Continuity and Differentiability

Q97 An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ………… FillBlank Q98 Derivative of ${x^2}$ w.r.t. ${x^3}$ is …………. FillBlank Q99 If $f(x) = |\cos x|,$ then ${f^\prime }\left( {\frac{\pi }{4}} \right)$ is equal to …….. FillBlank Q100 If $f(x) = |\cos x - \sin x|,$ then ${f^\prime }\left( {\frac{\pi }{3}} \right)$ is equal to………….. FillBlank Q101 For the curve $\sqrt x + \sqrt y = 1,\frac{{dy}}{{dx}}$ at $\left( {\frac{1}{4},\frac{1}{4}} \right)$ is………… FillBlank Q79 Find the values of $p$ and $q$, so that $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{{x^2} + 3x + p,{\rm{ if }}x \le 1}\\{qx + 2,{\rm{ }}\,\,\,\,\,\,\,\,\,\,\,{\rm{if }}x > 1}\end{array}} \right.$ is differentiable at $x = 1$} LA Q80 If ${x^m} \cdot {y^n} = {(x + y)^{m + n}}$, prove that

(i) $\frac{{dy}}{{dx}} = \frac{y}{x}$ and

(ii) $\frac{{{d^2}y}}{{d{x^2}}} = 0$ LA Q81 If $x = \sin t$ and $y = \sin pt,$ then prove that
$\left( {1 - {x^2}} \right)\frac{{{d^2}y}}{{d{x^2}}} - x\frac{{dy}}{{dx}} + {p^2}y = 0$ LA Q82 Find the value of $\frac{{dy}}{{dx}}$, if $y = {x^{\tan x}} + \sqrt {\frac{{{x^2} + 1}}{2}}$ LA Q83 If $f(x) = 2x$ and $g(x) = \frac{{{x^2}}}{2} + 1,$ then which of the following can be a discontinuous function? MCQ Q84 The function $f(x) = \frac{{4 - {x^2}}}{{4x - {x^3}}}$ is MCQ Q85 The set of points where the function $f$ given by $f(x) = |2x - 1|\sin x$ is differentiable is MCQ Q86 The function $f(x) = \cot x$ is discontinuous on the set MCQ Q87 The function $f(x) = {e^{|x|}}$ is MCQ Q88 If $f(x) = {x^2}\sin \frac{1}{x},$ where $x \ne 0,$ then the value of the function $f$ at $x = 0,$ so that the function is continuous at $x = 0,$ is MCQ Q89 If $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{mx + 1,}&{{\rm{ if }}\ x \le \frac{\pi}{2}}\\{(\sin x + n),}&{{\rm{ if }}\ x > \frac{\pi}{2}}\end{array}} \right.$ is continuous at $x = \frac{\pi}{2}$, find the relation between $m$ and $n$. MCQ Q90 If $f(x) = |\sin x|,$ then MCQ Q91 If $y = \log \left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right),$ then $\frac{{dy}}{{dx}}$ is equal to MCQ Q92 If $y = \sqrt {\sin x + y}$, then $\frac{{dy}}{{dx}}$ is equal to MCQ Q93 The derivative of ${\cos ^{ - 1}}\left( {2{x^2} - 1} \right)$ w.r.t. ${\cos ^{ - 1}}x$ is MCQ Q94 If $x = {t^2}$ and $y = {t^3},$ then $\frac{{{d^2}y}}{{d{x^2}}}$ is equal to MCQ Q95 The value of $c$ in Rolle's theorem for the function $f(x) = {x^3} - 3x$ in the interval $[0,\sqrt 3 ]$ is MCQ Q96 For the function $f(x) = x + \frac{1}{x},x \in [1,3],$ the value of $c$ for mean value theorem is MCQ Q $\sin (xy) + \frac{x}{y} = {x^2} - y$ SA Q1 Examine the continuity of the function $f(x) = {x^3} + 2{x^2} - 1$ at $x = 1$. SA Q2 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{3x + 5,}&{{\rm{ if }}x \ge 2}\\{{x^2},}&{{\rm{ if }}x < 2}\end{array}} \right.