Brushing up on functions and calculus

[DaveC426913]

It's now been 2 decades since I studied functions and calculaus in high school. I'm sure 75% of it will come back to me fairly quickly, but it seems to be written in a way that makes no sense to me.

This is what I'm studying:
http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/1.pdf

I'm in the exercises portion on p.7. My answers are bolded
1.1 Exercises
Starting from f(0) = 0 at constant velocity v, the distance function is f (t)= v.t .
When f (t) = 55t the velocity is v = v .
When f(t) = 55t + 1000 the velocity is still v
and the starting value is f (0)= 1000 .
In each case v is the slope of the graph of f.
When v is negative, the graphof f(t) goes downward.
In that case area in the t.-graph counts as absolute value (?).
Forward motion from f (0)= 0 to f (2)= 10 has v = 5 .
Then backward motion to f (4)= 0 has v = ? .

Did I get them right? I'm not sure about the second last one.

What is really confusing me is that this doesn't seem to be the way I learned it in H.S. The answers were very difficult for me to get. We always started with the formulae. I don't see any formula above - though I guess I've gotten the answers so far. But on that last one, how am I supposed to know what f(4) is if I don't even know what the formula is?

This is not grokking.

 

[DaveC426913]

OK:

Then backward motion to f (4)= 0 has v = -5

Got it.

It's still not grokking. I'm having to figure it out from first principles.

Question: is the idea here that the first thing I ought to do is DRAW the graph? I mean, that does make it easier to "see".

 

[DaveC426913]

Here is the question posed:

If T= 3 what is f(4)?

I don't understand what I'm being asked here. I know this is a fundamental principle; if I don't get this, I don't get functions.

OK, stream of consciousness:
OK, it doesn't say 3T, it says t=3.
OK, so T, 2T and 3T become 3, 6 and 9.
OK, so f(4) is when t=4.
Now, the slope from 3 < t <6 is 1/3.
So, at t=4, f(t) = 4/3?
Is that right?

 

[shmoe]

When f (t) = 55t the velocity is v = v .
When f(t) = 55t + 1000 the velocity is still v

I think they're after v=55 here.

In that case area in the t.-graph counts as absolute value (?).

It still counts as the distance traveled.

But on that last one, how am I supposed to know what f(4) is if I don't even know what the formula is?

They tell you what f(4) is!

Question: is the idea here that the first thing I ought to do is DRAW the graph?

If it helps, sure. Sketching some kind of graph is never wrong.

OK, stream of consciousness:
OK, it doesn't say 3T, it says t=3.
OK, so T, 2T and 3T become 3, 6 and 9.
OK, so f(4) is when t=4.
Now, the slope from 3 < t <6 is 1/3.

So far so good. You should be able to tell from the graph that f(4) can't be 4/3, the maximum of f(t) on this graph is 1. You have f(3)=0, for f(4) you move 1 unit to the right, how much do you move up?

 

[Office_Shredder]

When v is negative, the graphof f(t) goes downward.
In that case area in the t.-graph counts as absolute value (?)

I think displacement would be a better word than distance here, simply because negative distance is kind of a silly concept (and if the graph is going down, you just may get that).

 

[DaveC426913]

There's something I'm not getting here. I'm not getting the gestalt. I should see these things in my head, and be able to toss them around like blocks.

Let's look at just one example:
When f (t) = 55t

I know the left side is supposed to be spoken as "f of t", and I know f=function, and t = time. But what does the whole statement say and meanin verbose language?


It says "when...". So, today, f(t) could equal 55t and tomorrow f(t) could equal 105.7t? Is this the same function? Or a different one?

Shouldn't they be saying "in the function we're examining, f(t) IS 55t. Period. It can't be anything else because that would be a different function."

 

[DaveC426913]

So far so good. You should be able to tell from the graph that f(4) can't be 4/3, the maximum of f(t) on this graph is 1. You have f(3)=0, for f(4) you move 1 unit to the right, how much do you move up?

Right, I was thinking this was v(t).

OK, so
at ... f(3), d=0.
The slope is 1/3.
So at f(4), d = 1/3.

You know, I can get the answers, I'm just not getting why I'm getting the answers, if you know what I mean.

