We study the evolution of the population of an animal species within a nature reserve. The numbers of this population were recorded in different years. The collected data are presented in the following table:
Year
2000
2005
2010
2015
Number of individuals
50
64
80
100
To anticipate the evolution of this population, the reserve management chose to model the number of individuals as a function of time. For this, it uses a function, defined on the interval $[0; +\infty[$, where the variable $x$ represents the elapsed time, in years, from the year 2000. In its model, the image of 0 by this function equals 50, which corresponds to the number of individuals in the year 2000. Part A. Model 1 In this part, the reserve management makes the hypothesis that the function sought satisfies the following differential equation: $$y' = 0.05y - 0.5 \quad (E_1)$$
Solve the differential equation $(E_1)$ with the initial condition $y(0) = 50$.
Compare the results in the table with those that would be obtained with this model.
Part B. Model 2 In this part, the reserve management makes the hypothesis that the function sought satisfies the following differential equation: $$y' = 0.05y(1 - 0.00125y)$$ Let $f$ be the function defined on $[0; +\infty[$ by: $$f(x) = \frac{800}{1 + 15\mathrm{e}^{-0.05x}}$$ and $C$ its representative curve in an orthonormal coordinate system. Using computer algebra software, the following results were obtained. For the rest of the exercise, these results may be used without proof, except for question 5.
Prove that the function $f$ satisfies $f(0) = 50$ and that for all $x \in \mathbb{R}$: $$f'(x) = 0.05f(x)(1 - 0.00125f(x))$$ We admit that this function $f$ is the unique solution of $(E_2)$ taking the initial value of 50 at 0.
With this new model $f$, estimate the population size in 2050. Round the result to the nearest integer.
Calculate the limit of $f$ as $x \to +\infty$. What can be deduced about the curve $C$? Interpret this limit in the context of this concrete problem.
Justify that the function $f$ is increasing on $[0; +\infty[$.
Prove the result obtained in line 4 of the software.
We admit that the growth rate of the population of this species, expressed in number of individuals per year, is modeled by the function $f'$. a. Study the convexity of the function $f$ on the interval $[0; +\infty[$ and determine the coordinates of any inflection points of the curve $C$. b. The reserve management claims: ``According to this model, the growth rate of the population of this species will increase for a little more than fifty years, then will decrease''. Is the management correct? Justify.
4. To understand the relationship between annual family income and annual expenditure in a certain community, 5 families were randomly surveyed, and the following statistical data table was obtained:
Income $x$ (ten thousand yuan)
8.2
8.6
10.0
11.3
11.9
\begin{tabular}{ c } Expenditure $y$ (ten
thousand yuan)
& 6.2 & 7.5 & 8.0 & 8.5 & 9.8 \hline \end{tabular} Based on the table, the regression line equation $\hat { y } = \hat { b } x + \hat { a }$ can be obtained, where $\hat { b } = 0.76$ and $\hat { a } = \bar { y } - \hat { b } \bar { x }$. Based on this, the estimated annual expenditure for a family in this community with an income of 15 ten thousand yuan is A. 11.4 ten thousand yuan B. 11.8 ten thousand yuan C. 12.0 ten thousand yuan D. 12.2 ten thousand yuan
The figure below is a line graph of environmental infrastructure investment $y$ (in units of 100 million yuan) from 2000 to 2016 in a certain region. To predict the environmental infrastructure investment in 2018 for this region, two linear regression models for $y$ and time variable $t$ were established. Based on data from 2000 to 2016 (time variable $t$ takes values $1,2 , \cdots , 17$ respectively), Model (1) was established: $\hat { y } = - 30.4 + 13.5 t$. Based on data from 2010 to 2016 (time variable $t$ takes values $1,2 , \cdots , 7$ respectively), Model (2) was established: $\hat { y } = 99 + 17.5 t$. (1) Using each of these two models, find the predicted value of environmental infrastructure investment for 2018 in this region; (2) Which model's prediction do you think is more reliable? Explain your reasoning.
(12 points) The figure below is a line graph of environmental infrastructure investment $y$ (in units of 100 million yuan) from 2000 to 2016 in a certain region. To forecast the environmental infrastructure investment for 2018 in this region, two linear regression models were established for $y$ and time variable $t$. Based on data from 2000 to 2016 (time variable $t$ takes values $1, 2, \ldots, 17$ respectively), Model (1) is established: $\hat { y } = - 30.4 + 13.5 t$; based on data from 2010 to 2016 (time variable $t$ takes values $1, 2, \ldots, 7$ respectively), Model (2) is established: $\hat { y } = 99 + 17.5 t$. (1) Using each of these two models respectively, predict the environmental infrastructure investment for 2018 in this region; (2) Which model do you think gives a more reliable prediction? Explain your reasoning.
