Chapter 2, Differential Calculus Video Solutions, IIT JEE Super Course in Mathematics: Calculus

Chapter Questions

Problem 1

Evaluate the following limits (without applying L'Hospitals Rule):
(i) $\lim _{i+4} \frac{\sqrt{3-x}-\sqrt{3+x}}{x}$
(ii) $\lim _{n \rightarrow 4}\left(\frac{x^{5}-3125}{x-5}\right)$
(iii) $\lim _{n+4} \frac{\sqrt{1+x}-1}{x}$
(iv) $\lim _{6 \rightarrow+} \frac{\sin 5 x}{\tan 7 x}$
(v) $\lim _{x \rightarrow 1} \frac{1-\cos 7 \mathbf{x}}{1-\cos 9 x}$
(vi) $\lim _{n \rightarrow+} \frac{x \tan x}{1-\cos x}$
(vil) $\lim _{n \rightarrow \theta}\left(\frac{\tan 7 x-3 x}{7 x-\sin ^{2} x}\right)$
(viii) $\lim _{s+\infty}\left(\frac{\sin 2 x+\sin 6 x}{\sin 5 x-\sin 3 x}\right)$
(ix) $\lim _{n \rightarrow \infty}\left(\frac{\tan ^{-1} 2 x}{\sin 3 x}\right)$
(x) $\lim _{n \rightarrow \infty}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{b}}+\ldots+\frac{1}{3^{n}}\right)$
(xi) $\lim _{n \rightarrow 1} \frac{\sin (x-1)}{x^{7}+x-2}$
(xii) $\lim _{x \rightarrow \infty} 2 x\left(\sqrt{x^{2}+1}-x\right)$
(xiii) $\lim _{x \rightarrow \infty} \sqrt{\frac{x-\sin x}{x+\cos ^{1} x}}$
(xiv) $\lim _{t+1} \frac{\sin x-x}{x^{1}}$
(sv) $\lim _{x+1} \frac{2-\sqrt{2+x}}{\left[2^{y / 3}-(4-x)^{y / 3}\right]}$

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Problem 2

2. Check the continuity of the following functions:
(i) $f(x)=\left\{\begin{aligned} \frac{\sin ^{2} a x}{x^{2}}, \text { when } x \neq 0 & \text { at } x=0 \\ a^{2} \text { when } x=0 & \text { - } \end{aligned}\right.$
(ii) $f(x)=\left\{\begin{array}{ll}2-x, & x<2 \\ 2+1 & x \geq 2\end{array}\right.$ at $x=2$
(iii) $f(x)=\left\{\begin{array}{cl}5 x-4 & 0<x \leq 1 \\ 4 x^{2}-3 x & 1<x<2\end{array}\right.$ at $x=1$
(iv) $f(x)=\left\{\begin{array}{l}x^{3}\left(\frac{e^{1 / x}-e^{-1 / 2}}{e^{1 / x}+e^{-1 / 2}}\right), \quad x \neq 0 \\ 0 \quad, \quad x=0\end{array}\right.$ at $x=0$
(v) If $\mathrm{f}(\mathbf{x})=\left\{\begin{array}{ll}\frac{\cos 3 \mathrm{x}-1}{\sqrt{5 \mathrm{x}^{2}+1}-1} & \mathrm{x} \neq 0 \\ \lambda . & \text { , find the value of } \lambda \text { so that } \mathrm{f}(\mathbf{x}) \text { is continuous at } \mathrm{x}=0 .\end{array}\right.$

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Problem 3

Find the derivatives of the following functions:
(i) $y=\frac{\sec x-1}{\sec x+1}$
(ii) $y=\frac{1}{\log (\cos x)}$
(iii) $\left.y=\log \left(\log f \log \left(x^{\prime}\right)\right)\right)$
(iv) $y=\sin \left(\tan ^{-1} x\right)$
(v) $y=\sec ^{-1}\left(\frac{1}{2 x^{1}-1}\right)$
(vi) $y=\sin ^{-1}\left(\frac{\sqrt{1+x}+\sqrt{1-x}}{2}\right)$
(vii) $y=\tan ^{-1}\left(\frac{a \cos x-b \sin x}{b \cos x+a \sin x}\right)$
(viii) $y=\tan ^{-1} \frac{x \sqrt{2}}{\left(1-x^{1}\right)}+\log \left(\frac{1-x \sqrt{2}+x^{4}}{1+x \sqrt{2}+x^{2}}\right)$
(ix) $y=a \log \left(\frac{a+\sqrt{a^{2}-x^{2}}}{x}\right)-\sqrt{a^{2}-x^{2}}$ where, a is a constant
$(x) y=(1+\log x)^{y^{\prime}}$
(xi) $y=\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$
(xii) $y=\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}\right)$
(xiii) $y=\tan ^{-1} x+\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)+\tan ^{-1}\left(\frac{3 x-x^{2}}{1-3 x^{2}}\right)$
(xiv) $y=\tan ^{-1}\left(\frac{3 \sin x-3 \cos x}{3 \sin x+3 \cos x}\right)$
(xv) $f(x)=\sin (\log x)$ and $y=f\left(\frac{2 x+3}{3-2 x}\right)$

