Mathematics - Differentiability and Differentiation
Exam Duration: 45 Mins Total Questions : 30
Consider the function f(x)=|logex|,\(\forall\)x>0.Then,
- (a)
LHD does not exist at x=1
- (b)
RHD does not exist at x=1
- (c)
f is differentiable at x=1
- (d)
f is not differential at x=1
If \(\sqrt{1-x^2}-\sqrt{1-y^2}=a(x-y)\), then\(dx\over dy\)is equal to
- (a)
\(\sqrt{1-x^2\over1-y^2}\)
- (b)
\(\sqrt{1-y^2\over1-x^2}\)
- (c)
\(\sqrt{x^2-1\over1-y^2}\)
- (d)
\(\sqrt{y^2-1\over1-x^2}\)
Consider the function f(x) defined by\(f(x)=\begin{cases} \frac { { x(e }^{ -1/x }-{ e }^{ 1/x } }{ { e }^{ -1/x }+{ e }^{ 1/x } } ,x\neq 0 \\ o,\quad \quad \quad \quad x=0 \end{cases}\).Then,
- (a)
f is continuous and derivable at x=0
- (b)
f is continuous but not derivable at x=0
- (c)
f is not continuous at x=0
- (d)
None of the above
If f and g are differentiable functions in [0,1] satisfying f(0)=2=g(1), g(0)=0 and f(1)=6 then for some c\(\epsilon\)]o,1[
- (a)
2f'(c)=g'(c)
- (b)
2f'(c)=3g'(c)
- (c)
f'(c)=g'(c)
- (d)
f'(c)=2g'(c)
If y=sec(tan-1 x), then \(dy\over dx\) at x=1 is equal to
- (a)
\(1\over2\)
- (b)
1
- (c)
\(\sqrt{2}\)
- (d)
\(1\over\sqrt{2}\)
\(d^2x\over dy^2\)equals
- (a)
\(-({d^2y\over dx^2})^{-1}.({dy\over dx})^{-3}\)
- (b)
\(({d^2y\over dx^2}).({dy\over dx})\)
- (c)
\(-({d^2y\over dx^2}).({dy\over dx})^{-3}\)
- (d)
\(({d^2y\over dx^2})\)
If f:(-1,1)\(\rightarrow\)R is a differentiable function with f(0)=-1 and f'(0)=1.Let g(x)=[f(2f(x)+2)]2.Then, g'(0) is equal to
- (a)
4
- (b)
-4
- (c)
0
- (d)
-2
Consider the greatest integer function, defined by f(x)=[x],0\(\le \)x<2.Then,
- (a)
f is derivable at x=1
- (b)
f is not derivable at x=1
- (c)
f is derivable at x=2
- (d)
None of these
If sec \(\left( \frac { { x }^{ 2 }-2x }{ { x }^{ 2 }+1 } \right) =y,\) then \(\frac { dy }{ dx } \) is equal to
- (a)
\(\frac { { y }^{ 2 } }{ x^{ 2 } } \)
- (b)
\(\frac { 2y\sqrt { { y }^{ 2 }-1 } ({ x }^{ 2 }+x-1) }{ { (x }^{ 2 }+{ 1 })^{ 2 } } \)
- (c)
\(,\frac { { (x }^{ 2 }+x-1) }{ y\sqrt { { y }^{ 2 }-1 } } \)
- (d)
\(\frac { { x }^{ 2 }-{ y }^{ 2 } }{ { x }^{ 2 }+{ y }^{ 2 } } \)
If y = tan-1\(\left[ \frac { sinx+cosx }{ cosx-sinx } \right] \)then \(\frac { dy }{ dx } \) is equal to
- (a)
\(\frac { 1 }{ 2 } \)
- (b)
\(\frac { \pi }{ 4 } \)
- (c)
0
- (d)
1
If sec\(\left( \frac { x-y }{ x+y } \right) \)=a, then \(\frac{dy}{dx}\)
- (a)
-\(\frac{y}{x}\)
- (b)
\(\frac{x}{y}\)
- (c)
-\(\frac{x}{y}\)
- (d)
\(\frac{y}{x}\)
\(\frac { d }{ dx } \left( tan^{ -1 }\left( \frac { \sqrt { x } -\sqrt { a } }{ 1+\sqrt { xa } } \right) \right) ,x,a>0,\) is
- (a)
\(tan^{ -1 }\sqrt { x } +tan^{ -1 }\sqrt { a } \)
- (b)
\(\frac { 1 }{ 1+x } \)
- (c)
\(\frac { 1 }{ 1+x } +\frac { 1 }{ 1+a } \)
- (d)
\(\frac { 1 }{ 2\sqrt { x } (1+x) } \)
If y=\(log\left[ { e }^{ x }\left( \frac { x-1 }{ x+2 } \right) ^{ 1/2 } \right] \), then \(\frac{dy}{dx}\) is equal to
- (a)
7
- (b)
\(\frac{3}{x-2}\)
- (c)
\(\frac{3}{(x-1)}\)
- (d)
None of these
If xx = yy,, then \(\frac { dy }{ dx } \) is equal to
- (a)
\(-\frac { y }{ x } \)
- (b)
\(-\frac { x }{ y } \)
- (c)
\(1+log\left( \frac { x }{ y } \right) \)
- (d)
\(\frac { 1+logx }{ 1+logy } \)
If u=x2+y2 and x=s+3t, y=2x-t, then \(\frac { d^{ 2 }u }{ ds^{ 2 } } \) is equal to
- (a)
12
- (b)
32
- (c)
36
- (d)
10
The 2nd derivative of a sin3t with respect to a cos3t at t=\(\frac{\pi}{4}\) is
