1. Let \[f\left(x\right)=\begin{cases}x^{2} & if x\leq x_{0}\\ax+b & if x>x_{0} \end{cases}\]

The value of the coefficients a and b for which the
function is continuous and has a derivative at \[x_{0}\] , are

a) \[a=x_{0}, b=-x_{0}\]

b) \[a=2x_{0}, b=-x_0^2\]

c) \[a=x_0^2, b=-x_{0}\]

d) \[a=x_{0}, b=-x_0^2\]

Explanation: For f to be continuous everywhere, we must have

2. Given f' (2) = 6 and f' (1) = 4,
\[\lim_{h \rightarrow 0}\frac{f\left(2h+2+h^{2}\right)-f\left(2\right)}{f\left(h-h^{2}+1\right)-f\left(1\right)}\]

is equal to

a) 3/2

b) 3

c) 5/2

d) -3

Explanation:

3. Let \[f:R\rightarrow R\] be such that f (1) = 3 and
f' (1) = 6. Then \[\lim_{x \rightarrow 0}\left(\frac{f\left(1+x\right)}{f\left(1\right)}\right)^{1/x}\] equals

a) 1

b) \[e^{1/2}\]

c) \[e^{2}\]

d) \[e^{3}\]

Explanation:

4.The domain of the derivative of the function \[f\left(x\right)=\begin{cases}\tan^{-1}x &\mid x\mid \leq1\\\frac{1}{2}\left(\mid x\mid-1\right) & \mid x\mid > 1\end{cases}\]

a) \[R \sim\left\{0\right\}\]

b) \[R \sim\left\{1\right\}\]

c) \[R \sim\left\{-1\right\}\]

d) \[R \sim\left\{-1,1\right\}\]

Explanation:

5. If f (1) = 1, f' (1) = 3 then the derivative of
y = f (f ( f ( f (x))) at x = 1 is

a) 256

b) 16

c) 81

d) 27

Explanation: y'(x) = f'(f(f(f(x))) f'(f(f(x))) f'( f(x)) f'(x)

6. \[f\left(x\right)=\begin{cases}ax & x < 2\\ax^{2}-bx+3 & x \geq 2\end{cases}\]

If f is differentiable for all x then

a) a = 3/4, b = 9/4

b) a = 1, b = 2

c) a = 3/2, b = 9/2

d) a = 3/4, b = 9/2

Explanation: Since f is differentiable for all x, in particular

7. Let f and g be differentiable function such
that f' (x) = 2g(x) and g' (x) = – f (x), and let \[T\left(x\right)=\left(f\left(x\right)\right)^{2}-\left(g\left(x\right)\right)^{2}.\]

Then T' (x) is equal to

a) T(x)

b) 0

c) 2f (x) g(x)

d) 6 f (x) g(x)

Explanation:

8. Let f be a twice differentiable function
such that f" (x) = – f (x) and f' (x) = g(x). If \[h'\left(x\right)=\left[f\left(x\right)\right]^{2}+\left[g\left(x\right)\right]^{2}\] , h(1) = 7 and h(0) = 2, then h(3) is equal to

a) 11

b) 4

c) 14

d) 13

Explanation:

9. If \[y^{2}=P\left(x\right)\] is a polynomial of degree 3,

\[2\frac{d}{dx}\left(y^{3}\frac{d^{2}y}{dx^{2}}\right)\] is equal to

a) P(x) + P'(x)

b) P(x) P'(x)

c) P(x) P'''(x)

d) a constant

Explanation: From y

^{2}= P(x), we have 2yy

_{1}= P'(x), i.e.,

10. Let \[f\left(x\right)=\lim_{n \rightarrow \infty}\frac{\left(x^{2}+x+\frac{9}{4}+\cos\pi x\right)^{n}-1}{\left(x^{2}+x+\frac{9}{4}+\cos\pi x\right)^{n}+1}.\left(\frac{2\tan ^{n}x+x^{n}}{\tan^{n}x}\right)\]

then

a) f is differentiable for all \[x\epsilon R\]

b) f is continuous but not differentiable

c) f is discontinuous at all \[n\epsilon I\]

d) f is discontinuous only at finitely many points

Explanation: