Analyzing f(x) Graph: Identifying Points and Derivatives

[salma17]

Im working on my study guide..and I have no idea how to do these types of problems.Can anyone please just give me a hint on how I can approach these?
I attached the graph.

a few of the questions I am having trouble with are:

At which point is f(x) > 0?
a) A b) B c) C d) D e) None of these

At which point is f'(x)> 0?
a) A b) B c) C d) D e) None of these

At which point is f'(x) =0 ?
a) A b) B c) C d) D e) None of these

Id appreciate any feedback I get. thanks

 

[HallsofIvy]

For (a), you look at the graph! Or are you saying that you do not know what "f(x)> 0" means?

For (b) and (c), you have to know what f' means. It is the "slope of the tangent line". As long as the graph is increasing, f' is positive. As long as the graph is decreasing, f' is negative.

 

[salma17]

Yea, I don't know what it means. I learned all this material last year, and it was really easy to me back then, and now I have to know it for my next math class, and I forgot everything :/ So for a), its basically asking which point in the Y-axis is greater then 0? Which are letters A and H, right?

 

[salma17]

can anyone please help me with this??

 

[skiller]

For (a) ... For (b) and (c)

What?

Yea, I don't know what it means. I learned all this material last year, and it was really easy to me back then, and now I have to know it for my next math class, and I forgot everything :/ So for a), its basically asking which point in the Y-axis is greater then 0? Which are letters A and H, right?

Are you sure you've quoted the question correctly? Your multiple choice answers only include points A, B, C and D, but your attached graph also has points E, F, G and H.

 

[salma17]

Yes, that's how it is on my study guide. So I am thinking if any of the answer ends up to be either E,F,G or H, then it will be "none of these".

 

[skiller]

Yes, that's how it is on my study guide. So I am thinking if any of the answer ends up to be either E,F,G or H, then it will be "none of these".

If the options are A, B, C, D or None of these, and you think the possible correct answers are A and H, then why would you say "None of these"?

 

[salma17]

Im not saying that. above I said the answers for part a) should be A and H. in my previous post,im saying if an answer(in general) is either E,F,G,or H, then it will be none of these.
Anyways, am I right about part A?

 

[skiller]

Anyways, am I right about part A?

You haven't labelled any question as "Part A" etc, but yes you are correct for the first one.

For the others, do you understand what f'(x) means, as HallsofIvy asked? (He did actually explain what it meant, too.)

 

[salma17]

ok so for the question At which point is f'(x)> 0? , I look at the graph to see which point above 0 the graph is increasing at? which is at letter H, correct?

 

[haruspex]

ok so for the question At which point is f'(x)> 0? , I look at the graph to see which point above 0 the graph is increasing at?

Why would you only consider points above 0?

 

[salma17]

because the first part its asking what y-coordinate is greater than 0? or am i totally wrong?

 

[haruspex]

because the first part its asking what y-coordinate is greater than 0? or am i totally wrong?

That was the first part, this is the second part.

 

[salma17]

because it says >0 so greater then 0? I honestly have no idea how to even begin this one

 

[haruspex]

because it says >0 so greater then 0? I honestly have no idea how to even begin this one

The first part asked for f() > 0. That is, the function > 0.
The second part asks for f'() > 0. That is, the gradient greater than zero. That is independent of whether the function, f, is greater than zero. So it can be anywhere on the graph as long as it slopes from lower left to upper right.

 

[salma17]

ohh so I have to look at where the slope is increasing..so its from point F to H

 

[Mark44]

ohh so I have to look at where the slope is increasing

No, where the slope is positive. This is exactly the same as where f'(x) > 0.

This is not the same as "where the slope is increasing." None of the questions asked where the slope was increasing.

..so its from point F to H

 

[salma17]

I drew the tangents on each point, and I got positive slopes at points C,D,E and H. If this wrong, then I am drawing my tangents wrong and I'll need to youtube how to drank tangents.

 

[haruspex]

ohh so I have to look at where the slope is increasing..

Distinguish clearly in your mind the following:
- the places where f() > 0; that is the portion above the x axis.
- the places where f'() > 0; f'(x) is the slope at x. Where the slope > 0, f is increasing as x increases. The graph slopes from lower left to upper right. That can be a positive f getting larger, or a negative f getting less negative (or becoming positive).
- the places where f''() > 0; these sections are where the slope is increasing. Viewed from below the curve, the curve is convex. That can be a positive slope getting steeper, or a negative slope getting shallower (or becoming positive).
All 8 combinations (f being +/-, f' being +/-, f'' being +/-) can occur.

 

[Mark44]

I drew the tangents on each point, and I got positive slopes at points C,D,E and H. If this wrong, then I am drawing my tangents wrong and I'll need to youtube how to drank tangents.

The slopes of the tangent lines at D and H are positive, yes.

Pretty clearly, the slopes of the tangent lines at C and E are NOT positive. Why do you think they are?

 

[salma17]

When you draw tangent lines, don't they have to interesect at the origin of the graph? most likely no because that's what i did and why i got it wrong.

 

[Mark44]

When you draw tangent lines, don't they have to interesect at the origin of the graph?

No, not at all.

most likely no because that's what i did and why i got it wrong.

Think of the curve as a road that you're driving along. Wherever you are on the road, your eyes are looking straight ahead (or they should be). That direction is the same as the tangent line's direction.

 

[salma17]

Alright, so would the answer be point G, which is E)none of these for f ' (x) >0 since at G the slope is at X

 

[salma17]

^^ I meant H. and i think point G is f '(x)=0

 

[Mark44]

Alright, so would the answer be point G, which is E)none of these for f ' (x) >0 since at G the slope is at X

the slope is at X? What does that mean?

^^ I meant H. and i think point G is f '(x)=0

No. At G, f'(x) ≠ 0. Are you just guessing?

 

[salma17]

I read above that " f'(x) is the slope at x." by at X I mean its right on the x-axis

 

[Mark44]

I read above that " f'(x) is the slope at x."

Of course it is. f'(x) is the slope of the function at the point (x, f(x)). That doesn't tell me anything, though.

by at X I mean its right on the x-axis

Who is?
Please don't use "it".

 

[haruspex]

I read above that " f'(x) is the slope at x." by at X I mean its right on the x-axis

What do you mean by a slope "right on the x-axis"?
This can be a bit confusing because x is sometimes used to refer to the variable in general and sometimes to particular values. So let me clarify by using x₀ for a particular value.
f'(x₀) is the slope of f at x₀, i.e. the value of f'(x) evaluated at the point x = x₀. Pick a point x₀ on the x axis; draw a vertical line through there; see where that line intersects f(x); see what the slope of f is at that point. That is f'(x₀).
Example: f(x) = x². f'(x) = 2x. With x₀ = -1, f(-1) = 1, f'(-1) = -2.

 

[salma17]

Okkk I think that makes better sense to me. I did them again and got D for f 'x>0 and C for f 'x=0

 

[Mark44]

Okkk I think that makes better sense to me. I did them again and got D for f 'x>0 and C for f 'x=0

Good!