The function $f$ is defined by $f ( x ) = \sqrt { 25 - x ^ { 2 } }$ for $- 5 \leq x \leq 5$.
(a) Find $f ^ { \prime } ( x )$.
(b) Write an equation for the line tangent to the graph of $f$ at $x = - 3$.
(c) Let $g$ be the function defined by $g ( x ) = \left\{ \begin{array} { l } f ( x ) \text { for } - 5 \leq x \leq - 3 \\ x + 7 \text { for } - 3 < x \leq 5 . \end{array} \right.$
Is $g$ continuous at $x = - 3$ ? Use the definition of continuity to explain your answer.
(d) Find the value of $\int _ { 0 } ^ { 5 } x \sqrt { 25 - x ^ { 2 } } d x$.

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

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 $\lim _ { u \rightarrow \infty } \left[ 100 u - u \sum _ { k = 1 } ^ { 100 } f \left( 1 + \frac { k } { u } \right) \right]$.

27. Which of the following functions is differentiable at $x = 0$ :
(A) $\cos ( | x | ) + | x |$
(B) $\cos ( | x | ) - | x |$
(C) $\sin ( | x | ) + | x |$
(D) $\sin ( | x | ) - | x |$

Let the function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by $f ( x ) = x ^ { 3 } - x ^ { 2 } + ( x - 1 ) \sin x$ and let $g : \mathbb { R } \rightarrow \mathbb { R }$ be an arbitrary function. Let $f g : \mathbb { R } \rightarrow \mathbb { R }$ be the product function defined by $( f g ) ( x ) = f ( x ) g ( x )$. Then which of the following statements is/are TRUE?
(A) If $g$ is continuous at $x = 1$, then $f g$ is differentiable at $x = 1$
(B) If $f g$ is differentiable at $x = 1$, then $g$ is continuous at $x = 1$
(C) If $g$ is differentiable at $x = 1$, then $f g$ is differentiable at $x = 1$
(D) If $f g$ is differentiable at $x = 1$, then $g$ is differentiable at $x = 1$

If the function $g(x) = \begin{cases} k\sqrt{x+1}, & 0 \leq x \leq 3 \\ mx + 2, & 3 < x \leq 5 \end{cases}$ is differentiable, then the value of $k + m$ is:
(1) $2$
(2) $\frac{16}{5}$
(3) $\frac{10}{3}$
(4) $4$

Let $S = \left\{ t \in R : f ( x ) = | x - \pi | \cdot \left( e ^ { | x | } - 1 \right) \sin | x |\right.$ is not differentiable at $\left. t \right\}$. Then, the set $S$ is equal to:
(1) $\{ 0 , \pi \}$
(2) $\phi$ (an empty set)
(3) $\{ 0 \}$
(4) $\{ \pi \}$

The function $f ( x ) = \left| x ^ { 2 } - 2 x - 3 \right| \cdot \mathrm { e } ^ { 9 x ^ { 2 } - 12 x + 4 }$ is not differentiable at exactly :
(1) Four points
(2) Two points
(3) three points
(4) one point

$f , g : R \rightarrow R$ be two real valued function defined as $f ( x ) = \left\{ \begin{array} { c l } - | x + 3 | & , x < 0 \\ e ^ { x } & , x \geq 0 \end{array} \right.$ and $g ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } + k _ { 1 } x , & x < 0 \\ 4 x + k _ { 2 } , & x \geq 0 \end{array} \right.$, where $k _ { 1 }$ and $k _ { 2 }$ are real constants. If $gof$ is differentiable at $x = 0$, then $gof ( - 4 ) + gof ( 4 )$ is equal to
(1) $4 \left( e ^ { 4 } + 1 \right)$
(2) $2 \left( 2 e ^ { 4 } + 1 \right)$
(3) $4 e ^ { 4 }$
(4) $2 \left( 2 e ^ { 4 } - 1 \right)$

Let $f: (-2, 2) \rightarrow \mathbb{R}$ be defined by $f(x) = \begin{cases} x\lfloor x\rfloor, & 0 \leq x < 2 \\ (x-1)\lfloor x\rfloor, & -2 < x < 0 \end{cases}$ where $\lfloor x \rfloor$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2, 2)$ at which $y = f(x)$ is not continuous and not differentiable, then $m + n$ is equal to $\_\_\_\_$.

Q73. If $f ( x ) = \left\{ \begin{array} { l } x ^ { 3 } \sin \left( \frac { 1 } { x } \right) , x \neq 0 \\ 0 \quad , x = 0 \end{array} \right.$ then
(1) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 24 - \pi ^ { 2 } } { 2 \pi }$
(2) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 12 - \pi ^ { 2 } } { 2 \pi }$
(3) $f ^ { \prime \prime } ( 0 ) = 1$
(4) $f ^ { \prime \prime } ( 0 ) = 0$

Consider the function
$$f ( x ) = \left\{ \begin{array} { l l l } \operatorname { sen } x & \text { if } & x < 0 \\ x e ^ { x } & \text { if } & x \geq 0 \end{array} \right.$$
a) ( 0.75 points) Study the continuity and differentiability of $f$ at $x = 0$.\ b) (1 point) Study the intervals of increase and decrease of $f$ restricted to ( $- \pi , 2$ ). Prove that there exists a point $x _ { 0 } \in [ 0,1 ]$ such that $f \left( x _ { 0 } \right) = 2$.\ c) (0.75 points) Calculate $\int _ { - \frac { \pi } { 2 } } ^ { 1 } f ( x ) d x$.

Let $F ( x )$ be a polynomial with real coefficients and $F ^ { \prime } ( x ) = f ( x )$ . It is known that $f ^ { \prime } ( x ) > x ^ { 2 } + 1.1$ holds for all real numbers $x$. Select the correct options.
(1) $f ^ { \prime } ( x )$ is an increasing function
(2) $f ( x )$ is an increasing function
(3) $F ( x )$ is an increasing function
(4) $[ f ( x ) ] ^ { 2 }$ is an increasing function
(5) $f ( f ( x ) )$ is an increasing function