$at $x = 2$.} SA Q3 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\frac{{1 - \cos 2x}}{{{x^2}}},}&{{\rm{ if }}x \ne 0}\\{5,}&{{\rm{ if }}x = 0}\end{array}} \right.$at $x = 0$ } SA Q4 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\frac{{2{x^2} - 3x - 2}}{{x - 2}},}&{{\rm{ if }}x \ne 2}\\{5,}&{{\rm{ if }}x = 2}\end{array}} \right.$at $x = 2$ } SA Q5 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\frac{{|x - 4|}}{{2(x - 4)}},}&{{\rm{ if }}x \ne 4}\\{0,}&{{\rm{ if }}x = 4}\end{array}} \right.$at $x = 4$ } SA Q6
$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{|x|\cos \frac{1}{x},}&{{\rm{ if }}x \ne 0}\\{0,}&{{\rm{ if }}x = 0}\end{array}} \right.$at $x = 0$ } SA Q7 $$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{|x - a|\sin \frac{1}{{x - a}},}&{{\rm{ if }}x \ne 0}\\{0,}&{{\rm{ if }}x = a}\end{array}} \right.$$, at $x = a$ } SA Q8 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\frac{{{e^{1/x}}}}{{1 + {e^{1/x}}}},}&{{\rm{ if }}x \ne 0}\\{0,}&{{\rm{ if }}x = 0}\end{array}} \right.$at $x = 0$} SA Q9
$$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\frac{{{x^2}}}{2},}&{{\rm{ if }}0 \le x \le 1}\\{2{x^2} - 3x + \frac{3}{2},}&{{\rm{ if }}1 < x \le 2}\end{array}} \right.$$at $x = 1$ } SA Q10 $f(x) = |x| + |x - 1|$ at $x = 1$ SA Q11 $f(x) = \left\{ {\begin{array}{cccccccccccccccccccc}{3x - 8,{\rm{ if }}x \le 5}\\{2k,{\rm{ }}\,\,\,\,\,\,\,{\rm{if }}x > 5}\end{array}} \right.$at $x = 5$ } SA Q12 $f(x) = \left\{ {\begin{array}{cccccccccccccccccccc}{\frac{{{2^{x + 2}} - 16}}{{{4^x} - 16}},}&{{\rm{ if }}x \ne 2}\\{k,}&{{\rm{ if }}x = 2}\end{array}} \right.$at $x = 2$} SA Q13 $f(x) = \left\{ {\begin{array}{cccccccccccccccccccc}{\frac{{\sqrt {1 + kx} - \sqrt {1 - kx} }}{x},}&{{\rm{ if }} - 1 \le x < 0}\\{\frac{{2x + 1}}{{x - 1}},}&{{\rm{ if }}0 \le x \le 1}\end{array}} \right.$at $x = 0$} SA Q14 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\frac{{1 - \cos kx}}{{x\sin x}},}&{{\rm{ if }}x \ne 0}\\{\frac{1}{2},}&{{\rm{ if }}x = 0}\end{array}} \right.$at $x = 0$} SA Q15 Prove that the function $f$ defined by $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\frac{x}{{|x| + 2{x^2}}},}&{{\rm{ if }}x \ne 0}\\{k,}&{{\rm{ if }}x = 0}\end{array}} \right.$ remains discontinuous at $x = 0$, regardless the choice of $k$ } SA Q16 Find the values of $a$ and $b$ such that the function $f$ defined by
$$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\frac{{x - 4}}{{|x - 4|}} + a,}&{{\rm{ if }}x < 4}\\{a + b,}&{{\rm{ if }}x = 4}\\{\frac{{x - 4}}{{|x - 4|}} + b,}&{{\rm{ if }}x > 4}\end{array}} \right.$$.
is a continuous function at $x = 4$} SA Q17 If the function $f(x) = \frac{1}{{x + 2}}$, then find the points of discontinuity of the composite function $y = f\{ f(x)\}$ SA Q18 Find all points of discontinuity of the function $f(t) = \frac{1}{{{t^2} + t - 2}}$, where $t = \frac{1}{{x - 1}}$ SA Q19 Show that the function $f(x) = |\sin x + \cos x|$ is continuous at $x = \pi$.
