# Exponential function questions and answers

Recent questions in Exponential models
Exponential models

### The following question consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. When does population increase the fastest in the threshold logistic equation $$\displaystyle{P}'{\left({t}\right)}={r}{P}{\left({1}-{\frac{{{P}}}{{{K}}}}\right)}{\left({1}-{\frac{{{T}}}{{{P}}}}\right)}?$$

Exponential models

### 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?

Exponential models

### Find The Exponential Function $$\displaystyle{f{{\left({x}\right)}}}={a}^{{x}}$$ Whose Graph Is Given.

Exponential models

### 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. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline \hline f(x) & 13.98 & 17.84 & 20.01 & 22.7 & 24.1 & 26.15 & 27.37 & 28.38 & 29.97 & 31.07 & 31.43 \\ \hline \end{array}$$

Exponential models

### Graph each function and tell whether it represents exponential growth, exponential decay, or neither. $$\displaystyle{y}={\left({2.5}\right)}^{{{x}}}$$

Exponential models

### Transform the given differential equation or system into an equivalent system of first-order differential equations. $$\displaystyle{x}{''}+{2}{x}'+{26}{x}={82}{\cos{{4}}}{t}$$

Exponential models

### 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.

Exponential models

### 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.

Exponential models

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

Exponential models

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

Exponential models

### 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?

Exponential models

### 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. The models indicate that in 2013, Uganda's population will exceed Canada's.

Exponential models

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

Exponential models

### The table shows the populations P (in millions) of the United States from 1960 to 2000. Year 1960 1970 1980 1990 2000 Popupation, P 181 205 228 250 282 (a) Use the 1960 and 1970 data to find an exponential model P1 for the data. Let t=0 represent 1960. (c) Use a graphing utility to plot the data and graph models P1 and P2 in the same viewing window. Compare the actual data with the predictions. Which model better fits the data? (d) Estimate when the population will be 320 million.

Exponential models

### Transform the given differential equation or system into an equivalent system of first-order differential equations. $$\displaystyle{x}^{{{\left({4}\right)}}}+{3}{x}{''}+{x}={e}^{{{t}}}{\sin{{2}}}{t}$$

Exponential models

### 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. $$\begin{matrix} \text{x} & \text{f(x)}\\ \hline \text{−1−1} & \text{32 23}\\ \text{0} & \text{3}\\ \text{1} & \text{6}\\ \text{2} & \text{12}\\ \text{3} & \text{24}\\ \end{matrix}$$

Exponential models

### Transform the given differential equation or system into an equivalent system of first-order differential equations. $$\displaystyle{x}^{{{\left({4}\right)}}}+{6}{x}{''}-{3}{x}'+{x}={\cos{{3}}}{t}$$

Exponential models

### Transform the given differential equation or system into an equivalent system of first-order differential equations. $$\displaystyle{x}^{{{\left({3}\right)}}}-{2}{x}{''}+{x}'={1}+{t}{e}^{{{t}}}$$

Exponential models