122-- If $f(x)=(x-4)\sqrt[4]{x+3}$, then $\displaystyle\lim_{h\to 0}\dfrac{f^{2}(\Delta-h)-3f(\Delta-h)+2}{h(\Delta-h)}$ is which of the following?
(1) $\dfrac{13}{30}$ (2) $-\dfrac{5}{12}$ (3) $\dfrac{5}{6}$ (4) $-\dfrac{13}{15}$

Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. Then
(a) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(b) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(c) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$
(d) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$.

Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. Then
(a) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(b) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(c) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$
(d) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$.

Let $f(x) = a_0 + a_1 |x| + a_2 |x|^2 + a_3 |x|^3$, where $a_0, a_1, a_2, a_3$ are constants.
(A) $f(x)$ is differentiable at $x = 0$ whatever be $a_0, a_1, a_2, a_3$
(B) $f(x)$ is not differentiable at $x = 0$ whatever be $a_0, a_1, a_2, a_3$ Then
(C) $f(x)$ is differentiable at $x = 0$ only if $a_1 = 0$
(D) $f(x)$ is differentiable at $x = 0$ only if $a_1 = 0, a_3 = 0$

Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants.
(A) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(B) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ Then
(C) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$
(D) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$

Consider the function $f : \mathbb{R} \longrightarrow \mathbb{R}$, defined as follows: $$f(x) = \begin{cases} (x-1)\min\left\{x, x^2\right\} & \text{if } x \geq 0 \\ x\min\left\{x, \frac{1}{x}\right\} & \text{if } x < 0 \end{cases}$$ Then, $f$ is
(A) differentiable everywhere.
(B) not differentiable at exactly one point.
(C) not differentiable at exactly two points.
(D) not differentiable at exactly three points.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function which is differentiable at 0. Define another function $g : \mathbb { R } \rightarrow \mathbb { R }$ as follows:
$$g ( x ) = \begin{cases} f ( x ) \sin \left( \frac { 1 } { x } \right) & \text { if } x \neq 0 \\ 0 & \text { if } x = 0 \end{cases}$$
Suppose that $g$ is also differentiable at 0. Prove that
$$g ^ { \prime } ( 0 ) = f ^ { \prime } ( 0 ) = f ( 0 ) = g ( 0 ) = 0$$

8. Let $\alpha$ Î R. Prove that a function $\mathrm { f } : \mathrm { R } - - > \mathrm { R }$ is differentiable at $\alpha$ if and only if there is a function $\mathrm { g } : \mathrm { R }$ --> $R$ which is continuous at $\alpha$ and satisfies $f ( x ) - f ( \alpha ) = f ( x ) ( x - \alpha )$ for all $x \hat { I } R$.

21. The domain of the derivative of the function
$$f ( x ) = \left\{ \begin{array} { c } \tan ^ { - 1 } x , \text { if } | x | \leq 1 \\ \frac { 1 } { 2 } ( | x | - 1 ) , \text { if } | x | > 1 \end{array} \right\} \text { is }$$
(A) $\mathrm { R } - \{ 0 \}$
(B) $\mathrm { R } - \{ 1 \}$
(C) $\mathrm { R } - \{ - 1 \}$
(D) $\mathrm { R } - \{ - 1,1 \}$

13. $\lim _ { ( h \rightarrow 0 ) } \left( \mathrm { f } \left( 2 \mathrm {~h} + 2 + \mathrm { h } ^ { 2 } \right) - \mathrm { f } ( 2 ) \right) / \left( \mathrm { f } \left( \mathrm { h } - \mathrm { h } ^ { 2 } + 1 \right) - \mathrm { f } ( 1 ) \right)$, given that $f ^ { \prime } ( 2 ) = 6$ and $f ^ { \prime } ( 1 ) = 4$ :
(a) does not exists
(b) is equal to $- 3 / 2$
(c) is equal to $3 / 2$
(d) is equal to 3

Let $f(x) = a_0 + a_1|x| + a_2|x|^2 + a_3|x|^3$, where $a_0, a_1, a_2, a_3$ are constants. Then $f'(x)$ exists at $x = 0$ if and only if
(A) $a_1 = 0$
(B) $a_1 = 0$ and $a_2 = 0$
(C) $a_1 = 0$ and $a_3 = 0$
(D) $a_2 = 0$ and $a_3 = 0$

Let $a , b \in \mathbb { R }$ and $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by $f ( x ) = a \cos \left( \left| x ^ { 3 } - x \right| \right) + b | x | \sin \left( \left| x ^ { 3 } + x \right| \right)$. Then $f$ is
(A) differentiable at $x = 0$ if $a = 0$ and $b = 1$
(B) differentiable at $x = 1$ if $a = 1$ and $b = 0$
(C) NOT differentiable at $x = 0$ if $a = 1$ and $b = 0$
(D) NOT differentiable at $x = 1$ if $a = 1$ and $b = 1$