SA Q20 Examine the differentiability of ${\rm{f}}$, where ${\rm{f}}$ is defined by
$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{x[x],}&{{\rm{ if }}0 \le x < 2}\\{(x - 1)x,}&{{\rm{ if }}2 \le x < 3}\end{array}} \right.$at $x = 2$} SA Q21 $$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{{x^2}\sin \frac{1}{x},}&{{\rm{ if }}x \ne 0}\\{0,}&{{\rm{ if }}x = 0}\end{array}} \right.$$at $x = 0$ } SA Q22 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{1 + x,{\rm{ if }}x \le 2}\\{5 - x,{\rm{ if }}x > 2}\end{array}} \right.$at $x = 2$ } SA Q23 Show that $f(x) = |x - 5|$ is continuous but not differentiable at $x = 5$ SA Q24 A function $f:R \to R$ satisfies the equation $f(x + y) = f(x) \cdot f(y)$ for all $x,y \in R,$ $f(x) \ne 0$. Suppose that the function is differentiable at $x = 0$ and ${f^\prime }(0) = 2$, then prove that ${f^\prime }(x) = 2f(x)$ SA Q25 ${2^{{{\cos }^2}x}}$ SA Q26 $\frac{{{8^x}}}{{{x^8}}}$ SA Q27 $\log \left( {x + \sqrt {{x^2} + a} } \right)$ SA Q28 $\log \left[ {\log \left( {\log {x^5}} \right)} \right]$ SA Q29 $\sin \sqrt x + {\cos ^2}\sqrt x$ SA Q30 ${\sin ^n}\left( {a{x^2} + bx + c} \right)$ SA Q31 cos(tan$\sqrt{x+1}$ ) SA Q32 $\sin {x^2} + {\sin ^2}x + {\sin ^2}\left( {{x^2}} \right)$ SA Q33 ${\sin ^{ - 1}}\frac{1}{{\sqrt {x + 1} }}$ SA Q34 ${(\sin x)^{\cos x}}$ SA Q35 ${\sin ^m}x \cdot {\cos ^n}x$ SA Q36 ${(x + 1)^2}{(x + 2)^3}{(x + 3)^4}$ SA Q37 ${\cos ^{ - 1}}\left( {\frac{{\sin x + \cos x}}{{\sqrt 2 }}} \right), - \frac{\pi }{4} < x < \frac{\pi }{4}$ SA Q38 ${\tan ^{ - 1}}\sqrt {\frac{{1 - \cos x}}{{1 + \cos x}}} , - \frac{\pi }{4} < x < \frac{\pi }{4}$ SA Q39 ${\tan ^{ - 1}}(\sec x + \tan x),\frac{{ - \pi }}{2} < x < \frac{\pi }{2}$ SA Q40 ${\tan ^{ - 1}}\left( {\frac{{a\cos x - b\sin x}}{{b\cos x + a\sin x}}} \right),\frac{{ - \pi }}{2} < x < \frac{\pi }{2}$ and $\frac{a}{b}\tan x > - 1$ SA Q41 ${\sec ^{ - 1}}\left( {\frac{1}{{4{x^3} - 3x}}} \right),0 < x < \frac{1}{{\sqrt 2 }}$ SA Q42 ${\tan ^{ - 1}}\left( {\frac{{3{a^2}x - {x^3}}}{{{a^3} - 3a{x^2}}}} \right),\frac{{ - 1}}{{\sqrt 3 }} < \frac{x}{a} < \frac{1}{{\sqrt 3 }}$ SA Q43 ${\tan ^{ - 1}}\left[ {\frac{{\sqrt {1 + {x^2}} + \sqrt {1 - {x^2}} }}{{\sqrt {1 + {x^2}} - \sqrt {1 - {x^2}} }}} \right], - 1 < x < 1,x \ne 0$ SA Q44 $x = t + \frac{1}{t},y = t - \frac{1}{t}$ SA Q45 $x = {e^\theta }\left( {\theta + \frac{1}{\theta }} \right),y = {e^{ - \theta }}\left( {\theta - \frac{1}{\theta }} \right)$ SA Q46 $x = 3\cos \theta - 2{\cos ^3}\theta ,y = 3\sin \theta - 2{\sin ^3}\theta$ SA Q47 $\sin x = \frac{{2t}}{{1 + {t^2}}},\tan y = \frac{{2t}}{{1 - {t^2}}}$ SA Q48 $x = \frac{{1 + \log t}}{{{t^2}}},y = \frac{{3 + 2\log t}}{t}$ SA Q49 