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08:55

Problem 4

Find the derivatives of the following functions:
(1) $\left.y=x^{\left[s^{\prime}\right.}\right)$
(ii) $y=x^{2^{2}}$
(iii) $y=\sqrt{\tan x+\sqrt{\tan x+\sqrt{\tan x}+\ldots . u}}$
(iv) If $x^{3}=e^{x-1}$, show that $\frac{d y}{d x}=\frac{y^{2} \log x}{x^{2}}$
(v) If $x \sin (a+y)=\sin y, 8$ how that $\frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\operatorname{stn} a}$

Suman Saurav T.

Numerade Educator

Problem 5

Evaluate the following limits:
(1) $\lim _{n \rightarrow 0}\left(2-\frac{x}{a}\right)^{\ln \frac{\pi}{2 a}}$
(ii) $\lim _{x \rightarrow+(}\left(\frac{1}{x}-\frac{1}{e^{s}-1}\right)$
(iii) $\lim _{s \rightarrow 4}\left(e^{x x}+2 x\right)^{1 / x}$
(iv) $\lim _{s \rightarrow b}\left(\frac{\tan x}{x}\right)^{1 / x^{\prime}}$
(v) $\lim _{x \rightarrow-} \frac{x^{4} \sin \frac{1}{x}+2 x^{2}}{3+|x|^{\prime}}$
(vi) $\lim _{x \rightarrow \infty} x\left(\tan ^{-1} \frac{x+1}{x+2}-\frac{\pi}{4}\right)$
(vil) $\lim _{x \rightarrow-\infty}\left(\frac{x-2}{x+3}\right)^{2 x}$
(vilii) $\lim _{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+\mathrm{x}\right)^{Y}\right)$
(ix) $\left.\lim _{x \rightarrow+\left(\frac{\sin x}{\sin a}\right)}\right\}(x-e)$
(x) $\lim _{x \rightarrow 7 / 2} \frac{2^{-\cos k}-1}{x\left(x-\frac{\pi}{2}\right)}$

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Problem 6

(i) If $x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}=k$, show that $\frac{d^{2} y}{d x^{2}}=-\frac{k}{\left(1-x^{2}\right)^{3 / t}}$.
(ii) If $\mathrm{x}=\mathrm{s}(\mathrm{t}-\sin t), y=3(1-\cos \mathrm{t})$, find $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}$,
(iii) If $y=\sin \left(m \sin ^{-1} x\right)$, prove that $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+m^{2} y=0 .$
(iv) If $\cos ^{-1}\left(\frac{y}{b}\right)=\log \left(\frac{x}{n}\right)^{n}$, show that $x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+n^{1} y=0$.
(v) If $y=f(x)$ and $f$ is inversible, establish the result $\frac{d^{2} y}{d x^{1}}=-\frac{\frac{d^{2} x}{d y^{2}}}{\left(\frac{d x}{d y}\right)^{3}}$.
(vi) Use the resalt in (v) above to obtain $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ for the functions given below:
(a) $y=\sin ^{-1} x$
(b) $\sin y=x \sin (a+y)$

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02:49

Problem 7

Evaluate $\sum_{m=1}^{n} r x^{n-1}$ using Calculus.

Nidhi S.

Numerade Educator

Problem 8

Prove the inequalities
(i) $x-\log (1+x)>\frac{x^{2}}{2(1+x)^{2}}, x>0$
(ii) $1-\frac{x^{2}}{2}<\cos x<1-\frac{x^{2}}{2}+\frac{x^{4}}{24}$, for $0<x<\pi / 2$

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10:58

Problem 9

For the curve $x=a \cos ^{3} \theta, y=a \sin ^{3} \theta$, the tangent at $\theta$ ' meets the coordinate axes in $M$ and $N$. Find MN. Also find
(i) the length of tangent
(ii) the length of normal
(iii) the length of sub tangent and
(iv) the length of sub normal at the point ${ }^{*} \theta^{n}$

Malika S.

