Annual limousine production for the selected years is given in the table below.

Question

Annual limousine production for the selected years is given in the table below. (A) Let x represent time (in years) since 1980, and let y Limousines represent the corresponding production of limousines. Enter the data in a graphing calculator and find a cubic regression equation for the data Let L(x) denote the regression equation found in part (A) with coefficients rounded to the nearest hundredth. Find L(10) and L'(10). Year Produced 1980 1985 19904400 1995 2000 2100 6500 (B) 3900 5400 (C) Interpret L(18) and L (18) in this context

Explanation / Answer

Here R-code polynomial model is as;

x=c(1980,1985,1990,1995,2000)
y=c(2100,6500,4400,3900,5400)
d=data.frame(x,y)
l=lm(y~poly(x,3),d)
l

And the output is;

> l

Call:
lm(formula = y ~ poly(x, 3), data = d)

Coefficients:
(Intercept) poly(x, 3)1 poly(x, 3)2 poly(x, 3)3  
4460 1265 -1122 2688  

Thus the equation is;

L(x)= 2688x3 - 1122x2 + 1265x + 4460

Now,

L(10) = 2688(1000) - 1122(100) + 1265(10) + 4460

= 2688000 - 112200+ 12650 + 4460

= 2592910

Now

L' (x) = 3*2866x2 - 2*1122 x + 1265 (By taking derivative)

Thus,

L'(10) = 3*286600 - 2*11220 + 1265

= 838625

also

L(18)= (2688*18^3)-(1122*18^2)+(1265*18)+4460
= 15340118

and

L'(18) = (3*2688*18^2)-(2*1122*18)+(1265)
= 2573609

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