Question

Help with functions?

Answer 1

What functions?

Answer 2

The price of the cups, p(x), is a function of the number of x cups ordered. Using the table, determine the average rate of change for the first 1,000 cups ordered and then for all 3,000 cups ordered. What does this tell you about the function?

Answer 3

@whpalmer4 you busy?

Answer 4

Do you understand the concept of the average rate of change?

Answer 5

I suppose so. Its similar to the one you helped me with yesterday, with the cups to grams, right?

Answer 6

No, not really.

Answer 7

I believe I understand, that's just a bad example..

Answer 8

So how do you go about finding the average rate of change for the first 1,000 cups ordered?

Answer 9

Find the rate of change for the x and y values

Answer 10

I'm not sure where to start.. I thought I'd divide 1,000 by 44, but..

Answer 11

Do u want help?

Answer 12

Please.

Answer 13

@Drsuz98 are you going to?

Answer 14

Average rate of change of a function between points x = a and x = b is simply

\[\frac{f(b)-f(a)}{b-a}\]

Answer 15

In other words, it is the slope of the line that connects the two points.

Answer 16

So,\[\frac{f(1000) - (f44)}{1000-44}\]?

Answer 17

That can't be it.. I'd get 1 every time..

Answer 18

No. For the first 1000 cups ordered, you are looking for the average rate of change of f(x) from x=0 to x = 1000

Answer 19

cups ordered = x
price = y = f(x)

Answer 20

So, What are my f(b), and f(a)

Answer 21

Neither my mom, dad, or I are understanding what to input for a and b..

Answer 22

f(b) = f(x) where x=b
f(a) = f(x) where x=a

you're trying to find the average rate of change between two points. what are the two points that define the range "the first 1000"?

Answer 23

x=0 and x=1000, no?

Answer 24

No. 0, 500, 1000. theres a table up there.

Answer 25

I suppose if I' looking for 2 points, 0 ad 1000 are correct.

Answer 26

I swear I'm not retarded, and I'm paying attention.

Answer 27

\[\frac{ f(0)-f(1000) }{ 0-1000 }?\]

Answer 28

Yes, except that is what you would do to find the average rate of change going in the other direction :-)

\[\frac{f(1000)-f(0)}{1000-0}\]is what you want

Answer 29

to find the avg. rate of change of \(f(x)\) as \(x\) goes from \(a\) to \(b\).

Answer 30

OMG I am so srry I got busy with something else @RockerSk8er

Answer 31

\[\frac{f(b)-f(a)}{b-a}\]it's just the formula for the slope of a line, trust me!

Answer 32

though actually what you did is equivalent to just multiplying the numerator and denominator both by -1, so you would get the same answer. It's just typically written the way I wrote it.

Answer 33

@Drsuz98 Its all good. :) @whpalmer4 okay. so, my answer is 1?

Answer 34

remember, the slope of a line between two points \((x_1, y_1), (x_2,y_2)\) is \[m = \frac{y_2-y_1}{x_2-x_1}\]

well, we have here \(y = f(x)\) so you could write that as

\[m = \frac{f(x_2) - f(x_1)}{x_2-x_1}\]

No, you answer is not 1...well, maybe it is, but MY answer is not 1 :-)

Show me your work.

Answer 35

lol;)

Answer 36

Here's a picture of the entire table, plotted as points.

Answer 37

Gosh. Lol\[\frac{ 1000 -0}{ 1000-0 } = \frac{1000}{1000}=1\]

Answer 38

Sigh. You're not doing as I described. Take a piece of paper. Write a pair of columns, labeled respectively x and y=f(x). Under the x column, write down the numbers from the "cups" column of the table. Under the y=f(x) column, write down the numbers from the "price" column of the table.

Consult that table (and the column headers) as you do the problem again...

Answer 39

hint: 0 cups does not have a price of 0
1000 cups does not have a price of 1000

Answer 40

let me spell it out for you:

\[\text{av. rate of change for first 1000 } = \frac{\text{(price of 1000 cups)}-\text{(price of 0 cups)}}{\text{1000 cups} - \text{0 cups}}\]

Answer 41

y = f(x) = price
x = number of cups

Answer 42

the number of cups is the independent variable, and the price is the dependent variable

Answer 43

OHHH! ohmygod.. I'm stupid. ;-;

Answer 44

I was thinking more along the lines of standing a little too close to the tree to see the forest around you :-)

Answer 45

\[\frac{ 44.00 - 7.00 }{ 1000-0 }=\frac{37}{1000}=0.037\]

Answer 46

yes, very good. Now how about the rate of change for the first 3000 (or whatever it was the problem asked for)

Answer 47

@whpalmer4 \[\frac{188.00 - 44.00}{3000-1000}=\frac{9}{125}=0.072\]

Answer 48

does it not ask for the rate for "all 3000 cups ordered"? what you did (and unfortunately, did incorrectly) finds the rate for cups 1000-3000, not 0-3000 (although in this particular problem, the two are identical)

Answer 49

go look at your table again...try not to bang your head too hard on the desk when you see the error :-)

Answer 50

Jesus Christ. \[\frac{ 188.00-7.00 }{ 3000-0 }=\frac{181}{3000}=0.0603?\]

Answer 51

are you looking at the same table I'm looking at? go look at the entry for 3000 cups again.

Answer 52

you are getting closer, however :-)

Answer 53

I'm typing it wrong... \[\frac{118.00-7.00}{3000-0}=\frac{37}{3000}=0.037\]

Answer 54

there you go!

Answer 55

I'm sorry that I'm so difficult. My dyslexia is such a pain...

Answer 56

eh, as you know, I'm fairly patient with those making an honest effort, such as you are...

are you ready to go on with the discussion? yes, you've answered all the parts of the question above, but there's really more here to understand

Answer 57

Yes.

Answer 58

so as I mentioned earlier, the rate of change is the slope of the line tangent to the function. Let's say we have a graph of the length of your hair vs. time after a haircut. The y-intercept (the value of y where x = 0) is going to be your hair length as you walk out of the haircut. If we assume that your hair grows at the same rate all the time, then your hair length on a subsequent day will be just the length it was after the haircut, plus the rate it grows per day * the number of days since the haircut, right?

Answer 59

Yes.

Answer 60

that rate turns out to vary based on your ethnicity, but presumably that is fixed at conception, and doesn't change :-)

what we had with the cup problem was something where the rate of change was fixed: an extra 500 cups always added the same amount to the bill — can you figure out what that amount is?

Answer 61

I think you're asking me what each cup would cost, and how many 3500 cups would cost?
each cup cost $0.037, so 3500*0.037 = $129.50

Answer 62

well, I was asking a related question, which is how much does an additional 500 cups add to the price.

you could answer by saying 500*$0.037 = $18.50 but I was hoping that you would observe it directly from the table. Going from 500 to 1000 cups takes the price from $25.50 to $44, going from 1500 to 2000 takes it from $62.50 to $81 (again, a difference of $18.50), etc.

Answer 63

Because the rate of change is just the slope of the line, we can move from one spot to another by multiplying the rate of change (slope) by the difference in the x values. So, if we know the price at x=500, we can find the price at x=750 by multiplying (750-500)*($18.50/500) and adding that to the price at x = 500. Can you do that and tell me what the price for 750 cups would be? Check your answer against the graph that I posted earlier...

Answer 64

Sorry, mom made me stop and we went out of town...