Numerade Educator

07:18

Problem 10

Let $\mathrm{A}\left(\mathrm{p}^{2},-\mathrm{p}\right), \mathrm{B}\left(\mathrm{q}^{2}, \mathrm{q}\right), \mathrm{C}\left(\mathrm{r}^{2},-\mathrm{r}\right)$ be the vertices of a triangle $\mathrm{ABC}$. A parallelogram $\mathrm{AFDE}$ is drawn with $\mathrm{D}, \mathrm{E}, \mathrm{F}$ on
the line segments $\mathrm{BC}, \mathrm{CA}, \mathrm{AB}$ respectively. Using calculus, show that the maximum area of such a parallelogram is
$\frac{1}{4}(\mathrm{p}+\mathrm{q})(\mathrm{q}+\mathrm{r})(\mathrm{p}-\mathrm{r})$

Nidhi S.

Numerade Educator

09:38

Problem 11

$P_{n}(x)$ represents the nth degree polynomial in $x$ defined as $P_{n}(x)=\frac{1}{2^{n} n !} \frac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n}$. Find $P_{0}(x), P_{1}(x), P_{2}(x)$ and
$\mathrm{P}_{5}(\mathrm{x})$ and verify that $\left(1-\mathrm{x}^{2}\right) \mathrm{P}_{\mathrm{n}}^{\prime \prime}(\mathrm{x})-2 \mathrm{x} \mathrm{P}_{\mathrm{n}}^{\prime}(\mathrm{x})+\mathrm{n}(\mathrm{n}+1) \mathrm{P}_{\mathrm{a}}(\mathrm{x})=0$ where $\mathrm{n}=0,1,2,3 .$

Malika S.

Numerade Educator

For the curve defined as $y=\operatorname{cosec}^{-1}\left(\frac{1+x^{2}}{2 x}\right)+\sec ^{-1}\left(\frac{1+x^{2}}{1-x^{2}}\right)$ the set of points at which the tangent is parallel to the
$x$ -axis is
(a) $\bar{\phi}$
(b) $\mathrm{R}$
(c) $\mathrm{N}$
(d) $\mathrm{Z}$

Malika S.

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03:33

Problem 31

Directions: Each question contains Statement- 1 and Statement-2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement- 1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement 2 is False
(d) Statement- 1 is False, Statement- 2 is True

Statement 1 $\lim _{x \rightarrow 0}\left(\frac{\sin x-e^{x}+1}{x}\right)$ exists and $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1, \lim _{x \rightarrow 0} \frac{e^{x}-1}{x}=1$
and
Statement 2
If $\lim _{x \rightarrow 1}(f(x)-g(x))$ exists, then, both $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ exist.

Malika S.

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02:25

Problem 32

Statement 1 $\mathrm{f}(\mathrm{x})=[\mathrm{x}]^{2}-5[\mathrm{x}]+3,1<\mathrm{x}<4$, where [] represents the greatest integer function is not continuous at $\mathrm{x}=2$ and 3 but is bounded.
and

Statement 2
A continuous function in a closed interval I is bounded.

Malika S.

Numerade Educator

01:45

Problem 33

Statement 1
$\lim _{x \rightarrow \infty} \frac{\cos x}{x}=0$
and
Statement 2
$\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$

Malika S.

Numerade Educator

01:32

Let $\mathrm{f}(\mathrm{x})=\max \left(1-\mathrm{x}, \mathrm{x}^{2}-1\right)$. Then $\mathrm{f}$ is
(a) not continuous at $\mathrm{x}=1,-2$
(b) continuous and differentiable everywhere
(c) not differentiable at $\mathrm{x}=-2,1$
(d) continuous but not differentiable at $\mathrm{x}=1,-1$

Suman Saurav T.

Numerade Educator

03:46

Problem 116

If $f(x)=(1-x)^{-1},|x|<1$ then the value of $\frac{f^{\prime i}(0)}{f(0)}+\frac{f^{i i \prime}(0)}{f^{i}(0)}+\frac{f^{i r}(0)}{f^{i i}(0)}+\ldots+\frac{f^{n+1}(0)}{f^{n-1}(0)}$ is
(a) $\frac{n(n+1)(n+2)}{3}$
(b) $\frac{n(n+1)(2 n+1)}{6}$
(c) $\frac{n(n+1)(n+2)}{6}$
(d) None of the above

Malika S.

Numerade Educator

Directions: Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement- 1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True

If $f(x)=\max \left\{x^{2}-4,|x-2|,|x-4|\right\}$ then
(a) $\mathrm{f}(\mathrm{x})$ is continuous for all $\mathrm{x} \in \mathrm{R}$
(b) $\mathrm{f}(\mathrm{x})$ is differentiable except at $\mathrm{x}=\frac{-1 \pm \sqrt{33}}{2}$
(c) $\mathrm{f}(\mathrm{x})$ has a critical point at $\mathrm{x}=2$
(d) $\mathrm{f}(\mathrm{x})$ has no maximum

Suman Saurav T.

