Mathematics - Differentiability and Differentiation
Exam Duration: 45 Mins Total Questions : 30
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
Let y=f(ax)and f'(sin x)=logex.Then, for \({\pi\over2}
- (a)
\(a^x\ log_e\ a\ log_e(\pi-sin^{-1}\ a^x)\)
- (b)
\(a^x\ log_e\ a\ log_e(\pi+sin^{-1}\ a^x)\)
- (c)
\(a^x\ log_e\ a\ log_e(\sin^{-1}\ a^x)\)
- (d)
None of the above
Let \(f(x)=\lambda+\mu|x|+v|x|^2,\)where \(\lambda,\mu,v\) are real constants.Then, f'(0) exists, if
- (a)
\(\mu=0\)
- (b)
\(v=0\)
- (c)
\(\lambda=0\)
- (d)
\(\mu=v\)
Consider a function f:R\(\rightarrow\)R which satisfies the equation f(x+y)=f(x).f(y), \(\forall\) x, y\(\epsilon\)R, f(x)\(\neq\)0.Suppose that the function is differentiable at x=0 and f'(0)=2.Then f'(x) is equal to
- (a)
f(x)
- (b)
2f(x)
- (c)
\(1\over2\)f(x)
- (d)
None of these
Consider the function f(x), defined by \(f(x)=\begin{cases} x,\quad \quad\quad \quad \quad x\le 1 \\ { x }^{ 2 }+bx+c,\quad x>1 \end{cases}\) Let f'(x) exists finitely, \(\forall\ x\epsilon R\).Then,
- (a)
b=-1,c\(\epsilon\)R
- (b)
c=1,b\(\epsilon\)R
- (c)
b=1,c=-1
- (d)
b=-1,c=1
Let f(x)=x|x|,\(\forall\) x\(\epsilon\)R.Then,
- (a)
f is derivable at x=1
- (b)
f is not derivable at x=1
- (c)
f is continuous at x=1
- (d)
None of the above
If f(x)=\(\left| \begin{matrix} x+{ a }^{ 2 } & ab & ac \\ ab & x+b^{ 2 } & bc \\ ac & bc & x+c^{ 2 } \end{matrix} \right| \), then f'(x) is
- (a)
3x2+2x(a2+b2+c2)
- (b)
3x+2x2(a2+b2+c2)
- (c)
3x2+2x2(a2-b2-c2)
- (d)
None of the above
Let f(2) = 4 and f '(2) = 4. Then, \(=\underset { x\rightarrow 2 }{ lim } -\frac { xf(2)-2f(x) }{ x-2 } \)is given by
- (a)
2
- (b)
-2
- (c)
-4
- (d)
3
If f is a real - valued differentiable function satisfying \(\left| f(x)-f(y) \right| \le (x-y)^{ 2 },x,y\in R\) and f(0) =0, then f(1) equals
- (a)
1
- (b)
2
- (c)
0
- (d)
-1
If f(x) =- \(\sqrt { 25-x^{ 2 } } \) , then \(\underset { x\rightarrow 1 }{ lim } \frac { f(x)-f(1) }{ x-1 } \) is equal to
- (a)
\(\frac { 1 }{ 24 } \)
- (b)
\(\frac { 1 }{ 5 } \)
- (c)
\(-\sqrt { 24 } \)
- (d)
\(\frac { 1 }{ \sqrt { 24 } } \)
If xy2=ax2+bxy+y2, then find \(\frac{dy}{dx}\)
- (a)
\(\frac { 2ax+by+{ y }^{ 2 } }{ 2xy+bx+2y } \)
- (b)
\(\frac { 2ax+by-{ y }^{ 2 } }{ 2xy-bx-2y } \)
- (c)
\(\frac { ax+by+{ xy } }{ xy+x^{ 2 }+y^{ 2 } } \)
- (d)
\(\frac { 2x^{ 2 }+axy+{ y }^{ 2 } }{ x^{ 2 }+y^{ 2 }+2xy } \)
If y = sin-1 \(\left( \frac { \sqrt { x } -1 }{ \sqrt { x } +1 } \right) +sec^{ -1 }\left( \frac { \sqrt { x } +1 }{ \sqrt { x } -1 } \right) ,x>0,\) then \(\frac { dy }{ dx } \)is equal to
- (a)
1
- (b)
0
- (c)
\(\frac { \pi }{ 2 } \)
- (d)
None of these
Let f(x) =ex, g(x) = sin-1x and h(x) = f[g(x)],then\(\frac { h'(x) }{ h(x) } \) is equal to
- (a)
\({ e }^{ sin-1 }x\)
- (b)
\(\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } \)
- (c)
sin-1x
- (d)
\(\frac { 1 }{ (1-{ x }^{ 2 }) } \)