- (a)
\(\frac { 4\sqrt { 2 } }{ 3a } \)
- (b)
2
- (c)
\(\frac{1}{12a}\)
- (d)
None of these
If y=tan-1\(\left( \frac { log\left( \frac { e }{ { x }^{ 2 } } \right) }{ logex^{ 2 } } \right) +tan^{ -1 }\left( \frac { 3+2logx }{ 1-6logx } \right) \), then \(\left( \frac { { d }^{ 2 }y }{ dx^{ 2 } } \right) \) is equal to
- (a)
2
- (b)
1
- (c)
0
- (d)
-1
Let y=t10+1 and x=t8+1, then \(\frac { d^{ 2 }y }{ dx^{ 2 } } \) is equal to
- (a)
\(\frac{5}{2}\)t
- (b)
20t8
- (c)
\(\frac { 5 }{ 16{ t }^{ 6 } } \)
- (d)
None of these
If y = \((x+\sqrt { 1+x^{ 2 } } )^{ n, }\) then (1+x2)\(\frac { d^{ 2 }y }{ dx^{ 2 } } \) is equal to
- (a)
n2y
- (b)
-n2y
- (c)
-y
- (d)
2x2y
Let f(x) =\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\{ sinx,1-cosx,\quad for\quad x\ge 0\quad for\quad x\le 0\} \) and g(x)=ex Then the value of (gof)' (0) is
- (a)
1
- (b)
-1
- (c)
0
- (d)
None of these
y=sin- 1\(\left( \cfrac { \sqrt { x } -1 }{ \sqrt { x } +1 } \right) +sec^{ -1 }\left( \cfrac { \sqrt { x } +1 }{ \sqrt { x } -1 } \right) ,x>0\) \(\cfrac { dy }{ dx } \) is equal to
- (a)
1
- (b)
0
- (c)
\(\cfrac { \pi }{ 2 } \)
- (d)
None of these
A value of c for which the Mean value theorem holds for the function f(x) = logex on the interval [1,3] is
- (a)
2log3e
- (b)
\(\frac { 1 }{ 2 } { log }_{ e }3\)
- (c)
log3e
- (d)
loge3
The value of c in Mean value theorem for the function f(x) = x(x - 2), x \(\in \) [1, 2] is
- (a)
\(\frac { 3 }{ 2 } \)
- (b)
\(\frac { 2 }{ 3 } \)
- (c)
\(\frac { 1 }{ 2 } \)
- (d)
\(\frac { 5 }{ 2 } \)
The function f(x) =\(\frac { 4-{ x }^{ 2 } }{ 4x-{ x }^{ 3 } } \) is
- (a)
- (b)
discontinuous at only one point
- (c)
discontinuous at exactly two points
- (d)
discontinuous at exactly three points
- (e)
None of these
The function f(x) = cot x is discontinuous on the set
- (a)
{x=n\(\pi\); n\(\in \)Z}
- (b)
{x=2n\(\pi\);n\(\in \)Z}
- (c)
\(\left\{ x=(2n+1)\frac { \pi }{ 2 } ;n\in Z \right\} \)
- (d)
\(\left\{ x=\frac { n\pi }{ 2 } ;n\in Z \right\} \)
If f(x)=\(\begin{cases} mx+1 \\ sinx+n \end{cases}\begin{matrix} ifx\le \frac { \pi }{ 2 } \\ ifx>\frac { \pi }{ 2 } \end{matrix}\) is continous at x=\(\frac{\pi}{2}\), then
- (a)
m=1, n=0
- (b)
m=\(\frac{n\pi}{2}\)+1
- (c)
n=\(\frac{m\pi}{2}\)
- (d)
m=n=\(\frac{\pi}{2}\)
Statement-I: For x < 0,\(\frac { d }{ dx } (In|x|)=-\frac { 1 }{ x } \)
Statement-II: For x <0, |x| =-x
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -I.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement -I.
- (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.
If y=\(\sqrt{sinx+y}\), then is equal to
- (a)
\(\frac{cosx}{2y-1}\)
- (b)
\(\frac{cosx}{1-2y}\)
- (c)
\(\frac{sinx}{1-2y}\)
- (d)
\(\frac{sinx}{2y-1}\)
The value of c in Rolle's theorem for the function f(x) = x3 - 3x in the interval [0, \(\sqrt { 3 } \)] is
- (a)
1
- (b)
-1
- (c)
\(\frac { 3 }{ 2 } \)
- (d)
\(\frac { 1 }{ 3 } \)
Statement-I: If y =log10x + logex,then
\(\frac { dy }{ dx } =\frac { { log }_{ 10 }e }{ x } +\frac { 1 }{ x } \)
Statement-II:\(\frac { d }{ dx } ({ log }_{ 10 }x)=\frac { logx }{ log10 } \) and \(\frac { d }{ dx } ({ log }_{ e }x)=\frac { logx }{ loge } \)
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -I.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement -I.
- (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.