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

its the derivative?

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

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

i have no idea..HELP

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

slope?

I can't see your graph

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

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

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

The slope is a constant.

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

a line

yes

yea

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

straight line...so slope is 0

so you see a horizontal line?

yea

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

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

ok

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

1?

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

Where will that line cross the x=2 line?

1/2

= 1

As much as x changes, the function changes.

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

1/2

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

How much did x change from 0 to 2?

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

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

2?

Yes

2/2 = 1 = the slope.

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

its negative?

Negative what specifically?

slope

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

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

-1

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

It's not a trick question =)

The value of f' is the slope of f.

-1

Right.

1 is which?

-1

-2?

Is any of this making sense?

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

-1

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

f(3)=3

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

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

2,4,5

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

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

2?

Yes

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

just like change in x was 4-2

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

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

i hope you know that this is an antiderivative problem

I do.

oh cool

What is the slope of h?

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

4,8,10

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

And what is the derivative of h?

(remember that the derivative is the slope)

15

Very nearly.

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

5x+15

16?

Yes

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

Yes!

so thats the answer

Yes

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

Ok

f(7)

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

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

ok thankls