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

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

Rezat Reply

information

Eliyee

devaluation

Eliyee

WARKISA

hi guys good evening to all

Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

yes,thank you

Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

how do you save a country economic situation when it's falling apart

Lilia Reply

what is the difference between economic growth and development

Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

thank you so much 👍 sir

Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

types of unemployment

Yomi Reply

What is the difference between perfect competition and monopolistic competition?

Mohammed

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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