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# Calculus 1: exponential models solutions # Recent questions in Exponential models

Exponential models asked 2021-02-09

### The table gives the number of active Twitter users worldwide, semiannually from 2010 to 2016. \begin{array}{|c|c|} \hline \text{Years since} & \text{January 1, 2010} & \text{Twitter user} & \text{(millions)} \\ \hline 0 & 30 & 3.5 & 232 \\ \hline 0.5 & 49 & 4.0 & 255\\ \hline 1.0 & 68 & 4.5 & 284 \\ \hline 1.5 & 101 & 5.0 & 302 \\ \hline 2.0 & 138 & 5.5 & 307 \\ \hline 2.5 & 167 & 6.0 & 310 \\ \hline 3.0 & 204 & 6.5 & 317 \\ \hline \end{array} Use a calculator or computer to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models.

Exponential models asked 2021-01-31

### The table gives the midyear population of Japan, in thousands, from 1960 to 2010. $$\begin{array}{|c|c|}\hline \text{Year} & \text{Population} \\ \hline 1960 & 94.092 \\ \hline 1965 & 98.883 \\ \hline 1970 & 104.345 \\ \hline 1975 & 111.573 \\ \hline 1980 & 116.807 \\ \hline 1985 & 120.754 \\ \hline 1990 & 123.537 \\ \hline 1995 & 125.327 \\ \hline 2000 & 126.776 \\ \hline 2005 & 127.715 \\ \hline 2010 & 127.579 \\ \hline \end{array}$$ Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. [Hint: Subtract 94,000 from each of the population figures. Then, after obtaining a model from your calculator, add 94,000 to get your final model. It might be helpful to choose $$t=0$$ to correspond to 1960 or 1980.]

Exponential models asked 2021-01-19

### The annual sales S (in millions of dollars) for the Perrigo Company from 2004 through 2010 are shown in the table. $$\begin{array}{|c|c|} \hline \text{Year} & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 \\ \hline \text{Sales, S} & 898.2 & 1024.1 & 1366.8 & 1447.4 & 1822.1 & 2006.9 & 2268.9\\ \hline \end{array}$$ a) Use a graphing utility to create a scatter plot of the data. Let t represent the year, with t=4 corresponding to 2004. b) Use the regression feature of the graphing utility to find an exponential model for the data. Use the Inverse Property $$b=e^{\ln\ b}$$to rewrite the model as an exponential model in base e. c) Use the regression feature of the graphing utility to find a logarithmic model for the data. d) Use the exponential model in base e and the logarithmic model to predict sales in 2011. It is projected that sales in 2011 will be \$2740 million. Do the predictions from the two models agree with this projection? Explain.

Exponential models asked 2020-12-25

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