Question
43. Scientific Research In each year from 1983 to 2003, the percentage $y$ of research articles in Physical Review written by researchers in the United States can be approximated by$y=82-0.78 t-1.02 x$ percentage points $\quad(0 \leq t \leq 20)$where $t$ is the year since 1983 and $x$ is the percentage of articles written by researchers in Europe. $^{13}$ Calculate and interpret $\frac{\partial y}{\partial t}$ and $\frac{\partial y}{\partial x} .$
Step 1
This is denoted as $\frac{\partial y}{\partial t}$. In this case, we treat $x$ as a constant and differentiate $y$ with respect to $t$. The derivative of a constant is zero, so the derivative of $82$ with respect to $t$ is zero. The derivative of $-0.78t$ with Show more…
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In each year from 1983 to 2003, the percentage $y$ of research articles in Physical Review written by researchers in the United States can be approximated by $y=82-0.78 t-1.02 x$ percentage points $\quad(0 \leq t \leq 20)$ where $t$ is the year since 1983 and $x$ is the percentage of articles written by researchers in Europe. ${ }^{1}$ a. In 2003, researchers in Europe wrote $38 \%$ of the articles published by the journal that year. What percentage was written by researchers in the United States? b. In 1983 , researchers in the United States wrote $61 \%$ of the articles published that year. What percentage was written by researchers in Europe? c. What are the units of measurement of the coefficient of $t$ ?
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The percentage of research articles in the prominent journal Physical Review written by researchers in the United States during 1983-2003 can be modeled by A(t) = 25 + 36/(1+0.6(0.7)^-t), where t is times in years (t = 0 represents 1983). Numerically estimate lim(x→∑) A(t), and interpret the answer.
Scientific Research The percentage of research articles in the prominent journal Physical Review written by researchers in the United States can be modeled by $$ A(t)=25+\frac{36}{1+0.6(0.7)^{-t}} $$ where $t$ is time in years $(t=0$ represents 1983$) .{ }^{4}$ Numerically estimate $\lim _{t \rightarrow+\infty} A(t)$ and interpret the answer. HINT [See Example 6.]
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