Numerade Educator

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ such that for all $\mathrm{x}_{1}, \mathrm{x}_{2} \in \mathrm{R}, \mathrm{f}\left(\mathrm{x}_{2}\right)-\mathrm{f}\left(\mathrm{x}_{1}\right)|\leq| \mathrm{x}_{2}-\left.\mathrm{x}_{1}\right|^{3} .$ Prove that $\mathrm{f}(\mathrm{x})$ is a constant function.

Suman Saurav T.

Numerade Educator

(i) Find $S$, the sum of the infinite geometric series whose first term is $\mathrm{r}$ and common ratio is $\frac{1}{\mathrm{r}+1}$
(ii) Evaluate $\lim _{n \rightarrow \infty}\left\{\frac{\left[\sum_{r=1}^{2 n-1}\left(\mathrm{~S}_{t}-1\right)\right]^{3}}{\left[\sum_{r=1}^{2 n-1}\left(\mathrm{~S}_{t}-1\right)^{2}\right]^{2}}\right\}$

Suman Saurav T.

Numerade Educator

Problem 182

The function $\mathrm{f}(\mathrm{x})$ is defined as $\mathrm{f}(\mathrm{x})=\lceil\mathrm{x}\rceil+|2-\mathrm{x}|,-2<\mathrm{x}<3$ where $\lceil\mathrm{x}\rceil$ denotes the least integer greater than or equal to $\underline{x}$.
(i) Split the function $\mathrm{f}(\mathrm{x})$ into integral intervals
(ii) Discuss the continuity of $\mathrm{f}(\mathrm{x})$
(iii) Draw the graph of $\mathrm{f}(\mathrm{x})$.

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04:04

Problem 183

A function $\mathrm{y}=\mathrm{f}(\mathrm{x})$ is defined as $\mathrm{x}=3 \mathrm{t}-|\mathrm{t}|, \mathrm{y}=\mathrm{e}^{4 \mathrm{t}}$ for all $\mathrm{t}$
(i) Express $y$ in terms of $x$.
(ii) Discuss the continuity of $\mathrm{f}(\mathrm{x})$.

Suman Saurav T.

Numerade Educator

Directions: Each question contains Statement-1 and Statement-2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True

Statement 1 Let $f(x)=[x]+\left[x+\frac{1}{4}\right]+\left[x+\frac{1}{2}\right], x \in R$ where, [] denotes the greatest integer function. Then, the number of points of discontinuity of $\mathrm{f}(\mathrm{x})$ in $[-1,1]$ is 7
and
Statement 2 $[x+k]=[x]+k$ where, $k$ is an integer.

Suman Saurav T.

Numerade Educator

02:26

Problem 269

Directions: Each question contains Statement-1 and Statement-2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True

Statement 1
$\lim _{x \rightarrow 2} \frac{\sqrt{1-\cos (x-2)}}{(x-2)}=\frac{1}{\sqrt{2}}$
and
Statement 2
$\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1$

Suman Saurav T.

Numerade Educator

01:42

Let $F(x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\frac{x^{5}}{5}$
Statement 1
$\mathrm{F}(\mathrm{x})$ has exactly one zero at $\mathrm{x}=0 .$
and
Statement 2 $\mathrm{F}^{\prime}(\mathrm{x})$ is minimum at $\mathrm{x}=-1$.

Suman Saurav T.

Numerade Educator

05:11

Problem 274

Directions: Each question contains Statement-1 and Statement-2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True

Let $\mathrm{f}(x)=e^{2 x}(x-1)^{y / 5}$
Statement 1 $\mathrm{f}(\mathrm{x})$ has a maximum at $\mathrm{x}=\frac{3}{5}$.
and
Statement 2 $\mathrm{f}(\mathrm{x})$ is not differentiable at $\mathrm{x}=1$.

Suman Saurav T.

Numerade Educator

05:00

$\begin{array}{ll}\text { Column I } & \text { Column II }\end{array}$
(a) $\lim _{x \rightarrow 3} \frac{\left(x^{3}+27\right) \log (x-2)}{x^{2}-9}=$
(p) 12
(b) $\lim _{x \rightarrow 0}\left(\frac{\mathrm{e}^{\mathrm{x}}-1}{\mathrm{x}}\right)^{\left(\frac{x}{x+1-\epsilon^{*}}\right)}=$
(q) 8
(c) If $\lim _{x \rightarrow 0} \frac{x(a+\cos x)-b \sin x}{x^{3}}=1$ then a and b arerespectively
(r) 9
(d) If $\mathrm{f}(\mathrm{x})$ is a thrice differentiable function such that
(s) $\mathrm{e}^{-1}$ $\lim _{x \rightarrow 0} \frac{f(4 x)-3 f(3 x)+3 f(2 x)-f(x)}{x^{3}}=12$, then $f^{\prime \prime}(0)$
is equal to

Suman Saurav T.

Numerade Educator