If $x = {e^{\cos 2t}}$ and $y = {e^{\sin 2t}}$, then prove that $\frac{{dy}}{{dx}} = - \frac{{y\log x}}{{x\log y}}$ SA Q50 If $x = a\sin 2t(1 + \cos 2t)$ and $y = b\cos 2t(1 - \cos 2t),$ then show that ${\left( {\frac{{dy}}{{dx}}} \right)_{t = \pi /4}} = \frac{b}{a}$ SA Q51 If $x = 3\sin t - \sin 3t,y = 3\cos t - \cos 3t,$ then find $\frac{{dy}}{{dx}}$ at $t = \frac{\pi }{3}$ SA Q52 Differentiate $\frac{x}{{\sin x}}$ w.r.t. $\sin x$ SA Q53 Differentiate ${\tan ^{ - 1}}\frac{{\sqrt {1 + {x^2}} - 1}}{x}$ w.r.t. ${\tan ^{ - 1}}x,$ when $x \ne 0.$ SA Q55 $\sec (x + y) = xy$ SA Q56 ${\tan ^{ - 1}}\left( {{x^2} + {y^2}} \right) = a$ SA Q57 ${\left( {{x^2} + {y^2}} \right)^2} = xy$ SA Q58 If $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0,$ then show that $\frac{{dy}}{{dx}} \cdot \frac{{dx}}{{dy}} = 1$ SA Q59 If $x = {e^{x/y}},$ then prove that $\frac{{dy}}{{dx}} = \frac{{x - y}}{{x\log x}}$ SA Q60 If ${y^x} = {e^{y - x}}$, then prove that $\frac{{dy}}{{dx}} = \frac{{{{(1 + \log y)}^2}}}{{\log y}}$ SA Q61 If $y = {(\cos x)^{{{(\cos x)}^{{{(\cos x)}^{ - \infty }}}}}},$ then show that $\frac{{dy}}{{dx}} = \frac{{{y^2}\tan x}}{{y\log \cos x - 1}}.$ SA Q62 If $x\sin (a + y) + \sin a \cdot \cos (a + y) = 0,$ then prove that
$\frac{{dy}}{{dx}} = \frac{{{{\sin }^2}(a + y)}}{{\sin a}}$ SA Q63 If $\sqrt {1 - {x^2}} + \sqrt {1 - {y^2}} = a(x - y),$ then prove that $\frac{{dy}}{{dx}} = \sqrt {\frac{{1 - {y^2}}}{{1 - {x^2}}}}$ SA Q64 If $y = {\tan ^{ - 1}}x,$ then find $\frac{{{d^2}y}}{{d{x^2}}}$ in terms of $y$ alone SA Q65 $f(x) = x{(x - 1)^2}$ in [0,1] SA Q66 $f(x) = {\sin ^4}x + {\cos ^4}x$ in $\left[ {0,\frac{\pi }{2}} \right]$ SA Q67 $f(x) = \log \left( {{x^2} + 2} \right) - \log 3$ in [-1,1] SA Q68 $f(x) = x(x + 3){e^{ - x/2}}$ in [-3,0] SA Q69 $f(x) = \sqrt {4 - {x^2}}$ in [-2,2] SA Q70 Discuss the applicability of Rolle's theorem on the function given by $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{{x^2} + 1,}&{{\rm{ if }}0 \le x \le 1}\\{3 - x,}&{{\rm{ if }}1 \le x \le 2}\end{array}} \right.$} SA Q71 Find the points on the curve $y = (\cos x - 1)$ in $[0,2\pi ],$ where the tangent is parallel to $X$ -axis SA Q72 Using Rolle's theorem, find the point on the curve $y = x(x - 4),x \in [0,4],$ where the tangent is parallel to $X$-axis SA Q73 $f(x) = \frac{1}{{4x - 1}}$ in [1,4] SA Q74 $f(x) = {x^3} - 2{x^2} - x + 3{\mathop{\rm in}\nolimits} [0,1]$ SA Q75 $f(x) = \sin x - \sin 2x$ in $[0,\pi ]$ SA Q76 $f(x) = \sqrt {25 - {x^2}}$ in [1,5] SA Q77 Find a point on the curve $y = {(x - 3)^2},$ where the tangent is parallel to the chord joining the points (3,0) and (4,1) SA Q78 Using mean value theorem, prove that there is a point on the curve $y = 2{x^2} - 5x + 3$ between the points $A(1,0)$ and $B(2,1),$ where tangent is parallel to the chord AB. Also, find that point SA

