Integral Graph

[car0le_la]

Given the graph of f(x) on [-2,5] below and Find where F(x) is increasing:
decreasing:
Min:
Max:
CCU:
CCD:
P.O.I:

As you may know, there are these things known about integrals
1) F'(x)=d/dx = f(x)
2)F''(x)=f'(x)
3) increasing occurs when F'(x) is positive
4) decreasing occurs when F'(x) is negative
5) CCU when F''(x) is positive
6) CCD when F''(x) is negative
But my only problem is that I don't know how exactly I am supposed to start because the results could vary based on the integral function. If the integral f(x)dx has a bottom value of -2 or 0 could change the results. I don't know if I explained well but that's why I'm stuck.

 

[car0le_la]

When I say bottom and top value I mean this:
b


a
-------------------------------------------------------------
Where I know the b value might be x (whose highest value would be 5 due to the closed interval), but I don't know if a would 0 or -2.

 

[pka]

What don't you post answers and allow us to comment.
Here is one comment: on \([-2,0]\) the integral is negative area is is decreasing. WHY?

 

[Dr.Peterson]

Given the graph of f(x) on [-2,5] below and View attachment 18113Find where F(x) is increasing:
...
But my only problem is that I don't know how exactly I am supposed to start because the results could vary based on the integral function. If the integral f(x)dx has a bottom value of -2 or 0 could change the results. I don't know if I explained well but that's why I'm stuck.

When I say bottom and top value I mean this:
b
View attachment 18117

a
-------------------------------------------------------------
Where I know the b value might be x (whose highest value would be 5 due to the closed interval), but I don't know if a would 0 or -2.

It appears that you may be confusing definite and indefinite integrals.

What you are told is that F is an antiderivative (indefinite integral) of f. It is not a definite integral with limits a and b that are not known; that would be a number, not a function.

Antiderivatives can differ, but only by a constant. So two different F's will just be shifted up or down from one another, which will not affect the answers to any of the questions.

I would approach this by reversing the question: We are asked some things about a function F, knowing only the graph of its derivative, f.

What must be true of f (that is, F') if F is increasing on an interval? Where does that happen?

And so on.

 

[Steven G]

When f(x) is negative what does that tell you about the area under the curve? What about when f(x)>0?

If you have a negative area and then you add more negative is the area increasing, decreasing or staying the same. If the area is negative and then you start adding positive area is the area increasing, decreasing or staying the same? What if you add some more positive area? Then what happens if you start adding negative area??

 

[car0le_la]

When f(x) is negative what does that tell you about the area under the curve? What about when f(x)>0?

If you have a negative area and then you add more negative is the area increasing, decreasing or staying the same. If the area is negative and then you start adding positive area is the area increasing, decreasing or staying the same? What if you add some more positive area? Then what happens if you start adding negative area??

So when f(x) is negative we know that the area under the curve is negative and when f(x) is positive the area under curve is positive. Subsequently if f(x) continues on being negative, F(x) is decreasing since it is more negative area, while if f(x) continues on being positive, F(x) will begin to increase right? So we can obtain a sense of increase and decreasing and from this as well I can know when something has a min or max (becuase it is changing from either increasing to decreasing or increasing to decreasing). But how will I know from this the p.o.i? Also I need to not just find the x value of where the critical point occurs in F(x) but also the y value. How would this be possible?

 

[Steven G]

Yes, you have it down perfectly. That was just to see where F(x) is increasing/decreasing.

What do you think you need to know to do the other parts of the problem. What information do you need to find the Min, Max, POI, etc? How can you get this information. What is the relationship between f(x) and F(x)?

 

[car0le_la]

In order to obtain min x value, I need to look at the locations where the f(x) area under the curve changes from negative to positive.
In order to obtain the max x value, I need to look at the locations where the f(x) area under the curve changes from positive to negative.
But how would I be able to attain the y-value of these x-values?
Am I able to determine if F(x) is concave up based on if f(x) is increasing and vice versa for concave down?
By this same logic can I assume that p.o.i x value is determined from where f(x) has a critical value? But from there how would I determine the y value?
I know that the relationship between F(x) and f(x) is that the integral of f(x) is the function of F(x), but would I be able to obtain a y value from an indefinite integral?

