$x$	2	5
$f ( x )$	4	7
$f ^ { \prime } ( x )$	2	3

The table above gives values of the differentiable function $f$ and its derivative $f ^ { \prime }$ at selected values of $x$. If $\int _ { 2 } ^ { 5 } f ( x ) \, d x = 14$, what is the value of $\int _ { 2 } ^ { 5 } x \cdot f ^ { \prime } ( x ) \, d x$?
(A) 13
(B) 27
(C) $\frac { 63 } { 2 }$
(D) 41

The function $f$ is twice differentiable for $x > 0$ with $f ( 1 ) = 15$ and $f ^ { \prime \prime } ( 1 ) = 20$. Values of $f ^ { \prime }$, the derivative of $f$, are given for selected values of $x$ in the table above.

$x$	1	1.1	1.2	1.3	1.4
$f ^ { \prime } ( x )$	8	10	12	13	14.5

(a) Write an equation for the line tangent to the graph of $f$ at $x = 1$. Use this line to approximate $f ( 1.4 )$.
(b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate $\int _ { 1 } ^ { 1.4 } f ^ { \prime } ( x ) d x$. Use the approximation for $\int _ { 1 } ^ { 1.4 } f ^ { \prime } ( x ) d x$ to estimate the value of $f ( 1.4 )$. Show the computations that lead to your answer.
(c) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 1.4 )$. Show the computations that lead to your answer.
(d) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f ( 1.4 )$.

A certification body is commissioned to evaluate two heating devices, one from brand A and the other from brand B.
Parts 1 and 2 are independent.
Part 1: device from brand A Using a probe, the temperature inside the combustion chamber of a brand A device was measured. Below is a representation of the temperature curve in degrees Celsius inside the combustion chamber as a function of time elapsed, expressed in minutes, since the combustion chamber was ignited.
By reading the graph:

1. Give the time at which the maximum temperature is reached inside the combustion chamber.
2. Give an approximate value, in minutes, of the duration during which the temperature inside the combustion chamber exceeds $300 ^ { \circ } \mathrm { C }$.
3. We denote by $f$ the function represented on the graph. Estimate the value of $\frac { 1 } { 600 } \int _ { 0 } ^ { 600 } f ( t ) \mathrm { d } t$. Interpret the result.

Part 2: study of a function Let the function $g$ be defined on the interval $[0 ; + \infty [$ by: $$g ( t ) = 10 t \mathrm { e } ^ { - 0.01 t } + 20 .$$

1. Determine the limit of $g$ at $+ \infty$.
2. a. Show that for all $t \in \left[ 0 ; + \infty \left[ , \quad g ^ { \prime } ( t ) = ( - 0.1 t + 10 ) \mathrm { e } ^ { - 0.01 t } \right. \right.$. b. Study the variations of the function $g$ on $[0 ; + \infty [$ and construct its variation table.
3. Prove that the equation $g ( t ) = 300$ has exactly two distinct solutions on $[0 ; + \infty [$. Give approximate values to the nearest integer.
4. Using integration by parts, calculate $\int _ { 0 } ^ { 600 } g ( t ) \mathrm { d } t$.

Part 3: evaluation For a brand B device, the temperature in degrees Celsius inside the combustion chamber $t$ minutes after ignition is modelled on $[0 ; 600]$ by the function $g$.
The certification body awards one star for each criterion validated among the following four:

• Criterion 1: the maximum temperature is greater than $320 ^ { \circ } \mathrm { C }$.
• Criterion 2: the maximum temperature is reached in less than 2 hours.
• Criterion 3: the average temperature during the first 10 hours after ignition exceeds $250 ^ { \circ } \mathrm { C }$.
• Criterion 4: the temperature inside the combustion chamber must not exceed $300 ^ { \circ } \mathrm { C }$ for more than 5 hours.

Does each device obtain exactly three stars? Justify your answer.

Calculate the following two definite integrals. It may be useful to first sketch the graph. $$\int_{1}^{e^{2}} \ln|x|\, dx \qquad \int_{-1}^{1} \frac{\ln|x|}{|x|}\, dx$$

Two polynomial functions $f ( x ) , g ( x )$ satisfy for all real numbers $x$ $$f ( - x ) = - f ( x ) , \quad g ( - x ) = g ( x )$$ For the function $h ( x ) = f ( x ) g ( x )$, $$\int _ { - 3 } ^ { 3 } ( x + 5 ) h ^ { \prime } ( x ) d x = 10$$ What is the value of $h ( 3 )$? [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Find the value of $\int _ { 0 } ^ { \pi } x \cos ( \pi - x ) d x$. [3 points]

A function $f ( x )$ that is increasing and differentiable on the set of all real numbers satisfies the following conditions. (가) $f ( 1 ) = 1 , \int _ { 1 } ^ { 2 } f ( x ) d x = \frac { 5 } { 4 }$ (나) When the inverse function of $f ( x )$ is $g ( x )$, for all real numbers $x \geq 1$, $g ( 2 x ) = 2 f ( x )$. When $\int _ { 1 } ^ { 8 } x f ^ { \prime } ( x ) d x = \frac { q } { p }$, find the value of $p + q$. (Given that $p$ and $q$ are coprime natural numbers.) [4 points]