A study group at a school conducted seed germination experiments at 20 different temperature conditions to investigate the relationship between the germination rate $y$ of a certain crop seed and temperature $x$ (in ${}^{\circ}\mathrm{C}$). From the experimental data $\left( x _ { i } , y _ { i } \right) ( i = 1,2 , \cdots , 20 )$, a scatter plot was obtained. Based on this scatter plot, between $10^{\circ}\mathrm{C}$ and $40^{\circ}\mathrm{C}$, which of the following four regression equation types is most suitable as the regression equation type for the germination rate $y$ and temperature $x$? A. $y = a + b x$ B. $y = a + b x ^ { 2 }$ C. $y = a + b \mathrm { e } ^ { x }$ D. $y = a + b \ln x$
After years of environmental remediation, a certain region has transformed barren mountains into green mountains and clear waters. To estimate the total timber volume of a certain tree species in a forest area, 10 trees of this species were randomly selected. The cross-sectional area at the base (in $\mathrm { m } ^ { 2 }$ ) and timber volume (in $\mathrm { m } ^ { 3 }$ ) of each tree were measured, yielding the following data:
Sample number $i$
1
2
3
4
5
6
7
8
9
10
Total
Base cross-sectional area $x _ { i }$
0.04
0.06
0.04
0.08
0.08
0.05
0.05
0.07
0.07
0.06
0.6
Timber volume $y _ { i }$
0.25
0.40
0.22
0.54
0.51
0.34
0.36
0.46
0.42
0.40
3.9
It is calculated that $\sum _ { i = 1 } ^ { 10 } x _ { i } ^ { 2 } = 0.038 , ~ \sum _ { i = 1 } ^ { 10 } y _ { i } ^ { 2 } = 1.6158 , \sum _ { i = 1 } ^ { 10 } x _ { i } y _ { i } = 0.2474$ . (1) Estimate the average base cross-sectional area and average timber volume per tree of this species in the forest area; (2) Find the sample correlation coefficient between the base cross-sectional area and timber volume of this tree species (accurate to 0.01); (3) The base cross-sectional area of all trees of this species in the forest area was measured, and the total base cross-sectional area of all such trees is $186 \mathrm {~m} ^ { 2 }$ . Given that the timber volume of a tree is approximately proportional to its base cross-sectional area, use the above data to estimate the total timber volume of this tree species in the forest area. Note: Correlation coefficient $r = \frac { \sum _ { i = 1 } ^ { n } \left( x _ { i } - \bar { x } \right) \left( y _ { i } - \bar { y } \right) } { \sqrt { \sum _ { i = 1 } ^ { n } \left( x _ { i } - \bar { x } \right) ^ { 2 } \sum _ { i = 1 } ^ { n } \left( y _ { i } - \bar { y } \right) ^ { 2 } } } , \sqrt { 1.896 } \approx 1.377$ .
To understand whether IQ and brain volume are related, a small study used magnetic resonance imaging to measure the brain volume (in units of 10,000 pixels) of 5 people, along with their IQ listed in the table below:
Brain Volume $( X )$
90
95
91
88
106
$\mathrm { IQ } ( Y )$
90
100
112
80
103
It is known that the mean of $X$ in the table above is $\mu _ { X } = 94$ , the mean of $Y$ is $\mu _ { Y } = 97$ , and the correlation coefficient between brain volume ($X$) and IQ ($Y$) is $r _ { X , Y }$ . Based on the table above, determine which of the following options is most likely the value of $r _ { X , Y }$ ? (1) $r _ { X , Y } \leq - 1$ (2) $- 1 < r _ { X , Y } < - 0.5$ (3) $r _ { X , Y } = 0$ (4) $0 < r _ { X , Y } < 0.5$ (5) $r _ { X , Y } \geq 1$
A student derived an equation that two physical quantities $s$ and $t$ should satisfy. To verify the theory, he conducted an experiment and obtained 15 sets of data for the two physical quantities $(s_{k}, t_{k})$, $k = 1, \cdots, 15$. The teacher suggested that he first take the logarithm of $t_{k}$, and plot the corresponding points $\left(s_{k}, \log t_{k}\right)$, $k = 1, \cdots, 15$ on the coordinate plane; where the first data is the horizontal axis coordinate and the second data is the vertical axis coordinate. Using regression line analysis, the student verified his theory. The regression line passes through the origin with a positive slope less than 1. What is the relationship between $s$ and $t$ that the student obtained most likely to be which of the following options? (1) $s = 2t$ (2) $s = 3t$ (3) $t = 10^{s}$ (4) $t^{2} = 10^{s}$ (5) $t^{3} = 10^{s}$
