + \cdots = 0$ classified?

    By the discriminant $B^{2} - 4AC$: hyperbolic if $> 0$, parabolic if $= 0$, elliptic if $< 0$.

23. Write the one-dimensional wave equation and identify its type.

    $\dfrac{\partial^{2} u}{\partial t^{2}} = c^{2}\dfrac{\partial^{2} u}{\partial x^{2}}$; it is a hyperbolic PDE.

24. Write the one-dimensional heat (diffusion) equation and identify its type.

    $\dfrac{\partial u}{\partial t} = \alpha \dfrac{\partial^{2} u}{\partial x^{2}}$; it is a parabolic PDE.

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