Consumer Price Index An economy's consumer price index (CPI)...

Consumer Price Index An economy's consumer price index (CPI) is described by the function $$ I(t)=-0.2 t^{3}+3 t^{2}+100 \quad(0 \leq t \leq 11) $$ in year $t,$ where $t=0$ corresponds to 2003 a. At what rate was the CPI changing in $2008 ?$ In $2010 ?$ In $2013 ?$ b. What was the average rate of increase in the CPI over the period from 2008 to 2013 ?

Consumer Price Index An economy's consumer price index (CPI) is described by the function
$$
I(t)=-0.2 t^{3}+3 t^{2}+100 \quad(0 \leq t \leq 11)
$$
in year $t,$ where $t=0$ corresponds to 2003
a. At what rate was the CPI changing in $2008 ?$ In $2010 ?$ In $2013 ?$
b. What was the average rate of increase in the CPI over the period from 2008 to 2013 ?

Key Concepts

Derivative (Instantaneous Rate of Change)

The derivative of a function at a given point provides the instantaneous rate of change of the function at that point. It is used to determine how fast a quantity (represented by the function) is changing at any specific moment. In many applications, such as economics or physics, calculating the derivative helps in understanding and predicting the behavior of a system at particular instances.

Average Rate of Change

The average rate of change of a function over an interval is computed by taking the difference in function values at the endpoints of the interval and dividing by the length of the interval. This concept reflects the overall change in the function over a specified period and provides a simple way of summarizing the behavior of the function across that interval.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. These functions are important in calculus and algebra because of their well-behaved derivatives and the ease with which they can be manipulated, making them ideal for modeling various types of real-world phenomena.

Economic Indices (Consumer Price Index - CPI)

Economic indices like the Consumer Price Index (CPI) are statistical measures used to represent changes in the price level of a basket of consumer goods and services over time. These measures are important indicators of inflation and economic health, and understanding their rates of change can provide insights into economic trends and inform monetary policies.

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