Integrating a polynomial functions which describe the rate of change of a population over time to find and use an equation that describes the total population according to time.
", "licence": "None specified"}, "statement": "The growth rate, $r$ individuals per year, of a population in year $t$ after the start of observation is given by
$\\simplify {r=2*{a}t+{b}}$
a) The function for the growth rate, $r$, is the derivative of the function that describes the total population, $n$, in respect of time, $t$. So, to find an expression of the function $n$ we need to integrate $r$ in respect of $t$.
\nRemember, indefinite integration (integration without limits) is the reverse of differentiation.
\nTherefore,
\n\\[ \\begin{split} \\simplify{n=int({2*a}*t+{b},t)} &\\,= \\simplify{ {2*a}/2*t^2+{b}*t+c} \\\\ &\\,= \\simplify{{a}*t^2+{b}*t+c} \\end{split} \\]
To find the exact expression that describes the total population we need to find the value of $c$. To do so, we use the fact that when $t=0$ the total population is $\\var{c}$.
\\[ \\begin{split} \\simplify[!all]{{a}*0^2+{b}*0+c} &= \\var{c} \\\\ c&= \\var{c} \\end{split} \\]
\nThus, the expression that describes the total population in year $t$ is:
\n\\[ n= \\simplify{{a}*t^2+{b}*t+{c}} \\]
\nb) The function $n= \\simplify{{a}*t^2+{b}*t+{c}}$ gives us the total population at year t. Thus, to find the population at year $\\var{t_1}$, I need to substitute the value $t=\\var{t_1}$ in the formula.
\n\\[ \\begin{split} n&= \\simplify[!all]{{a}*{t_1}^2+{b}*{t_1}+{c}} \\\\ &= \\simplify[!all]{{a}*{t_1^2}+{b}*{t_1}+{c}} \\\\ &= \\simplify[!all]{{a*t_1^2}+{b*t_1}+{c}}\\\\ &=\\var{ansB} \\end{split} \\]
\nTherefore, when $t=\\var{t_1}$, the population is $n=\\var{ansB}$.
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\n$n=$[[0]]
\nGive your answer in the form $n=at^2+bt+c$.
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\nWhen $t=\\var{t_1}$, $n=$[[0]]
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