지수함수

For the following exercise, choose the correct answer choice.

A doctor injects a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there are 4.75 milligrams of dye remaining in the patient’s system. Which is an appropriate model for this situation?

• ⓐ $f(t)=13{(0.0805)}^{t}$
• ⓑ $f(t)=13{e}^{0.9195t}$
• ⓒ $f(t)=13{e}^{(-0.0839t)}$
• ⓓ $f(t)=\frac{4.75}{1+13{e}^{-0.83925t}}$

What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?

What is regression analysis? Describe the process of performing regression analysis on a graphing utility.

What might a scatterplot of data points look like if it were best described by a logarithmic model?

What does the y-intercept on the graph of a logistic equation correspond to for a population modeled by that equation?

For the following exercises, match the given function of best fit with the appropriate scatterplot in (the referenced figure) through (the referenced figure). Answer using the letter beneath the matching graph.

$y=10.209{e}^{-0.294x}$

For the following exercises, match the given function of best fit with the appropriate scatterplot in (the referenced figure) through (the referenced figure). Answer using the letter beneath the matching graph.

$y=5.598-1.912\ln (x)$

For the following exercises, match the given function of best fit with the appropriate scatterplot in (the referenced figure) through (the referenced figure). Answer using the letter beneath the matching graph.

$y=2.104{(1.479)}^{x}$

For the following exercises, match the given function of best fit with the appropriate scatterplot in (the referenced figure) through (the referenced figure). Answer using the letter beneath the matching graph.

$y=4.607+2.733\ln (x)$

For the following exercises, match the given function of best fit with the appropriate scatterplot in (the referenced figure) through (the referenced figure). Answer using the letter beneath the matching graph.

$y=\frac{14.005}{1+2.79{e}^{-0.812x}}$

To the nearest whole number, what is the initial value of a population modeled by the logistic equation $P(t)=\frac{175}{1+6.995{e}^{-0.68t}}?$ What is the carrying capacity?

Rewrite the exponential model $A(t)=1550{(1.085)}^{x}$ as an equivalent model with base $e.$ Express the exponent to four significant digits.

A logarithmic model is given by the equation $h(p)=67.682-5.792\ln (p).$ To the nearest hundredth, for what value of $p$ does $h(p)=62?$

A logistic model is given by the equation $P(t)=\frac{90}{1+5{e}^{-0.42t}}.$ To the nearest hundredth, for what value of t does $P(t)=45?$

What is the y-intercept on the graph of the logistic model given in the previous exercise?

For the following exercises, use this scenario: The population $P$ of a koi pond over $x$ months is modeled by the function $P(x)=\frac{68}{1+16{e}^{-0.28x}}.$

Graph the population model to show the population over a span of $3$ years.

For the following exercises, use this scenario: The population $P$ of a koi pond over $x$ months is modeled by the function $P(x)=\frac{68}{1+16{e}^{-0.28x}}.$

What was the initial population of koi?

For the following exercises, use this scenario: The population $P$ of a koi pond over $x$ months is modeled by the function $P(x)=\frac{68}{1+16{e}^{-0.28x}}.$

How many koi will the pond have after one and a half years?

For the following exercises, use this scenario: The population $P$ of a koi pond over $x$ months is modeled by the function $P(x)=\frac{68}{1+16{e}^{-0.28x}}.$

How many months will it take before there are $20$ koi in the pond?