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function. We say that $f$ has
$$\text{PROPERTY 1 if } \lim_{h\rightarrow 0} \frac{f(h) - f(0)}{\sqrt{|h|}} \text{ exists and is finite, and}$$
PROPERTY 2 if $\lim_{h\rightarrow 0} \frac{f(h) - f(0)}{h^2}$ exists and is finite.
Then which of the following options is/are correct?
(A) $f(x) = |x|$ has PROPERTY 1
(B) $f(x) = x^{2/3}$ has PROPERTY 1
(C) $f(x) = x|x|$ has PROPERTY 2
(D) $f(x) = \sin x$ has PROPERTY 2

If the function $f ( x ) = \left\{ \begin{array} { c c } k _ { 1 } ( x - \pi ) ^ { 2 } - 1 , & x \leq \pi \\ k _ { 2 } \cos x , & x > \pi \end{array} \right.$ is twice differentiable, then the ordered pair $\left( k _ { 1 } , k _ { 2 } \right)$ is equal to:
(1) $\left( \frac { 1 } { 2 } , 1 \right)$
(2) $( 1,0 )$
(3) $\left( \frac { 1 } { 2 } , - 1 \right)$
(4) $( 1,1 )$

Let $f ( x )$ be a differentiable function at $x = a$ with $f ^ { \prime } ( a ) = 2$ and $f ( a ) = 4$. Then $\lim _ { x \rightarrow a } \frac { x f ( a ) - a f ( x ) } { x - a }$ equals:
(1) $a + 4$
(2) $2 a - 4$
(3) $4 - 2 a$
(4) $2 a + 4$

4. (a) It is known that differentiation satisfies the four rules
(1) $\quad \frac { d } { d x } ($ constant $) = 0$,
(2) $\quad \frac { d } { d x } ( x ) = 1$,
(3) $\frac { d } { d x } ( a f ( x ) + b g ( x ) ) = a \frac { d f } { d x } + b \frac { d g } { d x }$ for any constants $a , b$, and
(4) $\frac { d } { d x } ( f ( x ) \cdot g ( x ) ) = \frac { d f } { d x } g ( x ) + f ( x ) \frac { d g } { d x }$. .From these rules alone show that, for $n = 1,2,3 , \ldots$,
$$\frac { d } { d x } \left( x ^ { n } \right) = n x ^ { n - 1 } .$$
Use Rule (4) to find the derivative of the function
$$f ( x ) = \frac { 1 } { x ^ { n } } .$$
(It will help you to notice that $x ^ { n } \cdot \frac { 1 } { x ^ { n } } = 1$.)
(b) A careless calculus student remembers Rules (1), (2) and (3) correctly, but thinks that Rule (4) says
$$\frac { d } { d x } ( f ( x ) \cdot g ( x ) ) = \frac { d f } { d x } + f ( x ) \frac { d g } { d x } .$$
What will he or she compute for
$$\frac { d } { d x } \left( x ^ { 4 } \right)$$

1. A set of 12 rods, each 1 metre long, is arranged so that the rods form the edges of a cube. Two corners, $A$ and $B$, are picked with $A B$ the diagonal of a face of the cube.

An ant starts at $A$ and walks along the rods from one corner to the next, never changing direction while on any rod. The ant's goal is to reach the corner $B$. A path is any route taken by the ant in travelling from $A$ to $B$.
(a) What is the length of the shortest path, and how many such shortest paths are there?
(b) What are the possible lengths of paths, starting at $A$ and finishing at $B$, for which the ant does not visit any vertex more than once (including $A$ and $B$ )?
(c) How many different possible paths of greatest length are there in (b)?
(d) Can the ant travel from $A$ to $B$ by passing through every other vertex exactly twice before arriving at $B$ without revisiting $A$ ? Give brief reasons for your answer. [Figure]

For a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$, $$f'(x) = 2x^{2} - 1, \quad f(2) = 4$$ Given this, what is the value of the limit $\displaystyle\lim_{x \rightarrow 2} \frac{f(x)-4}{x-2}$?
A) 3
B) 4
C) 5
D) 6
E) 7

$$\lim _ { x \rightarrow 1 } \frac { f ( x + 1 ) - 3 } { x - 1 } = 2$$
Given that, what is the value of the limit $\lim _ { x \rightarrow 2 } \frac { x \cdot f ( x ) - 6 } { x - 2 }$?
A) 5
B) 6
C) 7
D) 8
E) 9

I. $f ( x ) = x - 1$ II. $g ( x ) = | x - 1 |$ III. $h ( x ) = \sqrt [ 3 ] { ( x - 1 ) ^ { 2 } }$ Which of the following functions do not have a derivative at the point $x = 1$?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

$$\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x }$$
What is the value of this limit?
A) $\frac { \sqrt { 5 } } { 5 }$
B) $\frac { 2 \sqrt { 5 } } { 5 }$
C) $\frac { \sqrt { 5 } } { 10 }$
D) 0
E) $2 \sqrt { 5 }$