What is the average rate of change of the function f(m) from m=2 to m=8 and what does it represent

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What is the average rate of change of the function f(m) from m=2 to m=8 and what does it represent

let me know when you get it okay

Step-by-step explanation:

Part A) A reasonable domain is the interval [0,8]

Part B) At the beginning of the study the length of the fish was 3 cm (see the explanation)

Part C) The average rate of change is 0.40\ cm/month (see the explanation)

Read more on -

Step-by-step explanation:

a)

domain is the values for m

the starting point is m=0

f(0) = 3 (1) = 3

ending point is 5.98

5.98 = 3 (1.09)^m

5.98/3 = 1.09^m

taking the log of both sides

log1.09 (5.98/3) = log1.09 (1.09)^m

log1.09 (5.98/3) = m

8.00449 = m

since m is a little bigger than 8, i would have my domain run from 0 to 9.

so far i got a working on b)

Part A) A reasonable domain is the interval [0,8] [tex]0\leq m\leq 8[/tex]

Part B) At the beginning of the study the length of the fish was 3 cm (see the explanation)

Part C) The average rate of change is 0.40\ cm/month (see the explanation)

Step-by-step explanation:

Let

m ---> the number of months

f(m) ---> the length of the fish in cm

we have

[tex]f(m)=3(1.09^m)[/tex]

This is a exponential function of the form

[tex]f(m)=a(b^m)[/tex]

where

a is the initial value or y-intercept

b is the base of the exponential function

b=1+r

r is the percent average of change

In this problem we have

[tex]a=3\ cm\\b=1.09\\r=1.09-1=0.09\\r=0.09*100=9\%[/tex]

Part A) When the marine biologist concluded her study, the length of the fish was approximately 5.98 cm. What is a reasonable domain to plot the growth function?

For f(m)=5.98 cm

substitute the given value in the function

[tex]5.98=3(1.09^m)[/tex]

solve for m

[tex]1.993=(1.09^m)[/tex]

Apply log both sides

[tex]log(1.993)=log(1.09^m)\\log(1.993)=(m)log(1.09)\\m=log(1.993)/log(1.09)\\m=8\ months[/tex]

therefore

A reasonable domain is the interval [0,8]

[tex]0\leq m\leq 8[/tex]

Part B) What does the y-intercept of the graph of the function f(m) represent?

we know that

The y-intercept is the value of the function when the value of m is equal to zero

In this context the y intercept is the initial value of the function, or the starting size of the fish at month 0

For m=0

[tex]f(0)=3(1.09^0)\\f(0)=3\ cm[/tex]

therefore

At the beginning of the study the length of the fish was 3 cm

Part C) What is the average rate of change of the function f(m) from m = 2 to m =2, and what does it represent?

we know that

To find the average rate of change, we divide the change in the output value by the change in the input value

the average rate of change is equal to

[tex]\frac{f(b)-f(a)}{b-a}[/tex]

In this problem we have

[tex]a=2\\b=8\\f(a)=f(2)=3(1.09^2)=3.5643\ cm\\f(b)=f(8)=3(1.09^8)=5.9777\ cm[/tex]

Substitute

[tex]\frac{5.9777-3.5643}{8-2}=0.40\ cm/month[/tex]

The average rate of change is how fast the growth changed over the time period

The given equation is:

f(m) = 3(1.09)^m

where 'f(m)' represents the length of the fish after 'm' months.

Domain is considered as a set of all values of input. In this question, the input is number of months. The input should starts at 0 which will show the initial height and then as the number 'm' increases, the value of f(m) also increases.

Thus, the domain can be defined as m ≥ 0

y-intercept occurs at m=0. It means the height of the fish after 0 months, which can also be considered as initial height.

Calculate the value of function for m=2 and m=8

For m=2, f(2) = 3.564

For m=8, f(8) = 5.978

Average rate of change = (5.978-3.564) / (8-2)

Average rate of change = 0.402

It means the fish grows about 0.402cm each month in between 2 to 8 months

see below

step-by-step explanation:

f(m) = 3 (1.09)^m

a)

domain is the values for m

the starting point is m=0

f(0) = 3 (1) = 3

ending point is 5.98

5.98 = 3 (1.09)^m

5.98/3 = 1.09^m

taking the log of both sides

log1.09 (5.98/3) = log1.09 (1.09)^m

log1.09 (5.98/3) = m

8.00449 = m

since m is a little bigger than 8, i would have my domain run from 0 to 9 or 0 to 10

b )

the y intercept is the initial value of the function, or the starting size of the fish at month 0

c) to find the rate of change we need to find

f(8) = 3 (1.09)^8 = 5.977

f(4) = 3 (1.09)^4 = 4.235

aver rate of change = f(8) - f(4) 5.977 - 4.235 1.742

= = = .4355

8-4 4 4

the average rate of change is how fast the growth changed over the time period

207

Step-by-step explanation:

the answer maxed out