If y =aex +be-x + c where a,b, c are parameters, then y' is equal to
- (a)
aex - be-x
- (b)
aex + be-x
- (c)
-(aex + be-x)
- (d)
aex - bex
Derivative of the function f(x) = log5(log7x),x>7 is
- (a)
\(\frac { 1 }{ x(log5)(log7)(log_{ 7 }x) } \)
- (b)
\(\frac { 1 }{ x(log5)(log7) } \)
- (c)
\(\frac { 1 }{ x(logx) } \)
- (d)
None of these
If y=sinx+ex, then \(\frac { { d }^{ 2 }x }{ { dy }^{ 2 } } \) is equal to
- (a)
\(\frac { sinx-{ e }^{ x } }{ (cosx+{ e }^{ x })^{ 2 } } \)
- (b)
\(\frac { sinx-{ e }^{ x } }{ (cosx+{ e }^{ x })^{ 3 } } \)
- (c)
\(\frac { sinx-{ e }^{ x } }{ (cosx+{ e }^{ x }) } \)
- (d)
(-sinx+ex)-1
The derivative of f(tanx) w.r.t. g(secx) at x = \(\frac { \pi }{ 4 } ,\)where f' (1) = 2 and g' \(\left( \sqrt { 2 } \right) \) =4, is
- (a)
\(\frac { 1 }{ \sqrt { 2 } } \)
- (b)
\({ \sqrt { 2 } } \)
- (c)
1
- (d)
0
The derivative of ex3 with respect to log x is
- (a)
\({ e }^{ x^{ 3 } }\)
- (b)
\({ 3x }^{ 2 }2{ e }^{ x^{ 3 } }\)
- (c)
\({ 3x }^{ 3 }{ e }^{ x^{ 3 } }\)
- (d)
\({ 3x }^{ 2 }{ e }^{ x^{ 3 } }+3x^{ 2 }\)
The derivative of \({ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) \)with respect to \(cos^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 1+x^{ 2 } } \right) is\)
- (a)
-1
- (b)
1
- (c)
2
- (d)
4
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
If f(x)=\(\sqrt { 1+{ cos }^{ 2 }\left( x \right) ^{ 2 } } \),then the value of \(f'\left( \cfrac { \sqrt { \pi } }{ 2 } \right) \)
- (a)
\(\cfrac { \sqrt { \pi } }{ 6 } \)
- (b)
\(\sqrt { \cfrac { \pi }{ 6 } } \)
- (c)
\(\cfrac { 1 }{ \sqrt { 6 } } \)
- (d)
\(\cfrac { \pi }{ \sqrt { 6 } } \)
If x = f(i) and y = get(t), then \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \) is equal to
- (a)
\(\frac { g"(t) }{ f"(t) } \)
- (b)
\(\frac { g"(t)f'(t)-g'(t)f"(t) }{ (f'(t){ ) }^{ 3 } } \)
- (c)
\(\frac { g"(t)f'(t)-g'(t)f"(t) }{ (f'(t){ ) }^{ 2 } } \)
- (d)
none of these
If y2 = ax2 + bx + c, then \(\frac { d }{ dx } ({ y }^{ 3 }{ y }_{ 2 })\)=
- (a)
1
- (b)
-1
- (c)
\(\frac{4ac-b^{2}}{a^{2}}\)
- (d)
0
Let f(x) satisfy the requirements of Lagrange's mean value theorem in [0,2]. If f(0) =0 and \(f'(x)\le \frac { 1 }{ 2 } \) for all x in [0,2], then
- (a)
|f(x)|\(\le \) 2
- (b)
f(x)\(\le \) 1
- (c)
f(x) = 2x
- (d)
f(x) = 3 for atleast one x in [0, 2]
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)
\(\left\{ x=\frac { n\pi }{ 2 } ;n\in Z \right\} \)
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)
\(\frac{sinx}{2y-1}\)
If x = t2, y = t3, then \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \) is
- (a)
\(\frac { 3 }{ 2 } \)
- (b)
\(\frac { 3 }{ 4t } \)
- (c)
\(\frac { 3 }{ 2t } \)
- (d)
\(\frac { 3 }{ 4 } \)
For thefunction f(x) = x + \(\frac { 1 }{ x } \), x\(\in \)[1,3], the value of c for mean value theorem is
- (a)
1
- (b)
\(\sqrt { 3 } \)
- (c)
2
- (d)
None of these