Q1 Prove that the function $f(x) = 5x - 3$ is continuous at x$=$ 0, at $x = 0$ and at $x = 5.$ SA Q2 Examine the continuity of the function f(x)= $2{x^2} - 1$ at $x = 3.$ SA Q3 Examine the following functions for continuity :

(a)$f(x) = x - 5$

(b)$f(x) = \cfrac{1}{{x - 5}},x \ne 5$

(c) $f(x) = \cfrac{{{x^2} - 25}}{{x + 5}},x \ne - 5$

(d) $f(x) = |x - 5|$ SA Q4 Prove that the function $f(x) = {x^n}$ is continuous at x $=$ n, where n is a positive integer SA Q5 Is the function f defined by $f(x) = \left\{ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{x,}&{if}&{x \le 1}\\{5,}&{if}&{x > 1}\end{array}} \right.$ continuous at x $=$ 0 ? At x $=$ 1? At x $=$ 2 ? } SA Q6 $f(x) = \left\{ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{2x + 3,}&{if}&{x \le 2}\\{2x - 3,}&{if}&{x > 2}\end{array}} \right.$ } SA Q7 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{|x| + 3,}&{ifx \le - 3}\\{ - 2x,}&{if\; - 3 < x < 3}\\{6x + 2,}&{if\,x \ge 3}\end{array}} \right.$ } SA Q8 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\cfrac{{|x|}}{x},}&{if}&{x \ne 0}\\{0,}&{if}&{x = 0}\end{array}} \right.$ } SA Q9 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\cfrac{x}{{|x|}},if}&{x < 0}\\{ - 1,if}&{x \ge 0}\end{array}} \right.$ } SA Q10 . $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{x + 1,}&{ifx \ge 1}\\{{x^2} + 1,}&{ifx < 1}\end{array}} \right.$ } SA Q11 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{{x^3} - 3,}&{ifx \le 2}\\{{x^2} + 1,}&{ifx > 2}\end{array}} \right.$ } SA Q12 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{{x^{10}} - 1,}&{ifx \le 1}\\{{x^2},}&{ifx > 1}\end{array}} \right.$ SA Q13 Is the function defined by $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{x + 5,}&{ifx \le 1}\\{x - 5,}&{ifx > 1}\end{array}} \right.$ a continuous function?}} SA Q14 Discuss the continuity of the function f, where f is defined by $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{3,if}&{0 \le x \le 1}\\{4,if}&{1 < x < 3}\\{5,if}&{3 \le x \le 10}\end{array}} \right.$ }} SA Q15 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{2x,if}&{x < 0}\\{0,if}&{0 \le x \le 1}\\{4x,if}&{x > 1}\end{array}} \right.$ }} SA Q16 . $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{ - 2,if}&{x \le - 1}\\{2x,if}&{ - 1 < x \le 1}\\{2,if}&{x > 1}\end{array}} \right.$ SA Q17 Find the relationship between a and b so that the function f defined by

$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{ax + 1,}&{if}&{x \le 3}\\{bx + 3,}&{if}&{x > 3}\end{array}} \right.$ is continuous at x$=$ 3. SA Q18 For what value of $\lambda$ is the function defined by $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\lambda ({x^2} - 2x),}&{if}&{x \le 0}\\{4x + 1,}&{if}&{x > 0}\end{array}} \right.$ continuous at x$=$ 0?}}

What about continuity at x $=$ 1? SA Q19 Show that the function defined by g(x) $=$ x $-$ [x] is discontinuous at all integral points. Here, [x] denotes the greatest integer less than or equal to x. SA Q20 Is the function defined by $f(x) = {x^2} - \sin x + 5$ continuous at $x = \pi$? SA Q21 Discuss the continuity of the following functions :

(a) $f(x) = \sin x + \cos x$

(b) $f(x) = \sin x - \cos x$

(c) $f(x) = \sin x \cdot \cos x$ SA Q22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions. SA Q23 Find all points of discontinuity of f, where

$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\cfrac{{\sin x}}{x},}&{if}&{x < 0}\\{x + 1,}&{if}&{x \ge 0}\end{array}} \right.$ SA Q24 Determine if f defined by $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{{x^2}\sin \cfrac{1}{x},}&{if}&{x \ne 0}\\{0,}&{if}&{x = 0}\end{array}} \right.$ a continuous function?}} SA Q25 Examine the continuity off where f is defined by
$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\sin x - \cos x,}&{if}&{x \ne 0}\\{ - 1,}&{if}&{x = 0}\end{array}} \right.$ }} SA Q26 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\cfrac{{k\cos x}}{{\pi - 2x}},}&{if}&{x \ne \cfrac{\pi }{2}}\\{3,}&{if}&{x = \cfrac{\pi }{2}}\end{array}} \right.$ at $x = \cfrac{\pi }{2}$ . SA Q27 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{k{x^2}}&{if}&{x \le 2}\\{3,}&{if}&{x > 2}\end{array}} \right.$ at $x = 2.$ SA Q28 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{kx + 1,}&{if}&{x \le \pi }\\{\cos ,}&{if}&{x > \pi }\end{array}} \right.$ at $x = \pi .$ SA Q29 $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{kx + 1,}&{if}&{x \le 5}\\{3x - 5,}&{if}&{x > 5}\end{array}} \right.$ at $x = 5$ SA Q30 Find the values of a and b such that the function defined by $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{5,}&{if}&{x \le 2}\\{ax + b,}&{if}&{2 < x < 10}\\{21,}&{if}&{x \ge 10}\end{array}} \right.$ is a continuous function.}} SA Q31 Show that the function defined by $f(x) = \cos ({x^2})$ is a continuous function. SA Q32 Show that the function defined by $f(x) = |\cos x|$ is a continuous function. SA Q33 Examine that $sin|x|$ is a continuous function. SA Q34 Find all the points of discontinuity of f defined by $f(x) = |x| - |x + 1|.$ SA

