4.8 Fitting exponential models to data  (Page 9/12)

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Use a graphing utility to find an exponential regression formula $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and a logarithmic regression formula $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ for the points $\text{\hspace{0.17em}}\left(1.5,1.5\right)\text{\hspace{0.17em}}$ and Round all numbers to 6 decimal places. Graph the points and both formulas along with the line $\text{\hspace{0.17em}}y=x\text{\hspace{0.17em}}$ on the same axis. Make a conjecture about the relationship of the regression formulas.

Verify the conjecture made in the previous exercise. Round all numbers to six decimal places when necessary.

First rewrite the exponential with base e : $\text{\hspace{0.17em}}f\left(x\right)=1.034341{e}^{\text{0}\text{.247800x}}.\text{\hspace{0.17em}}$ Then test to verify that $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=x,$ taking rounding error into consideration:

$\begin{array}{ll}g\left(f\left(x\right)\right)\hfill & =4.035510\mathrm{ln}\left(1.034341{e}^{\text{0}\text{.247800x}}\text{\hspace{0.17em}}\right)-0.136259\hfill \\ \hfill & =4.03551\left(\mathrm{ln}\left(1.034341\right)+\mathrm{ln}\left({e}^{\text{0}\text{.2478}x}\text{\hspace{0.17em}}\right)\right)-0.136259\hfill \\ \hfill & =4.03551\left(\mathrm{ln}\left(1.034341\right)+\text{0}\text{.2478}x\right)-0.136259\hfill \\ \hfill & =0.136257+0.999999x-0.136259\hfill \\ \hfill & =-0.000002+0.999999x\hfill \\ \hfill & \approx 0+x\hfill \\ \hfill & =x\hfill \end{array}$

Find the inverse function $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)\text{\hspace{0.17em}}$ for the logistic function $\text{\hspace{0.17em}}f\left(x\right)=\frac{c}{1+a{e}^{-bx}}.\text{\hspace{0.17em}}$ Show all steps.

Use the result from the previous exercise to graph the logistic model $\text{\hspace{0.17em}}P\left(t\right)=\frac{20}{1+4{e}^{-0.5t}}\text{\hspace{0.17em}}$ along with its inverse on the same axis. What are the intercepts and asymptotes of each function?

The graph of $\text{\hspace{0.17em}}P\left(t\right)\text{\hspace{0.17em}}$ has a y -intercept at (0, 4) and horizontal asymptotes at y = 0 and y = 20. The graph of $\text{\hspace{0.17em}}{P}^{-1}\left(t\right)\text{\hspace{0.17em}}$ has an x - intercept at (4, 0) and vertical asymptotes at x = 0 and x = 20.

Exponential Functions

Determine whether the function $\text{\hspace{0.17em}}y=156{\left(0.825\right)}^{t}\text{\hspace{0.17em}}$ represents exponential growth, exponential decay, or neither. Explain

exponential decay; The growth factor, $\text{\hspace{0.17em}}0.825,$ is between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}1.$

The population of a herd of deer is represented by the function $\text{\hspace{0.17em}}A\left(t\right)=205{\left(1.13\right)}^{t},\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is given in years. To the nearest whole number, what will the herd population be after $\text{\hspace{0.17em}}6\text{\hspace{0.17em}}$ years?

Find an exponential equation that passes through the points and $\text{\hspace{0.17em}}\left(5,60.75\right).$

$y=0.25{\left(3\right)}^{x}$

Determine whether [link] could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

 x 1 2 3 4 f(x) 3 0.9 0.27 0.081

A retirement account is opened with an initial deposit of $8,500 and earns $\text{\hspace{0.17em}}8.12%\text{\hspace{0.17em}}$ interest compounded monthly. What will the account be worth in $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years? $42,888.18$ Hsu-Mei wants to save$5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with $\text{\hspace{0.17em}}7.5%\text{\hspace{0.17em}}$ APR, compounded daily, in order to reach her goal in $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ years?

Does the equation $\text{\hspace{0.17em}}y=2.294{e}^{-0.654t}\text{\hspace{0.17em}}$ represent continuous growth, continuous decay, or neither? Explain.

continuous decay; the growth rate is negative.

Suppose an investment account is opened with an initial deposit of $\text{\hspace{0.17em}}\text{10,500}\text{\hspace{0.17em}}$ earning $\text{\hspace{0.17em}}6.25%\text{\hspace{0.17em}}$ interest, compounded continuously. How much will the account be worth after $\text{\hspace{0.17em}}25\text{\hspace{0.17em}}$ years?

Graphs of Exponential Functions

Graph the function $\text{\hspace{0.17em}}f\left(x\right)=3.5{\left(2\right)}^{x}.\text{\hspace{0.17em}}$ State the domain and range and give the y -intercept.

domain: all real numbers; range: all real numbers strictly greater than zero; y -intercept: (0, 3.5);

Graph the function $\text{\hspace{0.17em}}f\left(x\right)=4{\left(\frac{1}{8}\right)}^{x}\text{\hspace{0.17em}}$ and its reflection about the y -axis on the same axes, and give the y -intercept.

The graph of $\text{\hspace{0.17em}}f\left(x\right)={6.5}^{x}\text{\hspace{0.17em}}$ is reflected about the y -axis and stretched vertically by a factor of $\text{\hspace{0.17em}}7.\text{\hspace{0.17em}}$ What is the equation of the new function, $\text{\hspace{0.17em}}g\left(x\right)?\text{\hspace{0.17em}}$ State its y -intercept, domain, and range.

$g\left(x\right)=7{\left(6.5\right)}^{-x};\text{\hspace{0.17em}}$ y -intercept: Domain: all real numbers; Range: all real numbers greater than $\text{\hspace{0.17em}}0.$

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this