Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 24

The table below shows how the average age of first marriage of Japanese woman varied in the last half of the 20th century.

$
\begin{array}{|c|c|c|c|c|}
\hline\\
t & A(t) && t & A(t)\\
\hline\\
1950 &23.0 &&1980 &25.2\\
1955 &23.8 &&1985 &25.5\\
1960 &24.4 &&1990 &25.9\\
1965 &24.5 &&1995 &26.3\\
1970 &24.2 &&2000 &27.0\\
1975 &24.7 &&&\\
\hline
\end{array}
$

a.) Use graphing calculator or computer to model those data with fourth-dgree polynomial.
b.) Use part (a) to find a model for $A'(t)$
c.) Determine the rate of change of marriage age for women in 1990.
d.) Graph the data points and the models for $A$ and $A'$

a.)


The model for $A(t) = -3\times 10^{-6} t^4 + 0.0244 t^3 - 72.331 t^2 + 95443 t - 5\times 10^7$

b.) Taking the derivative $A(t)$, we have
$ A'(t) = -3 \times 10^{-6} (4t^3) + 0.0244(3t^2) - 72.331(2t) + 95443 (1)$
$ A'(t) = -12 \times 10^{-6} t^3 + 0.0732t^2 - 144.662 t + 95443$

c.) when $t = 1990$,
$A'(1990) = - 12 \times 10^{-6} (1990)^3 + 0.0732(1990)^2 - 144.662(1990) + 95443$
$A'(1990) = 2877.752$

d.) For $A'(t)$

$
\begin{array}{|c|c|}
\hline\\
t & A'(t)\\
\hline\\
1950 & 2716.6\\
1955 & 2736.3135\\
1960 & 2756.1\\
1965 & 2776.1\\
1970 & 2796.2\\
1975 & 2816.4\\
1980 & 2836.8\\
1985 & 2857.2\\
1990 & 2877.752\\
1995 & 2898.3\\
2000 & 2919\\
\hline
\end{array}
$