Question

Let f(x) = x^2

Guess a formula for f'(x)

Answer 1

So I take it you haven't derived the formula for f'(x) yet?

Answer 2

no

Answer 3

Then the purpose of this question is to get you to think about what the answer might be. There is no absolutely correct answer.

But you can see three things. First, when x > 0, the slope to the graph of x^2 is positive. So f'(x) > 0 when x > 0.

When x = 0, the slope to the graph is flat, so f'(0) = 0.

When x < 0, the slope of the tangent to the graph is negative; it has negative slope.
So f'(x) < 0 for x < 0.

In summary
f'(x) > 0 when x > 0
f'(x) = 0 when x = 0
f'(x) < 0 when x < 0

Can you think of an elementary function that behaves like that?

Answer 4

Hint: How about some sort of straight line, y = mx + c?

Answer 5

so they just want you to do

f'(x) > 0 when x > 0
f'(x) = 0 when x = 0
f'(x) < 0 when x < 0
?

or do they want you to create a formula like all the others?

Answer 6

No, the question is asking you to guess what such a function is, explicitly. I'm suggesting to you that a straight line behaves like this.

Answer 7

i.e., f'(x) = mx + c. Figure out what values of m and c might make sense here.

Answer 8

you know that f'(0) = 0. That tells you something immediately.

Answer 9

ok, well I will try to figure this one out, and if i cant I will post another question.

Answer 10

A question for you. If f'(x) is a straight line, where does it intercept the y-axis?

Answer 11

x^2?

Answer 12

f'(0) = 0. So the line passes through the point (0,0). If y = mx + c is the equation of a straight line, what must c be equal to?

Answer 13

0?

Answer 14

Yes. Hence the straight line must be of the form y = mx. That is
f'(x) = mx.

Now we want to try and figure out m.

Answer 15

Now we know that
f'(x) > 0 if x > 0
and
f'(x) < 0 if x < 0.

For what values of m is that true? If f'(x) = mx, for what values of m are those two conditions satisfied?

Answer 16

0

Answer 17

No. if m = 0, then mx = 0 and hence f'(x) = 0

Answer 18

ahh, it would be x right
?

Answer 19

m = 1 would work, yes.

Answer 20

wouldn't it work either way?

Answer 21

Hence a reasonable guess for f'(x) would be

f'(x) = mx
= x

That is, a straight line, passing through the origin. It has the property that
f'(x) > 0 if x > 0
and
f'(x) < 0 if x < 0.

Answer 22

m=1 and m=x?

Answer 23

m is a constant.

Answer 24

oh, i forgot

Answer 25

m is the slope of the line.

Answer 26

it has been a while since i did real math, lol i am a bit rusty

Answer 27

ok. Hope that makes sense. Make sure you understand where these conditions came from:
f'(x) > 0 if x > 0
and
f'(x) < 0 if x < 0.

Answer 28

If you do, you'll be in good shape for your next class.

Answer 29

yes, looking at f(x)=x^2 and seeing what will and won't work..

Answer 30

in particular, looking at the graph of f(x) = x^2; looking at tangents to the graph; looking at the slopes of those tangents.

Answer 31

ok, now it makes sense. Thank you so much for your help.