Functions (continuity, derivability, etc): f (x) = x root (x (2-x)) on [0; 2]

[Ordimac]

Hello everyone !I'm French! I would like some helps in maths. Today, let's talk about maths. I train on Bac types (differentiability and continuity) to prepare for the tray (machine mode ). So, I want to appeal to the great masters of mathematics to explain to me an exercise that I do not understand!

Here is the statement:


Let the curve C, representative of the function f defined on [0; 2] by f (x) = x root (x (2-x)).


1) a. According to the graph, does the function f seem to continue on [0; 2]?
b. Justify your conjecture.


2) a. According to the graph, does the function f seem differentiable in 0? in 2 ?
b. Demonstrate your conjectures.


3) Draw the complete variation table of f.


4) a. Prove that the equation f (x) = 1 has two distinct solutions noted on [0; 2] and (with <)
b. To read approximate values of and.
c. Determine, using the calculator, a frame of amplitude 0.01. Is this consistent with the graphical reading of the previous question?

Here are my answers and research tracks:


1) a. The representative curve of f (C) on [0; 2] has a line which is "without lifting the pencil". So according to the graph, f is continuous on [0; 2].
b. I do not know how to justify mathematically speaking.


2) a. Here, big problem. I do not understand how one can know graphically if a function is derivable or not in a real.
b. There, I know that the rate and its limit must be calculated, but I do not know how to conclude.


3) For this question I calculated the derivative. I finally found that:
f '(x) = x (2-x) + (x-x 2) / x (2-x)
I then told myself that since x (2-x) is always positive, the sign of f '(x) depends on that of x-x2.
But I must have made a mistake since the values ​​that cancel x-x2 are 0 and 1, and that does not correspond to the graph ...


4) I confess that I have not done it yet, but I think to get out of it by using the intermediate value theorem and after, it's about calculator manipulation, so it should be fine.




Thank you very much in advance to those who will try to help me, I will be very grateful to them!
See you soon.

 

[mmm4444bot]

I'm French! I train on Bac types (differentiability and continuity) to prepare for the tray (machine mode ) …

Here is the statement:

… 3) Draw the complete variation table of f.

… 4) c. Determine, using the calculator, a frame of amplitude 0.01.

I'm not familiar with some of the terminology, here.

Can you explain in more detail the meaning of 'complete variation table'.

Does a "frame of amplitude 0.01" refer to the height of a graphing window?

I do not understand how one can know graphically if a function is derivable or not …

Is it possible to draw a tangent line with defined slope at each point along the curve?

Your textbook ought to have a section showing examples where derivatives are not defined; try looking up the word 'cusp' in the index, for one example. :cool:

 

[stapel]

I train on Bac types...

What is a "Bac type"?

...to prepare for the tray (machine mode).

What is "the tray"? What does "machine mode" mean in this context?

Let the curve C, representative of the function f defined on [0; 2] by f (x) = x root (x (2-x)).

Which radical is meant by "root"? What all is contained within that radical? Is the function something like the following?

. . . . .\(\displaystyle f(x)\, =\, x\, \sqrt{\strut x\, (2\, -\, x)\,}\)

Or something else?

1) a. According to the graph, does the function f seem to continue on [0; 2]?
b. Justify your conjecture.

By "to continue on", do you mean "to be continuous on", so that there are no points of discontinuity on the closed interval [0, 2]?

2) a. According to the graph, does the function f seem differentiable in 0? in 2 ?
b. Demonstrate your conjectures.

By "differentiable in", do you mean "differentiable at"?

3) Draw the complete variation table of f.

As mentioned in a previous reply, we don't know what a "complete variation table" is. My guess is that maybe this is a table of intervals of increase or decrease for the function (that is, a table listing the portions of [sub-intervals of] the interval [0, 2] in which the graph of the function f(x) is getting higher or else is getting lower, as you move along the x-axis from left to right). Is this correct?

4) a. Prove that the equation f (x) = 1 has two distinct solutions noted on [0; 2] and (with <)

What is meant by "and (with <)"?

b. To read approximate values of and.

What is meant by "approximate values of and"?

c. Determine, using the calculator, a frame of amplitude 0.01.

As mentioned previously, what is "a frame of amplitude 0.01"?

1) a. The representative curve...

What do you mean by "the reprsentative curve"?

...of f (C) on [0; 2] has a line which is "without lifting the pencil". So according to the graph, f is continuous on [0; 2].

Yes.

b. I do not know how to justify mathematically speaking.

What do you know about the relationship between a function which has a "jump" discontinuity at a point (that is, a function which jumps from one y-value on the left of a certain x-axis point to another y-value on the right of that same point), and the limits, from the left and right, of the function at that point?

What do you know about the relationship between a function with a "point" discontinuity (that is, a point where the function is not defined) and the value of the function at that point?

2) a. Here, big problem. I do not understand how one can know graphically if a function is derivable or not in a real.

What is meant by a function being "derivable or not" "in a real"?

b. There, I know that the rate and its limit must be calculated, but I do not know how to conclude.

What do you mean by "the rate" of the function? Which "limit" do you believe "must be calculated"? How far have you gotten in these processes? Where are you stuck (so that you "do not know how to conclude")?

3) For this question I calculated the derivative. I finally found that:
f '(x) = x (2-x) + (x-x 2) / x (2-x)

Please reply showing your work (rather than merely providing the final result), so we can see what's going on. For instance, how did you go from a function that had a radical, and get a derivative which did not contain a radical?

I then told myself that since x (2-x) is always positive, the sign of f '(x) depends on that of x-x2.

How did you conclude that 2x - x^2 is "always positive"? Or did you mean "always positive on the specified interval"?

Thank you!