Q1 $sin(x^2 + 5)$ SA Q2 cos(sin x) SA Q3 sin(ax + b) SA Q4 $\sec (\tan (\sqrt x ))$ SA Q5 $\cfrac{{\sin (ax + b)}}{{\cos (cx + d)}}$ SA Q6 $\cos {x^3} \cdot {\sin ^2}({x^5})$ SA Q7 $2\sqrt {\cot ({x^2})}$ SA Q8 $\cos (\sqrt x )$ SA Q9 Prove that the function f given by $f(x) = |x - 1|,x \in R$ is not differentiable at x $=$ 1. SA Q10 Prove that the greatest integer function defined by $f(x) = [x],0 < x < 3$ is not differentiable at x $=$ 1 and x $=$ 2. SA

Q1 2x + 3y $=$ sinx SA Q2 2x + 3y $=$ sin y SA Q3 $ax + b{y^2} = \cos y$ SA Q4 $xy + {y^2} = \tan x + y$ SA Q5 ${x^2} + xy + {y^2} = 100$ SA Q6 ${x^3} + {x^2}y + x{y^2} + {y^2} = 81$ SA Q7 ${\sin ^2}y + \cos xy = k$ SA Q8 . ${\sin ^2}x + {\cos ^2}y = 1$ SA Q9 $y = {\sin ^{ - 1}}\left( {\cfrac{{2x}}{{1 + {x^2}}}} \right)$ SA Q10 $y = {\tan ^{ - 1}}\left( {\cfrac{{3x - {x^3}}}{{1 - 3{x^2}}}} \right), - \cfrac{1}{{\sqrt 3 }} < x < \cfrac{1}{{\sqrt 3 }}$ SA Q11 $y = {\cos ^{ - 1}}\left( {\cfrac{{1 - {x^2}}}{{1 + {x^2}}}} \right),0 < x < 1.$ SA Q12 $y = {\sin ^{ - 1}}\left( {\cfrac{{1 - {x^2}}}{{1 + {x^2}}}} \right),0 < x < 1.$ SA Q13 $y = {\cos ^{ - 1}}\left( {\cfrac{{2x}}{{1 + {x^2}}}} \right), - 1 < x < 1.$ SA Q14 $y = {\sin ^{ - 1}}\left( {2x\sqrt {1 - {x^2}} } \right), - \cfrac{1}{{\sqrt 2 }} < x < \cfrac{1}{{\sqrt 2 }}.$ SA Q15 . $y = {\sec ^{ - 1}}\left( {\cfrac{1}{{2{x^2} - 1}}} \right),0 < x < \cfrac{1}{{\sqrt 2 .}}$ SA

Q1 $\cfrac{{{e^x}}}{{\sin x}}$ SA Q2 ${e^{{{\sin }^{ - 1}}x}}$ SA Q3 ${e^{{x^3}}}$ SA Q4 $\sin ({\tan ^{ - 1}}{e^{ - x}})$ SA Q5 $\log (\cos {e^x})$ SA Q6 ${e^x} + {e^{{x^2}}} + ..... + {e^{{x^5}}}$ SA Q7 $\sqrt {{e^{\sqrt x }}} ,x > 0$ SA Q8 $\log (\log x),x > 1$ SA Q9 $\cfrac{{\cos x}}{{\log x}},x > 0$ SA Q10 $\cos (\log x + {e^x}),x > 0$ SA

Q1 $cos{\rm{ }}x \cdot cos{\rm{ }}2x \cdot cos{\rm{ }}3x$ SA Q2 $\sqrt {\cfrac{{(x - 1)(x - 2)}}{{(x - 3)(x - 4)(x - 5)}}}$ SA Q3 ${(\log x)^{\cos x}}$ SA Q4 ${x^x} - {2^{\sin x}}$ SA Q5 ${(x + 3)^2} \cdot {(x + 4)^3} \cdot {(x + 5)^4}$ SA Q6 ${\left( {x + \cfrac{1}{x}} \right)^x} + {x^{\left( {1 + \cfrac{1}{x}} \right)}}$ SA Q7 ${(\log x)^x} + {x^{\log x}}$ SA Q8 ${(\sin x)^x} + {\sin ^{ - 1}}\sqrt x$ SA Q9 ${x^{\sin x}} + {(\sin x)^{\cos x}}$ SA Q10 ${x^{x\cos x}} + \cfrac{{{x^2} + 1}}{{{x^2} - 1}}$ SA Q11 ${(x\cos x)^x} + {(x\sin x)^{1/x}}$ SA Q12 ${x^y} + {y^x} = 1$ SA Q13 . ${y^x} = {x^y}$ SA Q14 ${(\cos x)^y} = {(\cos y)^x}$ SA Q15 $xy = {e^{(x - y)}}$ SA Q16 Find the derivative of the function given by
$f(x) = (1 + x)(1 + {x^2})(1 + {x^4})(1 + {x^8})$ and hence find$f'(1).$ SA Q17 Differentiate $({x^2} - 5x + 8)({x^3} + 7x + 9)$ in three ways mentioned below:

