Use the LOGarithm option of the REGression feature to find a logarithmic function of the form $y = a + b \ln (x)$ that best fits the data in the table.

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Use the logarithmic function to find the value of the function when $x = 10.$

$f (10) \approx 9.5$

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Graph the logarithmic equation on the scatter diagram.

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Use the intersect feature to find the value of $x$ for which $f (x) = 7.$

When $f (x) = 7,$ $x \approx 2.7.$

Graph of the intersection of a scattered plot with an estimation line and y=7.

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For the following exercises, refer to [link] .

x	f(x)
1	7.5
2	6
3	5.2
4	4.3
5	3.9
6	3.4
7	3.1
8	2.9

Use a graphing calculator to create a scatter diagram of the data.

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Use the LOGarithm option of the REGression feature to find a logarithmic function of the form $y = a + b \ln (x)$ that best fits the data in the table.

$f (x) = 7.544 - 2.268 \ln (x)$

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Use the logarithmic function to find the value of the function when $x = 10.$

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Graph the logarithmic equation on the scatter diagram.

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Use the intersect feature to find the value of $x$ for which $f (x) = 8.$

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For the following exercises, refer to [link] .

x	f(x)
1	8.7
2	12.3
3	15.4
4	18.5
5	20.7
6	22.5
7	23.3
8	24
9	24.6
10	24.8

Use a graphing calculator to create a scatter diagram of the data.

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Use the LOGISTIC regression option to find a logistic growth model of the form $y = \frac{c}{1 + a e^{- b x}}$ that best fits the data in the table.

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Graph the logistic equation on the scatter diagram.

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To the nearest whole number, what is the predicted carrying capacity of the model?

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Use the intersect feature to find the value of $x$ for which the model reaches half its carrying capacity.

When $f (x) = 12.5,$ $x \approx 2.1.$

Graph of the intersection of a scattered plot with an estimation line and y=12.

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For the following exercises, refer to [link] .

$x$	$f (x)$
0	12
2	28.6
4	52.8
5	70.3
7	99.9
8	112.5
10	125.8
11	127.9
15	135.1
17	135.9

Use a graphing calculator to create a scatter diagram of the data.

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Use the LOGISTIC regression option to find a logistic growth model of the form $y = \frac{c}{1 + a e^{- b x}}$ that best fits the data in the table.

$f (x) = \frac{136.068}{1 + 10.324 e^{- 0.480 x}}$

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Graph the logistic equation on the scatter diagram.

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To the nearest whole number, what is the predicted carrying capacity of the model?

about $136$

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Use the intersect feature to find the value of $x$ for which the model reaches half its carrying capacity.

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Extensions

Recall that the general form of a logistic equation for a population is given by $P (t) = \frac{c}{1 + a e^{- b t}},$ such that the initial population at time $t = 0$ is $P (0) = P_{0} .$ Show algebraically that $\frac{c - P (t)}{P (t)} = \frac{c - P_{0}}{P_{0}} e^{- b t} .$

Working with the left side of the equation, we see that it can be rewritten as $a e^{- b t} :$

$\frac{c - P (t)}{P (t)} = \frac{c - \frac{c}{1 + a e^{- b t}}}{\frac{c}{1 + a e^{- b t}}} = \frac{\frac{c (1 + a e^{- b t}) - c}{1 + a e^{- b t}}}{\frac{c}{1 + a e^{- b t}}} = \frac{\frac{c (1 + a e^{- b t} - 1)}{1 + a e^{- b t}}}{\frac{c}{1 + a e^{- b t}}} = 1 + a e^{- b t} - 1 = a e^{- b t}$

Working with the right side of the equation we show that it can also be rewritten as $a e^{- b t} .$ But first note that when $t = 0,$ $P_{0} = \frac{c}{1 + a e^{- b (0)}} = \frac{c}{1 + a} .$ Therefore,

$\frac{c - P_{0}}{P_{0}} e^{- b t} = \frac{c - \frac{c}{1 + a}}{\frac{c}{1 + a}} e^{- b t} = \frac{\frac{c (1 + a) - c}{1 + a}}{\frac{c}{1 + a}} e^{- b t} = \frac{\frac{c (1 + a - 1)}{1 + a}}{\frac{c}{1 + a}} e^{- b t} = (1 + a - 1) e^{- b t} = a e^{- b t}$

Thus, $\frac{c - P (t)}{P (t)} = \frac{c - P_{0}}{P_{0}} e^{- b t} .$

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Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

Petrus Reply

what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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