Few quick questions

TimeToShine

Hi everyone, I've a Calculus exam in a few weeks and I've done most of the past papers, there are just a few strange questions I'm not sure of, and it would really help if someone could guide me in the right direction.

1-Sketch the graph of the function f(x) = {x} (with those specific brackets)

2-Definition of the local maximum of f

3-Definition of derivative of f at point x

4-Give an example of a function which is not continuous.

5-Give an example of a function which has a critical point which is not a local max or min of the function.

6-Sketch 2x^2-3x-2 (I normally use completion of squares and they all work out, but it doesn't apply to this one and we weren't taught any other method.

I know some of the questions seem stupid, but I have difficulty articulating the concepts. I can differentiate and integrate almost anything but questions like that just throw me off the ball altogether.

Thanks.

bnt

When you plot something like f(x)=x, it might help you to replace f(x) with "y", since f(x) (the value of the function at x) will be on the plot's y-axis anyway. So the function can also be written as y=x, which you must have seen before, right?

re #2, the key word is "local": some functions, like f(x)=x³-2x, have values between ±∞ (positive or negative infinity), but also have "local" maxima and/or minima. Have a look at what you get if you plug this in to Wolfram Alpha, here. You can see a local maximum at x=-0.9, and a local minimum at x=0.9. In calculus terms, you can find these by differentiating the function, then finding the roots of that new function, since local maxima and minima have a slope of zero.

re #3: The derivative of a function at a point x is the slope of the function at point x. In general, taking the derivative of a function gives you another function which is the slope of the initial function. (Which ties in to #2 above: local maxima and minima have a slope of zero.) This is about as basic as Calculus ever gets - surely you knew this?

#4: one that springs to mind is f(x) = tan(x). Plot it (e,g, in Wolfram Alpha) and you'll see what I mean.

#5: a "critical point" is when the curvature changes. If you plot f(x)=x³, have a look at what happens at x=0

#6: you're being asked to plot it, not to solve it! Solving it can help with the plotting, since it gives you the roots but, in this case, does the function have (real) roots? I don't remember "completion of squares", but did you never learn the quadratic formula at all?

Try plugging some of these things in to Wolfram Alpha - see what happens. But this is basic Calculus, stuff you find in any text book, really! (Though I was in the same position as you just a few years ago.) I don't know to explain it without giving examples (sorry, Mods).

sponsoredwalk

Most likely f(x) = {x} is representing a sequence of natural numbers,
or possibly all of the integers. This would be a discrete graph, i.e. dots not
a connected line. That is just the standard notation in my experience.

You know a function is written as y = f(x). Well a better way to write a
function is f : R → R'. What this says is that you're taking
any number from R, which is the real numbers (basically every
number), and mapping it to another number in R'. In the case of a
sequence instead of going from R → R' the function is
f : N → R', where N : {1,2,3,...}, this way you can't get
1.5 on the x-axis, and the graph must necessarily be a bunch of dots.
Notice y can be any number!
If that doesn't make sense don't worry about it, it's just another way to
think about this.

A local maximum is something you really should know, I advise some
youtube videos on the calculus concept called max-min, i.e. maxima &
minima to learn how to deal with these ideas.

The definition of the derivative is self explanatory.

A non-continuous function is also self-explanatory.

A function with a critical point not being an extrema (max-min) is
best summed up in the graph of f(x) = x³, study this curve in a
calculus context & it will make maxima, minima & other kinds of
critical points make sense.

Sketching x² - 3x - 2 is a request to use calculus techniques to draw it.

Seeing as a lot of this is obviously fresh, in addition to your own
online research I will give you the links to some videos online.
Each set of videos is different in it's own way & I truly do advise
watching each.

www.khanacademy.org
www.justmathtutoring.com
http://www.math.armstrong.edu/faculty/hollis/calculusvideos/

After your own research you should come back here & make sure you've
got these concepts down, you've got a few weeks so it's plenty of
time.

Fremen

bnt wrote: »

When you plot something like f(x)=x, it might help you to replace f(x) with "y", since f(x) (the value of the function at x) will be on the plot's y-axis anyway. So the function can also be written as y=x, which you must have seen before, right?

Hm, I'd be careful there bnt. I suspect that function may be the fractional part of x. The notation {x} isn't really all that standard, so I'd double check your lecture notes, OP.

{x} is often taken to mean x - floor(x), where floor(x) is the largest integer less than x. In other words, set whatever comes before the decomal point to 0.
{sqrt(2)} = .41421...
The graph will look like a sawtooth.

Two and three will be in any calculus textbook.

4-Give an example of a function which is not continuous.

f(x) = 0 if x is less than 0, 1 if x is greater than or equal to 0 is not continuous on the interval [-1,1]

5-Give an example of a function which has a critical point which is not a local max or min of the function.

f(x) = x^3

6-Sketch 2x^2-3x-2 (I normally use completion of squares and they all work out, but it doesn't apply to this one and we weren't taught any other method.

Find the roots via the quadratic formula. Find the minimum with calculus. Join them all up with a smooth curve.

bnt

Fremen wrote: »

Hm, I'd be careful there bnt. I suspect that function may be the fractional part of x. The notation {x} isn't really all that standard, so I'd double check your lecture notes, OP.

I was trying to answer the OP's question as simply as possible; based on the tone of the questions, I suspect that talk of fractional parts and sequences might be a bit much. Same with plotting: the question was about how to plot a function, so "plot the function" is not a helpful answer!

MathsManiac

TimeToShine wrote: »

6-Sketch 2x^2-3x-2 (I normally use completion of squares and they all work out, but it doesn't apply to this one and we weren't taught any other method.

Although not especially convenient in this case, completing the square is not a bad way of graphng a quadratic function, since it gives you the roots (if any) and the turning point, by inspection.

It also makes it clear how every quadratic graph can be considered to be a transformation of y=x^2.

It does work in this case, as in all cases.

You should get f(x) = 2[(x - 3/4)^2 -25/16], which tells you that the min. turning point is at (3/4, -25/8) and the roots are (3/4) +/- (5/4).

TimeToShine

Thanks everyone, also for that f(x) = {x} it says the fractional part of x...what does that mean??

MathsManiac

TimeToShine wrote: »

Thanks everyone, also for that f(x) = {x} it says the fractional part of x...what does that mean??

Fremen explained it in post#4. The fractional part of 2.56 is 0.56, etc. You throw out the whole number and keep the rest. (What you do with negative numbers depends on the precise definition in your notes.)

TimeToShine

but this doesn't give a specific number, am i just supposed to pick any value for x and graph the fractional part?

Michael Collins

TimeToShine wrote: »

but this doesn't give a specific number, am i just supposed to pick any value for x and graph the fractional part?

Exactly - this is what'd you would do in general for any graph. Try it between 0 and +3 and see if you notice a pattern.