 

[Steven G]

Are you given the graph of f(x) or f'(x) or something else?

 

[pka]

Given the graph of f(x) on [-2,5] below and View attachment 18113Find where F(x) is increasing:
decreasing:
Min:
Max:
CCU:
CCD:
P.O.I:
View attachment 18114
As you may know, there are these things known about integrals

Lets suppose that \(F(x) = \int_{ - 2}^x {f(t)dt} \). On the interval \[-2.0] the function \(F\) is decreasing to a local minimum.
The function \(F\) is increasing on \([0,3]\). At \(x=3\) the function has a local maximum. On \([3, 5]\) the function is decreasing to another local minimum at \(x=5\)

 

[car0le_la]

Are you given the graph of f(x) or f'(x) or something else?

This is the graph of f(x)

Would the integral when trying to solve for y would it be
\(F(x) = \int_{ - 2}^x {f(x)dt} \) or \(F(x) = \int_{ 0}^x {f(x)dt} \)

 

[car0le_la]

This is what I have so far:
inc: (0,3)
dec: ( -2,9) u (3,5)
min: x=0
max: x=3
ccu: (-1,2) u (4,5)
ccd: (-2,-1) u (2,4)
p.o.i: x=-1,2,5
How would I obtain the y-value for the min, max and p.o.i based on the graph f(x) knowing that F(X) equals the integral of f(x).

 

[pka]

I have no idea what POI could mean. Read reply #10.

 

[Dr.Peterson]

This is what I have so far:
inc: (0,3)
dec: ( -2,9) u (3,5)
min: x=0
max: x=3
ccu: (-1,2) u (4,5)
ccd: (-2,-1) u (2,4)
p.o.i: x=-1,2,5
How would I obtain the y-value for the min, max and p.o.i based on the graph f(x) knowing that F(X) equals the integral of f(x).

P.o.i. means Point Of Inflection, right? And ccu means ConCave Up? It's worth being aware that not everyone uses the same abbreviations ...

But I think you are aware that you can't know any y-values (values of F(x)) unless they give you, say, one value and the areas of various regions in the graph provided. You don't indicate that you were given those things.

As I read the question, though, it doesn't ask for any; it just asks for "where F(x) ...", which means the values of x.

 

[car0le_la]

P.o.i. means Point Of Inflection, right? And ccu means ConCave Up? It's worth being aware that not everyone uses the same abbreviations ...

But I think you are aware that you can't know any y-values (values of F(x)) unless they give you, say, one value and the areas of various regions in the graph provided. You don't indicate that you were given those things.

As I read the question, though, it doesn't ask for any; it just asks for "where F(x) ...", which means the values of x.

This is all I was given:

It seems like I'll just leave the answer with just the x-values, (I tried to get the y-values since the professor usually wants them as part of the answer)

 

[car0le_la]

Thanks for all the help!!

 

[Steven G]

This is the graph of f(x)
View attachment 18166
Would the integral when trying to solve for y would it be
\(F(x) = \int_{ - 2}^x {f(x)dt} \) or \(F(x) = \int_{ 0}^x {f(x)dt} \)

The graph goes from x=-2 to x=5. So why would you not include x=-2 to x=2.
Please answer my question as that will easily solve this problem. Again, what is the relationship between f(x) and F(x)?

 

[Dr.Peterson]

The graph goes from x=-2 to x=5. So why would you not include x=-2 to x=2.
Please answer my question as that will easily solve this problem. Again, what is the relationship between f(x) and F(x)?

Are you disagreeing with what I said in post #4? I could always be wrong, but as I read the problem, F is not defined as a definite integral at all, but an indefinite integral, or antiderivative.

To put it almost equivalently, the answer to the question, "Would the integral when trying to solve for y be [MATH]F(x) = \int_{ - 2}^x {f(x)dt}[/MATH] or [MATH]F(x) = \int_{ 0}^x {f(x)dt}[/MATH] ?" is, it could be either, or anything else! Both are particular indefinite integrals, and both are defined for any x in the domain of f.