The velocity of a particle moving along a straight line is given by $v ( t ) = 1.3 t \ln ( 0.2 t + 0.4 )$ for time $t \geq 0$. What is the acceleration of the particle at time $t = 1.2$?
(A) - 0.580
(B) - 0.548
(C) - 0.093
(D) 0.660

If $f ( x ) = 7 x - 3 + \ln x$, then $f ^ { \prime } ( 1 ) =$
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

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 $t = 25$ ?

As a pot of tea cools, the temperature of the tea is modeled by a differentiable function $H$ for $0 \leq t \leq 10$, where time $t$ is measured in minutes and temperature $H ( t )$ is measured in degrees Celsius. Values of $H ( t )$ at selected values of time $t$ are shown in the table above. (a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time $t = 3.5$. Show the computations that lead to your answer. (b) Using correct units, explain the meaning of $\frac { 1 } { 10 } \int _ { 0 } ^ { 10 } H ( t ) d t$ in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate $\frac { 1 } { 10 } \int _ { 0 } ^ { 10 } H ( t ) d t$. (c) Evaluate $\int _ { 0 } ^ { 10 } H ^ { \prime } ( t ) d t$. Using correct units, explain the meaning of the expression in the context of this problem. (d) At time $t = 0$, biscuits with temperature $100 ^ { \circ } \mathrm { C }$ were removed from an oven. The temperature of the biscuits at time $t$ is modeled by a differentiable function $B$ for which it is known that $B ^ { \prime } ( t ) = - 13.84 e ^ { - 0.173 t }$. Using the given models, at time $t = 10$, how much cooler are the biscuits than the tea?

For every natural number $n$, the function $f_{n}$ is defined for all real $x$ in the interval $[0; 1]$ by:
$$f_{n}(x) = x + \mathrm{e}^{n(x-1)}$$
Let $\mathscr{C}_{n}$ denote the graph of the function $f_{n}$ in an orthogonal coordinate system.

Part A: generalities on the functions $\boldsymbol{f}_{\boldsymbol{n}}$

1. Prove that, for every natural number $n$, the function $f_{n}$ is increasing and positive on the interval $[0; 1]$.
2. Show that the curves $\mathscr{C}_{n}$ all have a common point A, and specify its coordinates.
3. Using the graphical representations, can one conjecture the behavior of the slopes of the tangent lines at A to the curves $\mathscr{C}_{n}$ for large values of $n$? Prove this conjecture.

Part B: evolution of $\boldsymbol{f}_{\boldsymbol{n}}(\boldsymbol{x})$ when $x$ is fixed

Let $x$ be a fixed real number in the interval $[0; 1]$. For every natural number $n$, we set $u_{n} = f_{n}(x)$.

1. In this question, assume that $x = 1$. Study the possible limit of the sequence $(u_{n})$.
2. In this question, assume that $0 \leq x < 1$. Study the possible limit of the sequence $(u_{n})$.

Part C: area under the curves $\mathscr{C}_{\boldsymbol{n}}$

For every natural number $n$, let $A_{n}$ denote the area, expressed in square units, of the region located between the $x$-axis, the curve $\mathscr{C}_{n}$ and the lines with equations $x = 0$ and $x = 1$ respectively. Based on the graphical representations, conjecture the limit of the sequence $(A_{n})$ as the integer $n$ tends to $+\infty$, then prove this conjecture.

Let $f$ be a real valued continuous function defined on $\mathbb{R}$ satisfying $$f'\left(\tan^{2}\theta\right) = \cos 2\theta + \tan\theta \sin 2\theta, \text{ for all real numbers } \theta.$$ If $f'(0) = -\cos\frac{\pi}{12}$ then find $f(1)$.

