The graph of the piecewise-defined function $f$ is shown in the figure above. The graph has a vertical tangent line at $x = - 2$ and horizontal tangent lines at $x = - 3$ and $x = - 1$. What are all values of $x , - 4 < x < 3$, at which $f$ is continuous but not differentiable?
(A) $x = 1$
(B) $x = - 2$ and $x = 0$
(C) $x = - 2$ and $x = 1$
(D) $x = 0$ and $x = 1$

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

For a differentiable function $f$, let $f ^ { * }$ be the function defined by $f ^ { * } ( x ) = \lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x - h ) } { h }$. (a) Determine $f ^ { * } ( x )$ for $f ( x ) = x ^ { 2 } + x$. (b) Determine $f ^ { * } ( x )$ for $f ( x ) = \cos x$. (c) Write an equation that expresses the relationship between the functions $f ^ { * }$ and $f ^ { \prime }$, where $f ^ { \prime }$ denotes the usual derivative of $f$.

The temperature of water in a tub at time $t$ is modeled by a strictly increasing, twice-differentiable function $W$, where $W ( t )$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t = 0$, the temperature of the water is $55 ^ { \circ } \mathrm { F }$. The water is heated for 30 minutes, beginning at time $t = 0$. Values of $W ( t )$ at selected times $t$ for the first 20 minutes are given in the table above.
(a) Use the data in the table to estimate $W ^ { \prime } ( 12 )$. Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.
(b) Use the data in the table to evaluate $\int _ { 0 } ^ { 20 } W ^ { \prime } ( t ) d t$. Using correct units, interpret the meaning of $\int _ { 0 } ^ { 20 } W ^ { \prime } ( t ) d t$ in the context of this problem.
(c) For $0 \leq t \leq 20$, the average temperature of the water in the tub is $\frac { 1 } { 20 } \int _ { 0 } ^ { 20 } W ( t ) d t$. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate $\frac { 1 } { 20 } \int _ { 0 } ^ { 20 } W ( t ) d t$. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning.
(d) For $20 \leq t \leq 25$, the function $W$ that models the water temperature has first derivative given by $W ^ { \prime } ( t ) = 0.4 \sqrt { t } \cos ( 0.06 t )$. Based on the model, what is the temperature of the water at time $t = 25$ ?

A continuous function $f$ is defined on the closed interval $- 4 \leq x \leq 6$. The graph of $f$ consists of a line segment and a curve that is tangent to the $x$-axis at $x = 3$, as shown in the figure above. On the interval $0 < x < 6$, the function $f$ is twice differentiable, with $f ^ { \prime \prime } ( x ) > 0$.
(a) Is $f$ differentiable at $x = 0$ ? Use the definition of the derivative with one-sided limits to justify your answer.
(b) For how many values of $a , - 4 \leq a < 6$, is the average rate of change of $f$ on the interval $[ a , 6 ]$ equal to 0 ? Give a reason for your answer.
(c) Is there a value of $a , - 4 \leq a < 6$, for which the Mean Value Theorem, applied to the interval $[ a , 6 ]$, guarantees a value $c , a < c < 6$, at which $f ^ { \prime } ( c ) = \frac { 1 } { 3 }$ ? Justify your answer.
(d) The function $g$ is defined by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$ for $- 4 \leq x \leq 6$. On what intervals contained in $[ - 4,6 ]$ is the graph of $g$ concave up? Explain your reasoning.

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 $x = a$.

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) $4$

The function $f ( x ) = 3 x ^ { 2 } - a x$ satisfies
$$\lim _ { n \rightarrow \infty } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } f \left( \frac { 3 k } { n } \right) = f ( 1 )$$
Find the value of the constant $a$. [4 points]

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

The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1]) = \tau$. Suppose that $f$ is differentiable on $[0,1]$.
Let $x\in[0,1]$, $(\alpha_n)_{n\geq 1}$ and $(\beta_n)_{n\geq 1}$ be two sequences of elements of $[0,1]$, convergent to $x$ and such that $\alpha_n \leq x \leq \beta_n$ and $\alpha_n < \beta_n$ for all $n$.
Show that the sequence with general term $\frac{f(\beta_n) - f(\alpha_n)}{\beta_n - \alpha_n}$ converges to $f'(x)$.

The integer part of the real number $x$ is denoted $[x]$. For every real $x$ and every non-zero natural number $n$: $r_n(x) = [2^n x] - 2[2^{n-1}x]$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1]) = \tau$.
Let $x\in[0,1]$.
a) If $x\in[0,1[$, by choosing: $$\alpha_n = \frac{r_1(x)}{2} + \cdots + \frac{r_n(x)}{2^n} \text{ and } \beta_n = \alpha_n + \sum_{k=n+1}^{\infty}\frac{1}{2^k},$$ prove that $f$ is not differentiable at $x$.
b) Prove that $f$ is not differentiable at $1$.