(i) by using product rule

(ii) by expanding the product to obtain a single polynomial,

(iil) by logarithmic differentiation.
Do they all give the same answer? SA Q18 If u, v and w are functions of x, then show that $\cfrac{d}{{dx}}(u \cdot v \cdot w) = \cfrac{{du}}{{dx}}v \cdot w + u \cdot \cfrac{{dv}}{{dx}} \cdot w + u \cdot v\cfrac{{dw}}{{dx}}$ in two ways-first by repeated application of product rule, second by logarithmic differentiation. SA

Q1 $x = 2a{t^2},y = a{t^4}$ SA Q2 $x = a\cos \theta ,y = b\cos \theta$ SA Q3 $x = \sin t,y = \cos 2t$ SA Q4 . $x = 4t,y = \cfrac{4}{t}$ SA Q5 $x = \cos \theta - \cos 2\theta ,y = \sin \theta - \sin 2\theta$ SA Q6 $x = a(\theta - \sin \theta ),y = a(1 + \cos \theta )$ SA Q7 $x = \cfrac{{{{\sin }^3}t}}{{\sqrt {\cos 2t} }},y = \cfrac{{{{\cos }^3}t}}{{\sqrt {\cos 2t} }}$ SA Q8 $x = a\left( {\cos t + \log \tan \cfrac{t}{2}} \right),y = a\sin t$ SA Q9 $x = a\sec \theta ,y = b\tan \theta$ SA Q10 $x = a(\cos \theta + \theta \sin \theta ),y = a(\sin \theta - \theta \sec \theta )$ SA Q11 If $x = \sqrt {{a^{{{\sin }^{ - 1}}t}}} ,y = \sqrt {{a^{{{\cos }^{ - 1}}}}} ,$ show that $\cfrac{{dy}}{{dx}} = - \cfrac{y}{x}.$ SA

Q1 $x^2 + 3x + 2$ SA Q2 ${x^{20}}$ SA Q3 $x \cdot \cos x$ SA Q4 log x SA Q5 ${x^3}\log x$ SA Q6 ${e^x}\sin 5x$ SA Q7 ${e^{6x}}\cos 3x$ SA Q8 ${\tan ^{ - 1}}x$ SA Q9 $\log (\log x)$ SA Q10 $\sin (\log x)$ SA Q11 lf $y = 5\cos x - 3\sin x,$ then prove that $\cfrac{{{d^2}y}}{{d{x^2}}} + y = 0$. SA Q12 If $y = {\cos ^{ - 1}}x,$ find $\cfrac{{{d^2}y}}{{d{x^2}}}$in terms of y alone. SA Q13 If $y = 3\cos (\log x) + 4\sin (\log x),$ then show that ${x^2}{y_2} + x{y_1} + y = 0$ SA Q14 If$y = A{e^{mx}} + B{e^{nx}}$ ,then show that
$\cfrac{{{d^2}y}}{{d{x^2}}} - (m + n)\cfrac{{dy}}{{dx}} + mny = 0$ . SA Q15 If $y = 500{e^{7x}} + 600{e^{ - 7x}},$ then show that $\cfrac{{{d^2}y}}{{d{x^2}}} = 49y.$ SA Q16 If ${e^y}(x + 1) = 1,$ then show that $\cfrac{{{d^2}y}}{{d{x^2}}} = {\left( {\cfrac{{dy}}{{dx}}} \right)^2}$ . SA Q17 If $y = {({\tan ^{ - 1}}x)^2},$ show that ${({x^2} + 1)^2}{y_2} + 2x({x^2} + 1){y_1} = 2.$ SA

Q1 Verify Rolle’s theorem for the function $f(x) = {x^2} + 2x - 8,x[ - 4,2]$ . SA Q2 Examine if Rolle's theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle's theorem from these examples?