 

[junior_J]

think of funtions as rules which take inputs and give outputs ... so for example ... f(x) = 5x is a rule ... where x may be oh say the set of all real numbers(input) and f(x) or say f(x)=y is the set of all output which themselves are real numbers as well :)

 

[shmoe]

It says "when...". So, today, f(t) could equal 55t and tomorrow f(t) could equal 105.7t? Is this the same function? Or a different one?

Shouldn't they be saying "in the function we're examining, f(t) IS 55t. Period. It can't be anything else because that would be a different function."

It's somewhat implied that within a given problem or set of problems that when you define f(t) to be some specific function that it doesn't change unless they explicitly gave it a new meaning. I would never have thought it necessary to mention this though.

 

[DaveC426913]

so for example ... f(x) = 5x is a rule

I would think that would just be y=5x.

What does the f(x) say/mean? And why does it need to be said explicitly?

 

[DaveC426913]

It's somewhat implied that within a given problem or set of problems that when you define f(t) to be some specific function that it doesn't change unless they explicitly gave it a new meaning. I would never have thought it necessary to mention this though.

Sure but look at the first two lines:

Starting from f(0) = 0 at constant velocity v, the distance function is f (t)= v.t .
When f (t) = 55t the velocity is v = 55.
When f(t) = 55t + 1000 the velocity is still v

I don't see any specific function defined there. In fact, line three, if I understand correctly, is actually a different function.

 

[shmoe]

I don't see any specific function defined there. In fact, line three, if I understand correctly, is actually a different function.

Sure, they define what they mean by f(t) freshly on each line.

Line 1 was telling you the general form of a function satisfying some conditions. Line 2 gave you a specific function ("When f(t)=55t...") satisfying these general conditions. Line 3 gave you a new function, ("When f(t)=55t+1000...") that didn't satisfy the conditions of general version of line 1.

Add subscripts to each new instance of f if it makes you more comfortable, but by the end of the book your count will be rather large. Having the same name can be handy though, line 4 was able to refer to the functions from the past few examples collectively as "f".

 

[arildno]

As I see it the symbol f(t) stands for the function value associated with the argument "t". The right-hand side of the equality f(t)=5t (that is, 5t), is the rule by which you may compute the function value f(t) by aid of "t".

(It might be times where the function values are known or specified otherwise; then the equality becomes an equation for those t's that produce the specified function value)

 

[HallsofIvy]

There's something I'm not getting here. I'm not getting the gestalt. I should see these things in my head, and be able to toss them around like blocks.

Let's look at just one example:
When f (t) = 55t

I know the left side is supposed to be spoken as "f of t", and I know f=function, and t = time. But what does the whole statement say and meanin verbose language?


It says "when...". So, today, f(t) could equal 55t and tomorrow f(t) could equal 105.7t? Is this the same function? Or a different one?

Shouldn't they be saying "in the function we're examining, f(t) IS 55t. Period. It can't be anything else because that would be a different function."

Are you saying that if I use f to represent one function today, I can never use it to represent a different function? The terminology "when f(t)= " is exactly the same as "if f(t)= " or "Suppose f(t)= ".

 

[junior_J]

think of f(x) as a "machine" ... f(x) signifies the fact that this "machine" takes x as input plus the fact that f(x) itself is the out put of the "machine". Its just notation , so y = f(x) = 5x .

 

[arildno]

Weell, I consider "y" to be the element in some set in which the range of f is a subset, so whenever we restrict ourself to that subset any y is equal to some function value f(x)

 

[junior_J]

yes yes the images of each x's is y ... that's another good way of putting it i guess

 

[DaveC426913]

Are you saying that if I use f to represent one function today, I can never use it to represent a different function? The terminology "when f(t)= " is exactly the same as "if f(t)= " or "Suppose f(t)= ".

Yeah. That wasn't what I was after.

"When" "if" and "suppose" is overkill. They should just say "f(t)=this".
By saying when, if or suppose, it implies (to me) that within the bounds of this specific question, f(t) can change arbitrarily.


Or looking at it differently, by saying "when f(t)=", they are suggesting that f(t) somehow has a life of its own, that it is the same f(t) in this question as it was in the last question, even though they have different equations. Like this:
Q2: When f(t) = 5t, then blah blah
Q3 But when f(t) = 100t, then blah blah