For the function $f ( x ) = x \ln x + 13 x$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

For the function $f ( x ) = 5 e ^ { 3 x - 3 }$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

For the function $f ( x ) = \cos x + 4 e ^ { 2 x }$, find the value of $f ^ { \prime } ( 0 )$. [3 points]

For the function $f ( x ) = 4 \sin 7 x$, find the value of $f ^ { \prime } ( 2 \pi )$. [3 points]

For the function $f ( x ) = \ln \left( x ^ { 2 } + 1 \right)$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

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

For a real number $t$, let $f(t)$ denote the slope of the line passing through the origin and tangent to the curve $y = \frac{1}{e^x} + e^t$. For the constant $a$ satisfying $f(a) = -e\sqrt{e}$, find the value of $f'(a)$. [3 points]
(1) $-\frac{1}{3}e\sqrt{e}$
(2) $-\frac{1}{2}e\sqrt{e}$
(3) $-\frac{2}{3}e\sqrt{e}$
(4) $-\frac{5}{6}e\sqrt{e}$
(5) $-e\sqrt{e}$

9. If the line $y = k x - 2$ is tangent to the curve $y = 1 + 3 \ln x$, then $k =$
A. $2$ B. $\frac { 1 } { 3 }$
C. $3$ D. $\frac { 1 } { 2 }$

Given is the function $f : x \mapsto \ln ( x - 3 )$ with maximal domain $D$ and derivative function $f ^ { \prime }$. (1a) [2 marks] State $D$ and the zero of $f$. (1b) [3 marks] Determine the point $x \in D$ for which $f ^ { \prime } ( x ) = 2$ holds.
Given is the function $g : x \mapsto \frac { 1 } { x ^ { 2 } } - 1$ defined in $\mathbb { R } \backslash \{ 0 \}$. (2a) [2 marks] State an equation of the horizontal asymptote of the graph of $g$ and the range of $g$.
(2b) [3 marks] Calculate the value of the integral $\int _ { \frac { 1 } { 2 } } ^ { 2 } g ( x ) \mathrm { dx }$.
A polynomial function $f$ defined in $\mathbb { R }$, which is not linear, with first derivative function $f ^ { \prime }$ and second derivative function $f ^ { \prime \prime }$ has the following properties:

• $f$ has a zero at $x _ { 1 }$.
• It holds that $f ^ { \prime } \left( x _ { 2 } \right) = 0$ and $f ^ { \prime \prime } \left( x _ { 2 } \right) \neq 0$.
• $f ^ { \prime }$ has a local minimum at the point $x _ { 3 }$.

Figure 1 shows the positions of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$. [Figure]
(3a) [2 marks] Justify that the degree of $f$ is at least 3.
(3b) [3 marks] Sketch a possible graph of $f$ in Figure 1.
Figure 2 shows the graph of the function $g$ defined in $\mathbb { R }$, whose only extreme points are $( - 1 \mid 1 )$ and $( 0 \mid 0 )$, as well as the point $P$.
[Figure]
Fig. 2
(4a) [2 marks] State the coordinates of the minimum point of the graph of the function $h$ defined in $\mathbb { R }$ with $h ( x ) = - g ( x - 3 )$.
Subtask Part A 4b $( 3 \mathrm { marks } )$ The graph of an antiderivative of $g$ passes through $P$. Sketch this graph in Figure 2.
Given is the function $f : x \mapsto 2 e ^ { - \frac { 1 } { 8 } x ^ { 2 } }$ defined in $\mathbb { R }$. Figure 3 shows the graph $G _ { f }$ of $f$, which has the x-axis as a horizontal asymptote.
[Figure]
Fig. 3

Let $n \in \mathbb{N}^*$.
a) Show that $\arccos(x) \sim \sqrt{2(1-x)}$ as $x \rightarrow 1$.
b) Deduce the calculation of $F_n'(1)$ and $F_n'(-1)$.

Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined as $$f(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Then which one of the following statements is correct?
(A) $f$ is not continuous at $x = 0$.
(B) $f$ is continuous but not differentiable at $x = 0$.
(C) $f$ is differentiable at $x = 0$ and $f'(0) = -\frac{1}{2}$.
(D) $f$ is differentiable at $x = 0$ and $f'(0) = \frac{1}{2}$.

