part a
answer: a reasonable domain is 
where n is a real number. so n can be between 0 and 10. both endpoints are included.
work shown:
n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.
f(n) = 12*(1.03)^n
16.13 = 12*(1.03)^n
16.13/12 = (1.03)^n
1.34416666666667 = (1.03)^n
(1.03)^n = 1.34416666666667
log[ (1.03)^n ] = log[ 1.34416666666667 ]
n*log[ 1.03 ] = log[ 1.34416666666667 ]
n = log[ 1.34416666666667 ]/log[ 1.03 ]
n = 10.0062999823929
this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.
note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)
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part b
answer: the y intercept is 12. it represents the starting height of the plant in cm.
work shown:
plug n = 0 into the f(n) function. simplify
f(n) = 12*(1.03)^n
f(0) = 12*(1.03)^0
f(0) = 12*(1)
f(0) = 12
on day n = 0, aka the starting point, the height f(n) is 12 cm
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part c
answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.
work shown:
compute f(3)
f(n) = 12*(1.03)^n
f(3) = 12*(1.03)^3
f(3) = 13.112724 < we'll use this later
compute f(10)
f(n) = 12*(1.03)^n
f(10) = 12*(1.03)^10
f(10) = 16.1269965521294 < we'll use this later
now use the formula below with a = 3 and b = 10
aroc = average rate of change
aroc = [ f(b) - f(a) ]/[ b - a ]
aroc = [ f(10) - f(3) ]/[ 10 - 3 ]
aroc = (16.1269965521294 - 13.112724)/(10 - 3)
aroc = 3.0142725521294/7
aroc = 0.43061036458991
round this however you need to
note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).
- 2b
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