Exponential function questions and answers

The table shows the annual service revenues R1 in billions of dollars for the cellular telephone industry for the years 2000 through 2006.

\(\begin{matrix} Year&2000&2001&2002&2003&2004&2005&2006\\ R_1&52.5&65.3&76.5&87.6&102.1&113.5&125.5 \end{matrix}\)
(a) Use the regression capabilities of a graphing utility to find an exponential model for the data. Let t represent the year, with t=10 corresponding to 2000. Use the graphing utility to plot the data and graph the model in the same viewing window.
(b) A financial consultant believes that a model for service revenues for the years 2010 through 2015 is \(\displaystyle{R}{2}={6}+{13}+{13},{9}^{{0.14}}{t}\). What is the difference in total service revenues between the two models for the years 2010 through 2015?

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.
\(\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline \hline f(x) & 13.98 & 17.84 & 20.01 & 22.7 & 24.1 & 26.15 & 27.37 & 28.38 & 29.97 & 31.07 & 31.43 \\ \hline \end{array}\)

A researcher is trying to determine the doubling time fora population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table. ​
\(\begin{array}{|c|c|} \hline \text{Time (hours)} & \text{Bacteria count (CFU/mL)}\\ \hline 0 & 37 \\ \hline 4 & 47 \\ \hline 8 & 63 \\ \hline 12 & 78 \\ \hline 16 & 105 \\ \hline 20 & 130 \\ \hline 24 & 173 ​\\ \hline \end{array}\) ​
Use a graphing calculator to find an exponential curve \(f(t)=a\times b^t\) that models the bacteria population t hours later.

The exponential growth models describe the population of the indicated country, A, in millions, t years after 2006. Canada \(A=33.1e0.009t\)

Uganda

\(A=28.2e0.034t\)

use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. By 2009, the models indicate that Canada's population will exceed Uganda's by approximately 2.8 million.

Determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data; for those that are exponential, find an exponential function that models the data.

\(\begin{array}{|cc|}\hline x & g(x) \\ \hline-1 & 6 \\ 0 & 1 \\ 1 & 0 \\ 2 & 3 \\ 3 & 10 \\ \hline\end{array}\)

Transform the single linear differential equation into a system of first-order differential equations.
\(\displaystyle{x}{''}-{3}{x}'+{2}{x}={t}^{{{2}}}\)

The table shows the annual service revenues R1 in billions of dollars for the cellular telephone industry for the years 2000 through 2006. Year 2000 2001 2002 2003 2004 2005 2006 R1 52.5 65.3 76.5 87.6 102.1 113.51 25.5
(a) Use the regression capabilities of a graphing utility to find an exponential model for the data. Let t represent the year, with t=10 corresponding to 2000. Use the graphing utility to plot the data and graph the model in the same viewing window.
(b) A financial consultant believes that a model for service revenues for the years 2010 through 2015 is \(\displaystyle{R}{2}={6}+{13}+{13},{9}^{{0.14}}{t}\). What is the difference in total service revenues between the two models for the years 2010 through 2015?

Transform the given differential equation or system into an equivalent system of first-order differential equations.
\(\displaystyle{t}^{{{3}}}{x}^{{{\left({3}\right)}}}-{2}{t}^{{{2}}}{x}{''}+{3}{t}{x}'+{5}{x}={\ln{{t}}}\)

Determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data; for those that are exponential, find an exponential function that models the data. \(\begin{matrix} \text{x} & \text{f(x)}\\ \hline \text{−1−1} & \text{32 23}\\ \text{0} & \text{3}\\ \text{1} & \text{6}\\ \text{2} & \text{12}\\ \text{3} & \text{24}\\ \end{matrix}\)