For the function $$f(x)=x\cos\frac{1}{x},\quad x\geq1,$$ (A) for at least one $x$ in the interval $[1,\infty),f(x+2)-f(x)<2$
(B) $\lim_{x\rightarrow\infty}f^{\prime}(x)=1$
(C) for all $x$ in the interval $[1,\infty),f(x+2)-f(x)>2$
(D) $f^{\prime}(x)$ is strictly decreasing in the interval $[1,\infty)$

If $y = \left( \frac { x } { x + 1 } \right) ^ { x } + x ^ { \left( \frac { x } { x + 1 } \right) }$, find $\frac { d y } { d x }$ at $x = 1$.
(1) $\frac { 1 } { 2 } + \ln 2$
(2) $1 + \frac { 1 } { 2 } \ln 2$
(3) $1 - \frac { 1 } { 2 } \ln 2$
(4) $\frac { 1 } { 2 } - \ln 2$

If $y(\alpha) = \sqrt { 2 \left( \frac { \tan \alpha + \cot \alpha } { 1 + \tan ^ { 2 } \alpha } \right) + \frac { 1 } { \sin ^ { 2 } \alpha } }$, $\alpha \in \left( \frac { 3 \pi } { 4 } , \pi \right)$, then $\frac { d y } { d \alpha }$ at $\alpha = \frac { 5 \pi } { 6 }$ is
(1) 4
(2) $\frac { 4 } { 3 }$
(3) $-4$
(4) $- \frac { 1 } { 4 }$

The derivative of $\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)$ at $x = \frac{1}{2}$ is:
(1) $\frac{2\sqrt{3}}{5}$
(2) $\frac{\sqrt{3}}{12}$
(3) $\frac{2\sqrt{3}}{3}$
(4) $\frac{\sqrt{3}}{10}$

If $y ( x ) = \cot ^ { - 1 } \left( \frac { \sqrt { 1 + \sin x } + \sqrt { 1 - \sin x } } { \sqrt { 1 + \sin x } - \sqrt { 1 - \sin x } } \right) , x \in \left( \frac { \pi } { 2 } , \pi \right)$, then $\frac { d y } { d x }$ at $x = \frac { 5 \pi } { 6 }$ is: (1) 0 (2) - 1 (3) $\frac { - 1 } { 2 }$ (4) $\frac { 1 } { 2 }$

The value of $\log _ { e } 2 \cdot \frac { \mathrm { d } } { \mathrm { d } x } \log _ { \cos x } \operatorname { cosec } x$ at $x = \frac { \pi } { 4 }$ is
(1) $- 2 \sqrt { 2 }$
(2) $2 \sqrt { 2 }$
(3) $-4$
(4) $4$

If $a = \lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 2 n } { n ^ { 2 } + k ^ { 2 } }$ and $f ( x ) = \sqrt { \frac { 1 - \cos x } { 1 + \cos x } } , x \in ( 0,1 )$, then:
(1) $2 \sqrt { 2 } f \left( \frac { a } { 2 } \right) = f ^ { \prime } \left( \frac { a } { 2 } \right)$
(2) $f \left( \frac { a } { 2 } \right) f ^ { \prime } \left( \frac { a } { 2 } \right) = \sqrt { 2 }$
(3) $\sqrt { 2 } f \left( \frac { a } { 2 } \right) = f ^ { \prime } \left( \frac { a } { 2 } \right)$
(4) $f \left( \frac { a } { 2 } \right) = \sqrt { 2 } f ^ { \prime } \left( \frac { a } { 2 } \right)$

$$f(x) = \ln\left(\sin^{2} x + e^{2x}\right)$$
Given this, what is $f'(0)$?
A) $e$
B) $1$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $2$