Consumer price index The U.S. consumer price index (CPI)...

Consumer price index The U.S. consumer price index (CPI) measures the cost of living based on a value of 100 in the years $1982-1984 .$ The CPI for the years $1995-2010$ (see figure) is modeled by the function $c(t)=151 e^{0.026 t},$ where $t$ represents years after 1995 a. Was the average growth rate greater between the years 1995 and $2000,$ or 2005 and $2010 ?$ b. Was the growth rate greater in $2000(t=5)$ or 2005 $(t=10) ?$ c. Use a graphing utility to graph the growth rate, for $0 \leq t \leq 15 .$ What does the graph tell you about growth in the cost of living during this time period? (GRAPH CAN'T COPY)

Consumer price index The U.S. consumer price index (CPI) measures the cost of living based on a value of 100 in the years $1982-1984 .$ The CPI for the years $1995-2010$ (see figure) is modeled by the function $c(t)=151 e^{0.026 t},$ where $t$ represents years after 1995
a. Was the average growth rate greater between the years 1995 and $2000,$ or 2005 and $2010 ?$
b. Was the growth rate greater in $2000(t=5)$ or 2005 $(t=10) ?$
c. Use a graphing utility to graph the growth rate, for $0 \leq t \leq 15 .$ What does the graph tell you about growth in the cost of living during this time period?
(GRAPH CAN'T COPY)

Key Concepts

Exponential Functions and Growth

Exponential functions are mathematical models that describe processes in which the rate of change of a quantity is proportional to its current value. These functions are widely used to represent growth in various contexts such as populations, investments, and economic indices, where small percentage increases can compound over time.

Consumer Price Index (CPI)

The Consumer Price Index is an economic indicator that measures the average change over time in the prices paid by urban consumers for a market basket of consumer goods and services. It is used to assess price changes associated with the cost of living, helping policymakers and economists evaluate inflation and economic performance.

Average Rate of Change

The average rate of change of a function over a specific interval provides a measure of how much the function's output changes per unit change in the input. This concept is crucial in comparing changes over different time periods, allowing for the analysis of trends such as inflation or economic growth across intervals.

Instantaneous Rate of Change

The instantaneous rate of change, often determined using derivatives, gives the rate at which a function is changing at any given point. In economic contexts, it is used to analyze how quickly quantities like consumer prices or economic indices are increasing or decreasing at a particular moment in time.

Graphical Analysis of Growth Rates

Graphing the rate of growth helps visualize changes over time, making patterns and trends clear. By plotting the instantaneous growth rate of an exponential function over a period, one can observe whether the rate is accelerating, decelerating, or remaining constant, which aids in understanding the dynamic behavior of economic indicators like the cost of living.

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