Find two real numbers $\alpha$ and $\beta$ such that: $$\forall n \in \mathbf{N}^*, \int_0^{\pi} (\alpha t^2 + \beta t) \cos(nt) \mathrm{d}t = \frac{1}{n^2}$$ then verify that if $t \in ]0, \pi]$, then: $$\forall n \in \mathbf{N}^*, \sum_{k=1}^n \cos(kt) = \frac{\sin\left(\frac{(2n+1)t}{2}\right)}{2\sin\left(\frac{t}{2}\right)} - \frac{1}{2}$$

The value of $$\int_{0}^{1} 4x^3 \left\{\frac{d^2}{dx^2}\left(1 - x^2\right)^5\right\} dx$$ is

Let $\int x ^ { 3 } \sin x \mathrm {~d} x = g ( x ) + C$, where $C$ is the constant of integration. If $8 \left( g \left( \frac { \pi } { 2 } \right) + g ^ { \prime } \left( \frac { \pi } { 2 } \right) \right) = \alpha \pi ^ { 3 } + \beta \pi ^ { 2 } + \gamma , \alpha , \beta , \gamma \in Z$, then $\alpha + \beta - \gamma$ equals :
(1) 48
(2) 55
(3) 62
(4) 47

Let $\mathrm { f } : \mathbf { R } \rightarrow \mathbf { R }$ be a twice differentiable function such that $f ( 2 ) = 1$. If $\mathrm { F } ( x ) = x f ( x )$ for all $x \in \mathbf { R }$, $\int _ { 0 } ^ { 2 } x \mathrm {~F} ^ { \prime } ( x ) \mathrm { d } x = 6$ and $\int _ { 0 } ^ { 2 } x ^ { 2 } \mathrm {~F} ^ { \prime \prime } ( x ) \mathrm { d } x = 40$, then $\mathrm { F } ^ { \prime } ( 2 ) + \int _ { 0 } ^ { 2 } \mathrm {~F} ( x ) \mathrm { d } x$ is equal to :
(1) 11
(2) 13
(3) 15
(4) 9

For a positive integer $n$ and a real number $a$, consider the function
$$f_n(a) = \int_0^{\pi} (\cos x + a\sin 2nx)^2\, dx$$
(1) When we transform $f_n(a)$ into
$$f_n(a) = \int_0^{\pi} \left\{\frac{1 + \cos \mathbf{L}\, x}{2} + a^2 \frac{1 - \cos \mathbf{M}\, nx}{2} + a(\sin(2n+1)x + \sin(2n-1)x)\right\} dx$$
and calculate the definite integral on the right side, we obtain
$$f_n(a) = \frac{\pi}{\mathbf{N}}\, a^2 + \frac{\mathbf{O}\, n}{\mathbf{P}\, n^2 - \mathbf{Q}}\, a + \frac{\pi}{\mathbf{R}}.$$
(2) Let $a_n$ denote the value of $a$ at which $f_n(a)$ is minimalized, and set $S_N = \sum_{n=1}^{N} \frac{a_n}{n}$.
Then
$$\begin{aligned} S_N &= -\frac{\mathbf{S}}{\pi} \sum_{n=1}^{N} \left(\frac{1}{2n - \mathbf{T}} - \frac{1}{2n + \mathbf{U}}\right) \\ &= -\frac{\mathbf{S}}{\pi} \left(\mathbf{V} - \frac{1}{\mathbf{W}N + \mathbf{U}}\right) \end{aligned}$$
Hence we obtain
$$\sum_{n=1}^{\infty} \frac{a_n}{n} = \lim_{N \to \infty} S_N = -\frac{\mathbf{Y}}{\pi}.$$

For the function $f$ whose graph is given above, $$\int_{1}^{3} \frac{x \cdot f'(x) - f(x)}{x^{2}}\, dx$$ What is the value of the integral?
A) $\frac{7}{2}$
B) $\frac{3}{2}$
C) $\frac{2}{3}$
D) $\frac{1}{3}$
E) $\frac{5}{4}$

f is a differentiable function on the set of real numbers and
$$\begin{aligned} & \int _ { 0 } ^ { 3 } f ( x ) d x = 2 \\ & \int _ { 0 } ^ { 3 } x f ^ { \prime } ( x ) d x = 1 \end{aligned}$$
Given this, what is the value of $\mathbf { f } \boldsymbol { ( } \mathbf { 3 } \boldsymbol { ) }$?
A) 0
B) 1
C) 2
D) 3
E) 4

For a function f defined on the set of real numbers and twice differentiable,
$$\begin{aligned} & f ( 1 ) = f ( 2 ) = 2 \\ & f ^ { \prime } ( 1 ) = f ^ { \prime } ( 2 ) = - 1 \end{aligned}$$
the following equalities are given.
Accordingly, what is the value of the integral $\int _ { 1 } ^ { 2 } x \cdot f ^ { \prime \prime } ( x ) d x$?
A) $- 1$
B) $- 2$
C) $- 3$
D) $\frac { - 1 } { 2 }$
E) $\frac { - 2 } { 3 }$

$\int _ { 1/2 } ^ { e } x \ln ( 2 x ) \, d x$\ What is the value of the integral?\ A) $\frac { e ^ { 2 } } { 2 }$\ B) $\frac { e ^ { 2 } - 1 } { 4 }$\ C) $\frac { e ^ { 2 } + 1 } { 16 }$\ D) 1\ E) 2