Gross domestic product The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070 (actual and projected).

Year	GDP	Year	GDP
2000	9143	2040	79,680
2005	12,145	2045	103,444
2010	16,174	2050	133,925
2015	21,270	2055	173,175
2020	27,683	2060	224,044
2025	35,919	2065	290,042
2030	46,765	2070	375,219
2035	61,100

Source: Social Security Administration Trustees Report

Assume the GDP can be modeled with the function

G ( t )   = 212.9 ( 0.2 t   +   5 ) 3 −   5016 ( 0.2 t   +   5 ) 2 +   8810.4 t   +   104 , 072

where G(t) is in billions of dollars and t is the number of years past 2000.

(a) Use the model to find and interpret the instantaneous rates of change of the GDP in 2005 and 2015.

(b) Use the data in the table to find the average rate of change of the GDP from 2005 to 2015.

(c) How well does your answer from part (b) approximate the instantaneous rate of change of GDP in 2010?

To calculate: The instantaneous rate of change of the GDP in 2005 and 2015 and interpret it. The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070. Assume that the GDP can be modeled with the function G(t)=212.9(0.2t+5)3−5016(0.2t+5)2+8810.4t+104,072 where G(t) is in billions of dollars and t is the number of years past 2000.

Given Information:

The table shows U.S. gross domestic product (GDP) in billions of dollars,

Formula used:

According to the power rule, if f(x)=xn, then,

f′(x)=nxn−1

According to the property of differentiation, if a function is of the form, g(x)=cf(x), then,

g′(x)=cf′(x)

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

f′(x)=u′(x)+v′(x)

According to the product rule, if f(x)=u(x)⋅v(x), then

f′(x)=u′(x)⋅v(x)+v′(x)⋅u(x)

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nun−1dudx

Calculation:

Consider the provided function,

G(t)=212.9(0.2t+5)3−5016(0.2t+5)2+8810.4t+104,072

Consider (0.2t+5) to be u,

G(t)=212.9(u)3−5016(u)2+8810.4t+104,072

Differentiate both sides with respect to t,

dGdt=ddt(212.9(u)3−5016(u)2+8810.4t+104,072)

Simplify using the power rule,

dGdt=ddt(212.9u3)−ddt(5016u2)+ddt(8810

To calculate: The average rate of change of the GDP from 2005 to 2015 using the data given. The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070. Assume that the GDP can be modeled with the function G(t)=212.9(0.2t+5)3−5016(0.2t+5)2+8810.4t+104,072 where G(t) is in billions of dollars and t is the number of years past 2000.

The wellness of average rate of change of the GDP from 2005 to 2015 approximates the instantaneous rate of change of GDP in 2020. The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070. Assume that the GDP can be modeled with the function G(t)=212.9(0.2t+5)3−5016(0.2t+5)2+8810.4t+104,072 where G(t) is in billions of dollars and t is the number of years past 2000.