Bob is studying the growth of a particular species of plant. He writes the following equation to show the height of the plant f(n), in cm, after n days: F(N)=10(1.02)^n
1. Starting from 0 days, plug x-values into our function to find out how many days it takes for the plant to grow to 11.04 cm
2. Write a compound inequality to show the reasonable domain of the plant growth:
my answer: f(n) = 10(1.02)n
3. What is the y-intercept of our function?
My answer: The height of the plant
4. What does the y-intercept of the graph of the function f(n) represent?
My Answer: i'm stuck
5. What is the average rate of change formula for exponential growth functions?
My Answer:???
6. Plug and chug n= 1 to n = 5 into your formula above. Show all work!
My Answer: f(n) = 10(1.02)n 11.04=10(1.02)5
7. What does the average rate of change answer represent?
My Answer: Growth per day.
11.04 = 10 (1.02)^n
I am not going to try lots of days, boring
log 11.04 = log 10 + n log 1.02
4.996 days
so five days
2. n>0 and <the life of the plant
3. when n = 0, height = 10
4. the height of the plant when we bought it
5. F(n+1) = F(n) * 1.02
6. calculate F(1) = 10(1.02) = 10.2
then F(2) = 10.2 * 1.02 =
F(3) = F(2) * 1.02 = etc
7. ratio of next day height to previous day height
OMG, I thought I knew this stuff better!!
Now i'm really lost!!
I guess I did not explain very well
in the first question for example it asks you to plug and chug from 0 to 11.04
I did not bother but
first day
10*1.02 = 10.2
second day
10.2 * 1.02 = 10.4
third day
10.4 * 1.02 = 10.61
fourth day
10.61 * 1.02 = etc until you get to 11.04
10.82
fifth day
10.82*1.02 = 11.04 done
Is # 6 like this
F(1)= 10(1.02) = 10.2
F(2)= 10.2 x 1.02 = 10.40
F(3=10.404 x 1.02 = 10.61
then continue? So, the final answer is 11.04?
Yes, it told you to keep doing it until you get to 11.04, that was after 5 days
(I cheated and took the log of both sides so solve directly)
I believe 1 and 6 are the same question asked different ways.
The important thing here is that the rate of change is proportional to your present value, so the change keeps changing.
The most common example is compound interest
if interest rate is 2% per year compounded yearly
then
Yn+1 = Yn + .02 Yn = Yn (1.02)
and in general
Y = Yinitial * (1.02)^n
4. The y-intercept of the graph of the function f(n) represents the initial height of the plant when n = 0. In this case, the y-intercept of the function f(n) = 10(1.02)^n is 10, so it represents the initial height of the plant as 10 cm.
5. The average rate of change formula for exponential growth functions is given by the formula: (f(b) - f(a))/(b - a), where f(b) and f(a) represent the values of the function at two different points b and a, and (b - a) represents the difference between those points in the independent variable. In this case, for the function f(n) = 10(1.02)^n, the average rate of change formula would be (f(b) - f(a))/(b - a).
6. To find the values of the function f(n) for n = 1 to n = 5, you can plug in these values into the formula f(n) = 10(1.02)^n one by one.
For n = 1: f(1) = 10(1.02)^1 = 10.2
For n = 2: f(2) = 10(1.02)^2 = 10.404
For n = 3: f(3) = 10(1.02)^3 = 10.61208
For n = 4: f(4) = 10(1.02)^4 = 10.8243216
For n = 5: f(5) = 10(1.02)^5 = 11.040826432
7. The average rate of change answer represents the rate at which the plant is growing on average per day. It indicates the change in height of the plant over a given time period, divided by the length of that time period. In this case, since we used a difference of 1 day for each calculation, the average rate of change represents the average growth of the plant in centimeters per day.