Question

Find the average rate of change of the function: f(x)=x^2
over the interval 2<=x<=5 (<= means less than or equal to)

Answer 1

Use the formula for slope\[m=\frac{y_2-y_1}{x_2-x_1}\] where y2,y1 and x2,x1 are the y and x-coordinates of f(x) = x^2 in [2, 5]

Answer 2

So x and y for the function are 0,0. So:
5-0 5
--- = --- = 2.5 ?
2-0 2

@navk

Answer 3

In (x1, y1) and (x2, y2) x1 and y1 are the x-values 2 and 5 of the interval. and y1, y2 are the y-values of the function at those x values so thats not right.

Answer 4

Would it be:

5-2 3
--- = --- = 0 ?
0-0 0

@navk I am unsure how to go about setting up the equation to solve.

Answer 5

plug in x = 2 in the function f(x) to get f(2) = 2^2 = 4. this is the y-value of the point. so you get one point (2, 4) (which means x1 = 2 and y1 = 4). Can you find the other point simiarly for x = 5 ?

Answer 6

So we have 2,4.

Then:
f(5) = 5^2 = 25 = 5,25.

so then:

25-2 23
---- = ----
5-4 1


@navk

Answer 7

DEFINITION: A function is a process by which every input is associated with exactly one output. When create a process (or series of steps) to do a certain task we are often creating a function. If we want to use it over and over again then to make our lives easier we give it a name. It helps us remember the name when it has something to do with the process that is being described.

The Average Rate of Change function describes the average rate at which one quanity is changing with respect to something else changing.

You are already familiar with some average rate of change calculations:

(a) Miles per gallon - calculated by dividing the number of miles by the number of gallons used

(b) Cost per killowatt - calculated by dividing the cost of the electricity by the number of killowatts used

(c) Miles per hour - calculated by dividing the numebr of miles traveled by the number of hours it takes to travel them.



In general, an average rate a change function is a process that calculates the the amount of change in one item divided by the corresponding amount of change in another. Using function notation, we can define the Average rate of Change of a function f from a to x as

Answer 8

You got it almost right! Just jumbled up the x and y values in the formula. It should be like this:\[m=\frac{25-4}{5-2}\]

Answer 9

A is the name of this average rate of change function
x - a represents the change in the input of the function f
f(x) - f(a) represents the change in the function f as the input changes from a to x

You might have noticed that the Average Rate of Change function looks a lot like the formula for the slope of a line. In fact, if you take any two distinct points on a curve, \[(x _{1},y _{1})\] and \[(x _{2},y _{2})\] the slope of the line connecting the points will be the average rate of change from \[x _{1},y _{1}\] and \[x _{2},y _{2}\]

Answer 10

OH! Okay:

21
---- = 7
3

Answer 11

@Muzzack Please don't copy paste

Answer 12

@mkbwashere thats right! Good to go :)

Answer 13

Example 1: Find the slope of the line going through the curve \[f(x)=\frac{ 1 }{ 3 } x ^{2}-4\] as x changes from 3 to 0.

Answer 14

Thank you so much for the help!!! @navk