Question

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- USCALCI 3.4.167. Smo Nocas I Ask Your Tear chyby decade in the 19 century. Consider the table, which reflects the population (in millions of Population of a city Years since 10 Population (millions) 1339 () Using a car or a computer program, find the best resu ntomes p ionn e auction of the number of years since 1000 (Round alumical values to five decimal places) C) Find the device of the Gution from part(). Explain the physical meaning of the drive c) ed the second d ate of the artion from part(). Explain the physical meaning of the second derivative There which the o tion points OSCALCI 3.4.16 y su vou Toucher

CALCULUS

Answer 1

Solution:

a) Using the given data, we can find the required best fit line using the Excel data analysis tool. Following is the required snapshot of the regression results:

Years since Population 1800 (millions) SUMMARY OUTPUT 0.8796 1.039 1.264 1.516 1.66 Regression Statistics Multiple F 0.969174 R Square 0.939298 Adjusted! 0.93171 Standard E 0.325192 Observati 10 2.634 3.272 3.912 4.422 ANOVA of Regressio Residual Total SSMS F gnificance F 1 13.09087 13.09087 123.7909 3.81E-06 8 0.845998 0.10575 9 13.93686 Coefficier Standard Et Stat P-value Lower 959 Upper 95% Lower 95.1 Upper 95.0% Intercept 0.427482 0.19416 2.201697 0.058838 -0.02025 0.875216 -0.02025 0.875216 X Variable 0.039834 0.00358 11.12614 3.81E-06 0.031578 0.04809 0.031578 0.04809

From the above image, we can see that intercept term is = 0.427482, and the slope coefficient is 0.039834. So, the best fit linear function becomes:

P(t) = 0.4275 + 0.0398*t

b) Using the above equation, we can find the derivative of equation, P'(t) as:

P'(t) = dP(t)/dt

P'(t) = 0.0398

The physical meaning of this derivative becomes that: As t increases, the population increases (due to positive sign of P'(t)) by 0.0398 millions (for 10 years increase in t).

c) For second derivative: note that the above equation P'(t) is independent of t. So, the double derivative will be 0.

This means that rate at which the population changes/increases/grows is constant.

(Drill downs options presented would be better for accuracy of answers)