Let $f$ be the piecewise-linear function defined by $$f ( x ) = \begin{cases} 2 x - 2 & \text { for } x < 3 \\ 2 x - 4 & \text { for } x \geq 3 \end{cases}$$ Which of the following statements are true? I. $\lim _ { h \rightarrow 0 ^ { - } } \frac { f ( 3 + h ) - f ( 3 ) } { h } = 2$ II. $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( 3 + h ) - f ( 3 ) } { h } = 2$ III. $f ^ { \prime } ( 3 ) = 2$
(A) None
(B) II only
(C) I and II only
(D) I, II, and III

Which of the following is equivalent to the definite integral $\int _ { 2 } ^ { 6 } \sqrt { x } \, d x$ ?
(A) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 4 } { n } \sqrt { \frac { 4 k } { n } }$
(B) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 6 } { n } \sqrt { \frac { 6 k } { n } }$
(C) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 4 } { n } \sqrt { 2 + \frac { 4 k } { n } }$
(D) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 6 } { n } \sqrt { 2 + \frac { 6 k } { n } }$

O limite $\displaystyle\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$ é igual a
(A) 0 (B) 1 (C) 2 (D) 4 (E) indefinido

Consider the following function: $$f(x) = \begin{cases} x^{2} \cos\left(\frac{1}{x}\right), & x \neq 0 \\ a, & x = 0 \end{cases}$$ (a) Find the value of $a$ for which $f$ is continuous. Use this value of $a$ to calculate the following.
(b) $f'(0)$.
(c) $\lim_{x \rightarrow 0} f'(x)$.

Define the right derivative of a function $f$ at $x = a$ to be the following limit if it exists. $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( a + h ) - f ( a ) } { h }$, where $h \rightarrow 0 ^ { + }$ means $h$ approaches 0 only through positive values.
Statements
(1) If $f$ is differentiable at $x = a$ then $f$ has a right derivative at $x = a$.
(2) $f ( x ) = | x |$ has a right derivative at $x = 0$.
(3) If $f$ has a right derivative at $x = a$ then $f$ is continuous at $x = a$.
(4) If $f$ is continuous at $x = a$ then $f$ has a right derivative at $x = a$.

The value of $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } - 1 } { \sqrt { x + 3 } - 2 }$ is? [2 points]
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11

When the function $f ( x )$ is $$f ( x ) = \begin{cases} 1 - x & ( x < 0 ) \\ x ^ { 2 } - 1 & ( 0 \leqq x < 1 ) \\ \frac { 2 } { 3 } \left( x ^ { 3 } - 1 \right) & ( x \geqq 1 ) \end{cases}$$ which of the following statements in are correct? [3 points]
Remarks ㄱ. $f ( x )$ is differentiable at $x = 1$. ㄴ. $| f ( x ) |$ is differentiable at $x = 0$. ㄷ. The minimum natural number $k$ such that $x ^ { k } f ( x )$ is differentiable at $x = 0$ is 2.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a polynomial function $f ( x )$, if $\lim _ { x \rightarrow 2 } \frac { f ( x + 1 ) - 8 } { x ^ { 2 } - 4 } = 5$, find the value of $f ( 3 ) + f ^ { \prime } ( 3 )$. [3 points]

For the function $f ( x ) = x ^ { 2 } + 5$, what is the value of $\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h }$? [2 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

For the function $f(x) = x^3 + 9x + 2$, find the value of $\lim_{x \rightarrow 1} \frac{f(x) - f(1)}{x - 1}$. [3 points]

For the function $f ( x ) = 2 x ^ { 2 } + a x$, when $\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h } = 6$, what is the value of the constant $a$? [3 points]
(1) $-4$
(2) $-2$
(3) $0$
(4) $2$
(5) $4$

For the function $f ( x ) = x ^ { 3 } + x + 1$, let $g ( x )$ be its inverse function. What is the value of $g ^ { \prime } ( 1 )$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 2 } { 5 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 4 } { 5 }$
(5) 1

For the function $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 3$, find the value of $f ^ { \prime } ( 2 )$. [3 points]

For the function $f(x) = 2x^3 - 5x^2 + 3$, find the value of $\lim_{h \rightarrow 0} \frac{f(2+h) - f(2)}{h}$. [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the function $f(x) = x^{3} - 8x + 7$, what is the value of $\lim_{h \rightarrow 0} \frac{f(2+h) - f(2)}{h}$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the function $f ( x ) = 3 x ^ { 3 } + 4 x + 1$, what is the value of $\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h }$? [2 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$.
For all $x \in [x_0, 1]$, we define $$h(x) = \sqrt{|f(x) - f(x_0)|}$$
(a) Show that the function $h$ defines a bijection from $[x_0, 1]$ to $[0, h(1)]$.
(b) Show that the application $h$ is differentiable at $x_0$ from the right, and that $h'_+(x_0) = \sqrt{\frac{f''(x_0)}{2}}$.

