http://edugen.wiley.com/edugen/courses/crs4196/art/qb/qu/ch0/EAT_1227298660014_0_33367326491234830.gif
Thats the graph of f' and assume f is continuous and f(0)=0
Find f(3)

- anonymous

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- anonymous

Well, what does the value of f' tell you about f?

- anonymous

its the derivative?

- anonymous

Sweethert, keep it in chat please. this is for mmonish's question.

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## More answers

- anonymous

Yes, but what does the derivative tell you? What does it mean to be the derivative?

- anonymous

i have no idea..HELP

- anonymous

What is the derivative of the function f(x) = 5x + 3?

- anonymous

5

- anonymous

slope?

- myininaya

I can't see your graph

- anonymous

Right. The derivative is the slope of the curve at any given x value.

- anonymous

So, if the derivative is a constant, what does that mean about the curve of the original function?

- anonymous

@myinninaya: i don't know y
@polpak: the slope is ?

- anonymous

The slope is a constant.

- anonymous

What does the graph of f(x) = 5x+3 look like?

- anonymous

a line

- myininaya

yes

- anonymous

Right, and you can see that if you take the derivative of a line you will get a constant value for the slope.
\(y = mx + b \implies y' = m\)

- anonymous

yea

- anonymous

So look at the graph of f'. What is f' from x=0 to x=2?

- anonymous

straight line...so slope is 0

- myininaya

so you see a horizontal line?

- anonymous

yea

- anonymous

No, the value of f' over that interval is constantly 1. f' = 1 on the interval from 0 to 2.

- anonymous

Which means that f must have a constant slope of 1 over that interval.

- anonymous

ok

- anonymous

So if f(0) = 0, what will f(2) be?

- anonymous

1?

- anonymous

No. Imagine a line with a slope of 1, that goes through the origin.

- anonymous

Where will that line cross the x=2 line?

- anonymous

1/2

- anonymous

Each time x increases by 1, the function increases by 1. That's what it means to have a slope of 1. \(\frac{\text{change in f(x)}}{\text{change in x}}\)

- anonymous

= 1

- anonymous

As much as x changes, the function changes.

- anonymous

So if f(0) = 0, f(2) = ?

- anonymous

1/2

- anonymous

\(\frac{\Delta f}{\Delta x} = 1.\)
\(\Delta x = 2 \implies \Delta f =?\)

- anonymous

How much did x change from 0 to 2?

- anonymous

2

- anonymous

The function must also have changed by the same amount if the ratio equals 1

- anonymous

So how much does the function change over that same interval?

- anonymous

2?

- anonymous

Yes

- anonymous

2/2 = 1 = the slope.

- anonymous

So now we know f(2). What do we know about the slope over the interval from x=2 to x=3?

- anonymous

its negative?

- anonymous

Negative what specifically?

- anonymous

slope

- anonymous

We have a graph of f'. The value of f' is the slope of f.

- anonymous

What is the value of f' from x=2 to x=3?

- anonymous

-1

- anonymous

What is the slope of f from x=2 to x = 3?

- anonymous

It's not a trick question =)

- anonymous

The value of f' is the slope of f.

- anonymous

-1

- anonymous

Right.

- anonymous

So if the slope of f is -1 from 2 to 3. And the
\(\frac{\text{Change of f}}{\text{Change of x}}\) = Slope = -1,
What is the change in x?
What is the change in f?

- anonymous

1

- anonymous

1 is which?

- anonymous

x

- anonymous

Right. So
\[\frac{\text{Change in f}}{\text{Change in x}} = -1\]
\[\frac{\text{Change in f}}{1} = -1\]
Change in f =?

- anonymous

-1

- anonymous

Right. So f(2) = 2, and from 2 to 3, the slope of f is -1, then the value of f will change by -1, what is f(3)?

- anonymous

-2?

- anonymous

Is any of this making sense?

- anonymous

No f changes by -1. It goes down 1.

- anonymous

f goes down 1 every time x goes up 1. That's what it means to have a -1 slope. Just like having a slope of 1 means that f goes up 1 each time x goes up 1.

- anonymous

-1

- anonymous

So if f(2) = 2, and x goes up 1, what will f do?

- anonymous

f(3)=3

- anonymous

Your answers are all over the place. I'm not sure what is confusing you about what I'm asking, but clearly something is out of sync. Do you understand what I'm saying about slope?

- anonymous

If I have a line g(x) = x. What is the slope?

- anonymous

1

- anonymous

Ok, and what is g(2) and g(4) and g(5)?

- anonymous

2,4,5

- anonymous

Ok. And on the interval from x=2 to x=4, how much does x increase?

- anonymous

2

- anonymous

And g(x) over that interval increases how much?

- anonymous

2?

- anonymous

Yes

- anonymous

Change in g = g(4)-g(2)

- anonymous

just like change in x was 4-2

- anonymous

So as x increases by 1 , g increases by 1. That is what it means to have a slope of 1.

- anonymous

If we have h(x) = 2x, What is the slope?

- anonymous

i hope you know that this is an antiderivative problem

- anonymous

I do.

- anonymous

But there are some small holes in your understanding that make solving these very easy. If I fill the holes you can answer all these in less than a minute.

- anonymous

oh cool

- anonymous

What is the slope of h?

- anonymous

2

- anonymous

So what is h(2), h(4), and h(5)?

- anonymous

4,8,10

- anonymous

So each time x increases by 1, h increases by how much?

- anonymous

2

- anonymous

And what is the derivative of h?

- anonymous

2

- anonymous

So if I tell you that j(0) = 1, and the derivative of j (also called j') = 5, Can you tell me what j(3) =?

- anonymous

(remember that the derivative is the slope)

- anonymous

15

- anonymous

Very nearly.

- anonymous

That would be the correct answer if j(0) = 0

- anonymous

5x+15

- anonymous

But because j(0) = 1, we need to start at one, and add 15 because the change in x is 3 and the slope is 5.

- anonymous

16?

- anonymous

Yes

- anonymous

Now. If I tell you that f(0) = 0 and the derivative ( also called slope, and f') is 1 from 0 to 2, what is f(2) again?

- anonymous

2

- anonymous

And now if we know that f(2) = 2, and f' = -1 from 2 to 3, what is f(3)?

- anonymous

1

- anonymous

Yes!

- anonymous

so thats the answer

- anonymous

Yes

- anonymous

ok dont help me solve this but let me know if i did it correctly

- anonymous

Ok

- anonymous

f(7)

- anonymous

oh pellet my boss is here...ill ttyl..thanks a ton

- anonymous

lol, k. You can repost the answer here and I'll check back

- anonymous

ok thankls

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