Given the function g(x) = 6^x, section a is from x = 1 to x = 2 and section b is from x = 3 to x = 4. part a: find the average rate of change in each section. part b: how many times greater is the average rate of change of section b than section a?

Part a: the average rate of change over a certain interval is the slope of the line that connects the left and right points  on a graph. i have included a picture from my notes of this concept.so for you question, the average rate of change over a section iswhere  are the x-coordinates of the boundaries  of the section/interval.for section a,  . (the order doesn't matter; we could have also done . as long as we have the boundaries of the section.) and the value of  at those x-coordinates can be calculated: so the average rate of change for section a  isfor section b,  . and the value of  at those x-coordinates can be calculated: so the average rate of change for section b ispart b: the average rate of change of section b is 30 times greater than section a. if you multiply the average rate of change of section a by 30, you'll get section b's avg rate of change.

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step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

1 the 1st term, 3 the 2nd term, 6 the 3rd term,etc.

note the following:

from the 1st term (1) to the 2nd term (3), we go up by 2,from the 2nd term (3) to the 3rd term (6), we go up by 3from the 3rd term (6) to the 4th term (10), we go up by 4, andfrom the 4th term (10) to the 5th term (15), we go up by 5.

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

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