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Differential Calculus - Limits and continuity

Hey guys,

I've got a set of problems that I am unsure of if I have done the right way. The first one (exercise 35) below is a supposedly pretty simple task. For a) I have written:

P(x)=

P(0 < x < 501)= 0.80x

P(501 < x < 1001) = 0.72x

P( x > 1000) = 0.65x

Is there a way I can write this as one equation (and perhaps including limits somehow?)

Also, for a) and b), how is this different? Since there are no other elements (like fixed costs etc)

What's c) really asking for?

Attachment 24740

Attachment 24741

For exercise 44, I think I have found the correct way to read the graph, but I am not certain.

a) 3

b) -1

c) 3

d) -1

e) 1

f) 1

g) -2

h) -2

i) infinity+

j) infinity-

k) Does not exist

l) Does not exist

m) 5

n) 5

p) 5

q) 2

r) 4

s) 2

Attachment 24742

Attachment 24743

I also have a problem to grasp what the difference between

m) lim f(x) x -> 6-

n) lim f(x) x -> 6+

p) lim f(x) x -> -6

q) f(6)

If someone could help out here it would be awesome! I am not really looking for someone to do my homework, because we are going to have several tests and exams where this will be a respectable part. I hope also the fact that I am spending saturday night of my first weekend at college doing math shows some kind of dedication (I hope!).

Thanks in advance

Re: Differential Calculus - Limits and continuity

a) and b) are not the same. a) is asking you the cost per CD, while b) is asking you for the total cost of purchasing x cds.

For each of these, you will need to use a hybrid function.

For c) it's asking you to evaluate a left hand limit and a right hand limit.

Re: Differential Calculus - Limits and continuity

Thanks for the answer,

How do you write a hybrid function? How can I get all the three prices into one function?

The math book we are using is a $10 ebook which I find extremely hard to understand, at the same time is english not my native language. It does not have an example with a cost function

Re: Differential Calculus - Limits and continuity

Just write down a function for each of the three pieces, then put them together, something like this...

$\displaystyle \displaystyle \begin{align*} y = \begin{cases} f(x) \textrm{ if } a < x \leq b \\ g(x) \textrm{ if } b < x \leq c \\ h(x) \textrm{ if } c < x \leq d \end{cases} \end{align*}$

Re: Differential Calculus - Limits and continuity

So basically the one I posted in post 1?

P(x)=

f(x) if 0 < x < 501 = 0.80x

g(x) if 501 < x < 1001 = 0.72x

h(x) if x > 1000 = 0.65x

Correct?

And still, since the exercise does not say anything about other costs, so I can't see how a) and b) are different.. Cost per CD = 0.80 if it's under 500, and total cost will be 0.80*x etc.

Also could need a little bit help on exercise 44

Re: Differential Calculus - Limits and continuity

Quote:

Originally Posted by

**leezgo** So basically the one I posted in post 1?

P(x)=

f(x) if 0 < x < 501 = 0.80x

g(x) if 501 < x < 1001 = 0.72x

h(x) if x > 1000 = 0.65x

Correct?

And still, since the exercise does not say anything about other costs, so I can't see how a) and b) are different.. Cost per CD = 0.80 if it's under 500, and total cost will be 0.80*x etc.

Also could need a little bit help on exercise 44

You just answered your own question. The cost per CD = 0.80 if it's under 500. But the TOTAL cost after purchasing x CDs would be 0.80x if it's under 500. Now finish.

Re: Differential Calculus - Limits and continuity

Thanks,

a) P(x) =

if blablabla = 0.80

if blablabla = 0.72

if blablabla = 065

b)

C(x) = x * P(x)

__

So far so good! What about the limits on c though, you say a left hand and a right hand limit, should I just identify where it is discontinuity from the right and left side? Really confused on this one

Re: Differential Calculus - Limits and continuity

Quote:

Originally Posted by

**leezgo** Thanks,

a) P(x) =

if blablabla = 0.80

if blablabla = 0.72

if blablabla = 065

b)

C(x) = x * P(x)

__

So far so good! What about the limits on c though, you say a left hand and a right hand limit, should I just identify where it is discontinuity from the right and left side? Really confused on this one

If you approach x = 1000 from the left, what value does the cost approach? If you approach x = 1000 from the right, what value does the cost approach?

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Re: Differential Calculus - Limits and continuity

We are talking what y values we land on when we reach x= 1000 from each side, right?

If so, then around 600 (the green circle) from right, and 700 (the red circle?) from left?

Attachment 24744

This is really helpful!