WBJEE Mathematics - Calculus and Trigonometry Flashcards

24 sample cards from the Mathematics - Calculus and Trigonometry deck

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

1. Simplify $2\tan^{-1}x$ as a single inverse function (give the standard "$2\tan^{-1}x$" results).

   $2\tan^{-1}x = \tan^{-1}\!\dfrac{2x}{1-x^{2}} = \sin^{-1}\!\dfrac{2x}{1+x^{2}} = \cos^{-1}\!\dfrac{1-x^{2}}{1+x^{2}}$ (for appropriate ranges of $x$).

2. State the odd/negative-argument identities for $\sin^{-1}(-x)$, $\cos^{-1}(-x)$, and $\tan^{-1}(-x)$.

   $\sin^{-1}(-x) = -\sin^{-1}x$, $\quad \tan^{-1}(-x) = -\tan^{-1}x$, $\quad \cos^{-1}(-x) = \pi - \cos^{-1}x$.

3. State the sine rule (law of sines) for a triangle with sides $a, b, c$ and circumradius $R$.

   $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R$.

4. State the cosine rule for side $a$, and the rearrangement giving $\cos A$.

   $a^{2} = b^{2} + c^{2} - 2bc\cos A$, equivalently $\cos A = \dfrac{b^{2}+c^{2}-a^{2}}{2bc}$.

5. State the projection formula for side $a$ of a triangle.

   $a = b\cos C + c\cos B$ (and cyclically $b = c\cos A + a\cos C$, $c = a\cos B + b\cos A$).

6. Give three expressions for the area of a triangle: the $\frac{1}{2}ab\sin C$ form and Heron's formula.

   Area $\Delta = \dfrac{1}{2}ab\sin C = \dfrac{1}{2}bc\sin A = \dfrac{1}{2}ca\sin B$, and Heron's formula $\Delta = \sqrt{s(s-a)(s-b)(s-c)}$ with $s = \dfrac{a+b+c}{2}$.

7. State the half-angle formula for $\sin\dfrac{A}{2}$ in a triangle in terms of $s$ and the sides.

   $\sin\dfrac{A}{2} = \sqrt{\dfrac{(s-b)(s-c)}{bc}}$, where $s = \dfrac{a+b+c}{2}$.

8. State the half-angle formulas for $\cos\dfrac{A}{2}$ and $\tan\dfrac{A}{2}$ in a triangle.

   $\cos\dfrac{A}{2} = \sqrt{\dfrac{s(s-a)}{bc}}$, $\quad \tan\dfrac{A}{2} = \sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}}$, with $s=\dfrac{a+b+c}{2}$.

9. How are the area $\Delta$, circumradius $R$, and inradius $r$ related to the sides and $s$?

   $R = \dfrac{abc}{4\Delta}$ and $r = \dfrac{\Delta}{s}$, where $\Delta$ is the area and $s$ the semi-perimeter.

10. Define a function and distinguish one-one (injective), onto (surjective), and bijective functions.

    A function $f:A\to B$ assigns each element of $A$ exactly one element of $B$. One-one (injective): distinct inputs give distinct outputs. Onto (surjective): every element of $B$ is an image. Bijective: both one-one and onto.

11. What is the domain and range of $f(x) = \sqrt{x}$ and of $f(x) = \log x$?

    $\sqrt{x}$: domain $[0,\infty)$, range $[0,\infty)$. $\log x$: domain $(0,\infty)$, range $\mathbb{R}$ (all reals).

12. Define an even function and an odd function, with a defining symmetry property.

    Even: $f(-x) = f(x)$ (graph symmetric about the $y$-axis), e.g. $\cos x$, $x^{2}$. Odd: $f(-x) = -f(x)$ (graph symmetric about the origin), e.g. $\sin x$, $x^{3}$.

13. State the standard limit $\displaystyle\lim_{x\to 0}\frac{\sin x}{x}$ and $\displaystyle\lim_{x\to 0}\frac{\tan x}{x}$.

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

14. State the standard limit $\displaystyle\lim_{x\to 0}\frac{a^{x}-1}{x}$ and $\displaystyle\lim_{x\to 0}\frac{e^{x}-1}{x}$.

    $\displaystyle\lim_{x\to 0}\frac{a^{x}-1}{x} = \ln a$ and $\displaystyle\lim_{x\to 0}\frac{e^{x}-1}{x} = 1$.

15. State the standard limits $\displaystyle\lim_{x\to 0}\frac{\log(1+x)}{x}$ and $\displaystyle\lim_{x\to 0}(1+x)^{1/x}$.

    $\displaystyle\lim_{x\to 0}\frac{\log(1+x)}{x} = 1$ and $\displaystyle\lim_{x\to 0}(1+x)^{1/x} = e$.

16. State the standard limit $\displaystyle\lim_{x\to a}\frac{x^{n}-a^{n}}{x-a}$.

    $\displaystyle\lim_{x\to a}\frac{x^{n}-a^{n}}{x-a} = n\,a^{n-1}$.

17. State L'Hospital's rule and the indeterminate forms to which it directly applies.

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

18. What is the condition for a function $f$ to be continuous at a point $x = a$?

    $f$ is continuous at $a$ if $\displaystyle\lim_{x\to a}f(x)$ exists, $f(a)$ is defined, and $\displaystyle\lim_{x\to a^{-}}f(x) = \lim_{x\to a^{+}}f(x) = f(a)$.

19. What is the relationship between differentiability and continuity of a function at a point?

    If $f$ is differentiable at $a$, then $f$ is continuous at $a$. The converse is false: continuity does not imply differentiability (e.g. $f(x)=|x|$ at $x=0$).

20. State the definition of the derivative of $f$ at $x$ (first-principles / limit definition).

    $f'(x) = \displaystyle\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$, provided the limit exists.

21. State the product rule and quotient rule for differentiation.

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

22. State the chain rule for differentiating a composite function $y = f(g(x))$.

    $\dfrac{dy}{dx} = f'(g(x))\cdot g'(x)$, i.e. if $y = f(u)$ and $u = g(x)$ then $\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}$.

23. Give the derivatives of $\sin x$, $\cos x$, $\tan x$, $e^{x}$, and $\ln x$.

    $\dfrac{d}{dx}\sin x = \cos x$; $\dfrac{d}{dx}\cos x = -\sin x$; $\dfrac{d}{dx}\tan x = \sec^{2}x$; $\dfrac{d}{dx}e^{x} = e^{x}$; $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$.

24. Give the derivatives of the inverse functions $\sin^{-1}x$, $\tan^{-1}x$, and $\sec^{-1}x$.

    $\dfrac{d}{dx}\sin^{-1}x = \dfrac{1}{\sqrt{1-x^{2}}}$; $\dfrac{d}{dx}\tan^{-1}x = \dfrac{1}{1+x^{2}}$; $\dfrac{d}{dx}\sec^{-1}x = \dfrac{1}{|x|\sqrt{x^{2}-1}}$.

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Mathematics - Calculus and Trigonometry flashcards FAQ