(i) f(x) $=$ [x] for x $\in$ [5, 9]

(ii) f(x) $=$ [x] for x$\in$[-2, 2]

(iii) f(x) $=$ ${x^2} -$1 for x$\in$ [1, 2]. SA Q3 If $f:[ - 5,5] \to R$ is a differentiable function and if $f'(x)$does not vanish anywhere, then prove that $f( - 5) \ne f(5)$. SA Q4 Verify Mean Value Theorem, if $f(x) = {x^2} - 4x - 3$ in the interval [a,b], where a $=$ 1 and b $=$ 4. SA Q5 Verify Mean Value Theorem, if $f(x) = {x^3} - 5{x^2} - 3x$ in the interval [a, b], where a $=$ 1 and b $=$ 3. SA Q6 Examine the applicability of Mean Value Theorem for all three functions given in the above question 2. SA

Q1 ${(3{x^2} - 9x + 5)^9}$ SA Q2 ${\sin ^3}x + {\cos ^6}x$ SA Q3 ${(5x)^{3\cos 2x}}$ SA Q4 ${\sin ^{ - 1}}(x\sqrt x ),0 \le x \le 1.$ SA Q5 $\cfrac{{{{\cos }^{ - 1}}\left( {\cfrac{x}{2}} \right)}}{{\sqrt {2x + 7} }}, - 2 < x < 2.$ SA Q6 ${\cot ^{ - 1}}\left[ {\cfrac{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} }}{{\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right],0 < x < \cfrac{\pi }{2}.$ SA Q7 ${(\log x)^{\log x}},x > 1$ SA Q8 cos(a cos x + b sin x), for some constant a and b. SA Q9 ${(\sin x - \cos x)^{(\sin x - \cos x)}},\cfrac{\pi }{4} < x < \cfrac{{3\pi }}{4}.$ SA Q10 ${x^x} + {x^a} + {a^x} + {a^a},$ for some fixed $a > 0$ and $x > 0.$ SA Q11 ${x^{{x^2} - 3}} + {(x - 3)^{{x^2}}},$ for $x > 3.$ SA Q12 Find $\cfrac{{dy}}{{dx}},ify = 12(1 - \cos t),x = 10(t - \sin t), - \cfrac{\pi }{2} < t < \cfrac{\pi }{2}.$ SA Q13 Find,$\cfrac{{dy}}{{dx}},ify = {\sin ^{ - 1}}x + {\sin ^{ - 1}}\sqrt {1 - {x^2}} , - 1 \le x \le 1.$ SA Q14 If $x\sqrt {1 + y} + y\sqrt {1 + x} = 0,$ for $- 1 < x <$1 , prove that $\cfrac{{dy}}{{dx}} = - \cfrac{1}{{{{(1 + x)}^2}}}$ SA Q15 If ${(x - a)^2} + {(y - b)^2} = {c^2}$, for some $c > 0$, prove that $$\cfrac{{{{\left[ {1 + {{\left( {\cfrac{{dy}}{{dx}}} \right)}^2}} \right]}^{3/2}}}}{{\cfrac{{{d^2}y}}{{d{x^2}}}}}$$ is a constant independent of a and b. SA Q16 If cos y$=$x cos(a + y), with cos a$\ne \pm$ 1, prove that $\cfrac{{dy}}{{dx}} = \cfrac{{{{\cos }^2}(a + y)}}{{\sin a}}$. SA Q17 If $x = a(\cos t + t\sin t)$ and $y = a(\sin t - t\cos t),$ find $\cfrac{{{d^2}y}}{{d{x^2}}}.$ SA Q18 If $f(x) = |x{|^3},$ show that f$''(x)$ exists for all real x and find it

. SA Q19 Using mathematical induction prove that $\cfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$ for all positive integers it. SA Q20 Using the fact that $\sin (A + B) = \sin A\cos B + \cos A\sin B$ and the differentiation, obtain the sum formula for cosines. SA Q21 Does there exist a function which is continuous everywhere but not differentiable at exactly two points?
Justify your answer. SA Q22 If $y = \left| {\begin{array}{llllllllllllllllllll}{f(x)}&{g(x)}&{h(x)}\\l&m&n\\a&b&c\end{array}} \right|,$ prove that

$\cfrac{{dy}}{{dx}} = \left| {\begin{array}{llllllllllllllllllll}{f'(x)}&{g'(x)}&{h'(x)}\\l&m&n\\a&b&c\end{array}} \right|$ . SA Q23 If $y = {e^{a{{\cos }^{ - 1}}x}}, - 1 \le x \le 1$, show that $(1 - {x^2})\cfrac{{{d^2}y}}{{d{x^2}}} - x\cfrac{{dy}}{{dx}} - {a^2}y = 0.$ SA