Let $A$ be a real square matrix of size $n$ and $b$ an element of $\mathbb{R}^n$. Let $f$ be the map from $\mathbb{R}^n$ to $\mathbb{R}^n$ defined by $$\forall x \in \mathbb{R}^n \quad f(x) = Ax + b$$ Show that $f$ is of class $C^1$ and specify its Jacobian matrix $J_f(x)$ at every point $x$ of $\mathbb{R}^n$.

Let $f$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ satisfying $f(0) = 0$. For $t$ real and $j$ an integer in $\llbracket 1, n \rrbracket$, we denote by $t_j$ the element $(0, \ldots, 0, t, 0, \ldots, 0)$ of $\mathbb{R}^n$, the real number $t$ being in position $j$.
We admit that if functions $\varphi_1, \varphi_2, \ldots, \varphi_n$ are continuous on $\mathbb{R}$ and take values in $\mathbb{R}^n$, then the function $\Phi$ defined on $\mathbb{R}$ by: $$\Phi(t) = \operatorname{det}(\varphi_1(t), \varphi_2(t), \ldots, \varphi_n(t))$$ is continuous on $\mathbb{R}$.
Using question I.B.2 and the multilinearity of the determinant, show that in a neighbourhood of 0 $$\operatorname{det}\left(f(t_1), f(t_2), \ldots, f(t_n)\right) = t^n \mathrm{jac}_f(0) + \mathrm{o}\left(t^n\right)$$

Let $f$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ satisfying $f(0) = 0$. For $t$ real and $j$ an integer in $\llbracket 1, n \rrbracket$, we denote by $t_j$ the element $(0, \ldots, 0, t, 0, \ldots, 0)$ of $\mathbb{R}^n$, the real number $t$ being in position $j$.
Deduce that $$\lim_{t \to 0} \frac{\operatorname{det}\left(f(t_1), \ldots, f(t_n)\right)}{\operatorname{det}\left(t_1, \ldots, t_n\right)} = \mathrm{jac}_f(0)$$

Let $f$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ satisfying $f(0) = 0$. For $t$ real and $j$ an integer in $\llbracket 1, n \rrbracket$, we denote by $t_j$ the element $(0, \ldots, 0, t, 0, \ldots, 0)$ of $\mathbb{R}^n$, the real number $t$ being in position $j$.
In the case $n = 2$ (respectively $n = 3$), give a geometric interpretation of the absolute value of the Jacobian of $f$ at 0 using areas of parallelograms (respectively volumes of parallelepipeds).

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $\mathrm{D}_{i,j} f_k(x)$ or $\frac{\partial^2 f_k}{\partial x_i \partial x_j}(x)$, or also $f_{i,j,k}(x)$.
Justify that, for all $x$ in $\mathbb{R}^n$ and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, we have $f_{i,j,k}(x) = f_{j,i,k}(x)$.

112- If $f(x) = (x^2 - x - 2)\sqrt[3]{x^2 - 7x}$, then what is $\displaystyle\lim_{h \to 0} \dfrac{f(-1+h) - f(-1)}{h}$?
(1) $-6$ (2) $-3$ (3) $-\dfrac{3}{2}$ (4) $-\dfrac{3}{4}$

119. The right-hand derivative of the function with the rule $f(x) = ([x] - |x|)^5\!\sqrt{9x}$, at the point $x = -3$, is equal to:
(1) $-\dfrac{16}{3}$ (2) $-5$ (3) $-4$ (4) $\dfrac{7}{3}$
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120. The tangent line to the curve of function $f$ at a point of length 3 on it, has equation $x + 2y = 7$. If $g(x) = \dfrac{1}{x} f^{-1}(x)$, then $g'(2)$ is which of the following?
(1) $-\dfrac{7}{4}$ (2) $-\dfrac{5}{4}$ (3) $-\dfrac{3}{4}$ (4) $\dfrac{1}{4}$

118. If $f$ is differentiable at $x = 4$ and $\displaystyle\lim_{x \to 4} \frac{f(x) + 5}{x - 4} = \frac{-3}{2}$, then $\dfrac{f(2x)}{x}$ at $x = 2$ equals:
(1) $-\dfrac{1}{4}$ (2) $-\dfrac{1}{2}$ (3) $\dfrac{1}{4}$ (4) $\dfrac{1}{2}$