A laboratory collected a large number of two similar species $A$ and $B$, recording their body length $x$ (in centimeters) and body weight $y$ (in grams). The average body lengths of species $A$ and $B$ are $\overline{x_{A}} = 5.2$ and $\overline{x_{B}} = 6$ respectively, with standard deviations 0.3 and 0.1 respectively. Let the average body weights of species $A$ and $B$ be $\overline{y_{A}}$ and $\overline{y_{B}}$ respectively. If the regression lines of body weight $y$ on body length $x$ for species $A$ and $B$ are $L_{A}: y = 2x - 0.6$ and $L_{B}: y = 1.5x + 0.4$ respectively, with correlation coefficients 0.6 and 0.3 respectively. An individual $P$ with body length 5.6 centimeters and body weight 8.6 grams is discovered. Select the correct options. (1) $\overline{y_{A}} < \overline{y_{B}}$ (2) The standard deviation of body weight for species $A$ is less than that for species $B$ (3) For species $A$, the absolute difference between individual $P$'s body weight and the average body weight $\overline{y_{A}}$ is greater than one standard deviation (4) The distance from point $(5.6, 8.6)$ to line $L_{A}$ is less than its distance to line $L_{B}$ (5) The distance from point $(5.6, 8.6)$ to point $(\overline{x_{A}}, \overline{y_{A}})$ is less than its distance to point $(\overline{x_{B}}, \overline{y_{B}})$
A company collected data on the number of customers $x$ (in units of 100 people) and sales revenue $y$ (in units of 10,000 yuan) from 8 branches last week, obtaining 8 data points $(x, y)$ as follows: $(3,3), (3,5), (3,2), (4,4), (5,8), (6,7), (8,12), (8,7)$. These 8 points are plotted on the coordinate plane, and the regression line equation for $y$ with respect to $x$ is determined to be $y = \frac { 5 } { 4 } x - \frac { 1 } { 4 }$. The company wants to analyze from another perspective. The 8 data points are sorted separately from smallest to largest for the number of customers and sales revenue, resulting in new 8 data points $(x, y)$ as follows: $(3,2), (3,3), (3,4), (4,5), (5,7), (6,7), (8,8), (8,12)$. Let the regression line equation for $y$ with respect to $x$ for the new 8 data points be $y = m x + b$, where $m, b$ are real numbers. Based on the above, select the correct option. (1) $m = \frac { 5 } { 4 }$ and $b = - \frac { 1 } { 4 }$ (2) $m > \frac { 5 } { 4 }$ and $b > - \frac { 1 } { 4 }$ (3) $m > \frac { 5 } { 4 }$ and $b < - \frac { 1 } { 4 }$ (4) $m < \frac { 5 } { 4 }$ and $b > - \frac { 1 } { 4 }$ (5) $m < \frac { 5 } { 4 }$ and $b < - \frac { 1 } { 4 }$
A certain alloy is composed of two metals, A and B. A student wants to know the relationship between the metal ratio and the wavelength of the alloy. He conducted an experiment measuring ``the wavelength $y$ (in nanometers) of an alloy with A comprising $x\%$'' and plotted 20 data points $(x_k, y_k)$, $k = 1, \cdots, 20$, on the $xy$ plane. The regression line (best-fit line) is $y = 21.3 x - 40$. To comply with submission standards, the report must be described as ``the wavelength $v$ (in micrometers) of an alloy with B comprising $u\%$''. He converted the data $(x_k, y_k)$ to $(u_k, v_k)$, $k = 1, \cdots, 20$, and obtained the regression line on the $uv$ plane as $v = a u + b$. Given that 1 nanometer $= 10 ^ { - 9 }$ meter and 1 micrometer $= 10 ^ { - 6 }$ meter. Select the correct options. (1) $u _ { k } = 100 - x _ { k } , k = 1 , \cdots , 20$ (2) $v _ { k } = 1000 y _ { k } , k = 1 , \cdots , 20$ (3) The standard deviation of $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots , u _ { 20 }$ equals the standard deviation of $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots , x _ { 20 }$ (4) $b = 2.09$ (5) The student found another data point $(u _ { 21 } , v _ { 21 })$ satisfying $v _ { 21 } = a u _ { 21 } + b$; if these 21 data points $(u_k, v_k)$, $k = 1, \cdots, 21$, are plotted on the $uv$ plane, the regression line is still $v = a u + b$