From now on, $f$ denotes an infinitely differentiable function from $[ 0,1 ]$ to $\mathbb { R }$. We assume that there exists a unique point $x _ { 0 } \in \left[ 0,1 \left[ \right. \right.$ where $f ^ { \prime }$ vanishes. We also assume that $f ^ { \prime \prime } \left( x _ { 0 } \right) > 0$. We are also given an infinitely differentiable function $g : [ 0,1 ] \rightarrow \mathbb { R }$.
For all $x \in \left[ x _ { 0 } , 1 \right]$, we define $$h ( x ) = \sqrt { \left| f ( x ) - f \left( x _ { 0 } \right) \right| }$$
(a) Show that the function $h$ defines a bijection from $\left[ x _ { 0 } , 1 \right]$ to $[ 0 , h ( 1 ) ]$.
(b) Show that the map $h$ is differentiable at $x _ { 0 }$ on the right, and that $h ^ { \prime } \left( x _ { 0 } \right) = \sqrt { \frac { f ^ { \prime \prime } \left( x _ { 0 } \right) } { 2 } }$.

Let $[a, b]$ be a compact interval of $\mathbb{R}$ and $f$ a function continuous on $[a, b]$ and differentiable on $]a, b[$, with real values. Suppose that $f'(x)$ has a finite limit $\ell$ as $x \rightarrow a^{+}$. Show that $f$ is right-differentiable at $a$ and specify the value of $f'(a)$.

Suppose $f : \mathbb { R } \rightarrow \mathbb { R }$ is a function given by
$$f ( x ) = \begin{cases} 1 & \text { if } x = 1 \\ e ^ { \left( x ^ { 10 } - 1 \right) } + ( x - 1 ) ^ { 2 } \sin \left( \frac { 1 } { x - 1 } \right) & \text { if } x \neq 1 \end{cases}$$
(a) Find $f ^ { \prime } ( 1 )$.
(b) Evaluate $\lim _ { u \rightarrow \infty } \left[ 100 u - u \sum _ { k = 1 } ^ { 100 } f \left( 1 + \frac { k } { u } \right) \right]$.

The limit $$\lim _ { x \rightarrow 0 } \frac { 1 } { x } \left( \cos ( x ) + \cos \left( \frac { 1 } { x } \right) - \cos ( x ) \cos \left( \frac { 1 } { x } \right) - 1 \right)$$ (A) equals 0.
(B) equals $\frac { 1 } { 2 }$.
(C) equals 1.
(D) does not exist.

Let $g : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function with $g ( 0 ) = 0 , g ^ { \prime } ( 0 ) = 0$ and $g ^ { \prime } ( 1 ) \neq 0$. Let $$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { | x | } g ( x ) , & x \neq 0 \\ 0 , & x = 0 \end{array} \right.$$ and $h ( x ) = e ^ { | x | }$ for all $x \in \mathbb { R }$. Let $( f \circ h ) ( x )$ denote $f ( h ( x ) )$ and $( h \circ f ) ( x )$ denote $h ( f ( x ) )$. Then which of the following is (are) true?
(A) $f$ is differentiable at $x = 0$
(B) $\quad h$ is differentiable at $x = 0$
(C) $f \circ h$ is differentiable at $x = 0$
(D) $h \circ f$ is differentiable at $x = 0$

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

The value of $\lim_{x \rightarrow 0} \frac{(1 - \cos 2x)(3 + \cos x)}{x \tan 4x}$ is equal to
(1) 1
(2) 2
(3) $-\frac{1}{4}$
(4) $\frac{1}{2}$

Let $f : R \rightarrow R$ be a function such that $| f ( x ) | \leq x ^ { 2 }$, for all $x \in R$. Then, at $x = 0 , f$ is
(1) differentiable but not continuous
(2) neither continuous nor differentiable
(3) continuous as well as differentiable
(4) continuous but not differentiable