Question

A scientist 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

Part A: When the scientist concluded his study, the height of the plant was approximately 11.04 cm. What is a reasonable domain to plot the growth function?

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

Part C: What is the average rate of change of the function f(n) from n = 1 to n = 5, and what does it represent?

Answer 1

I'm assuming you mean \(f(n)=10(1.02)^n\)? If not, it shouldn't change the general idea behind the other parts to this question.

Anyway, since height is measured as a function of time in days, it stands to reason you can only have a non-negative number of days (unless you reasonably define what negative time refers to). A possible candidate for the domain is then the non-negative integers, \(\mathbb{N}\cup\{0\}\). Alternatively, if you measure throughout the day (say, on the hour), you can consider \(\left\{x\in\mathbb{Q}|x=\dfrac{n}{24},n\in\mathbb{N}\cup\{0\}\right\}\). The possibilities are virtually endless.

The \(y\) axis gives you the value of \(f(n)\) for any value of \(n\). The intercept occurs when \(n=0\). What does this mean for your function?

The average rate of change of a function continuous over an interval \([a,b]\) is given by \(\dfrac{f(b)-f(a)}{b-a}\). It should be obvious what this represents because the name gives it away.