Question

For the graphed function f(x) = (2)x + 2 + 1, calculate the average rate of change from x = 0 to x = 2.
@ranga

Answer 1

@shamil98

Answer 2

@kc_kennylau

Answer 3

What is (2)x + 2 +1?
\[2x + 2 + 1\]
or
\[x^2 + 2x + 1\]

this looks to me to be an exponential function of the form
\[2^{x+2} + 1\]

Answer 4

ah nevermind... I answered my own question

Answer 5

lol it took you soo long to type that? yes this is a exponential function.

Answer 6

you need to know 2 things here to do this derivative: the chain rule and that:
\[dA^x = ln(A)A^xdx\]

\[d(2^{x+2} + 1) = ln(2)2^{x+2}dx\]

Answer 7

ahh just realized you don't want the calculus version of this.

Answer 8

So how do I apply these rules?

Answer 9

if you want the "average rate of change" then you want to construct a line from
\[(0, 2^2+1) = (0,5) \rightarrow (2,2^4+1) = (2,17)\]
and figure out the slope

Answer 10

Average rate of change between x = 0 to x = 2 is: { f(2) - f(0) } / { 2 - 0 }

Answer 11

@polaris_s0i do i solve it ?
and @ranga so the slope is 2?

Answer 12

Yeah @ranga has it
\[slope = \frac{y_2 - y_1}{x_2-x_1}\]
and slope is the average rate of change.

Answer 13

what are the two y

Answer 14

because there are 2 points:
\[(x_1,y_1) = (0,5) and (x_2,y_2)=(2,17)\]

Answer 15

f(x) = 2^(x+2) + 1
f(2) = 2^(2+2) + 1 = 2^4 + 1 = 17
f(0) = 2^(0+2) + 1 = 2^2 + 1 = 5
Average rate of change between x = 0 to x = 2 is: { f(2) - f(0) } / { 2 - 0 }
= (17-5) / 2 = 12 / 2 = 6

Answer 16

what is this process called @ranga

Answer 17

I don't know if there is any special name. What it does, is approximates a straight line between the two points instead of a curve and finds the slope of the straight line. That is the average rate of change. The derivative gives the instant rate of change at a point. Thus, f'(2) is the instantaneous rate of change at x = 2.
f'(0) is the instantaneous rate of change at x = 0.
But (f(2) - f(0)) / (2-0) is the average rate of change between x = 0 and x = 2.

Graphically it is:

Answer 18

you are calculating the distance in y direction over the distance in x.
rise over run.

Answer 19

so slope equals 6

Answer 20

yes.