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Continuity question

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Jeremiah Mburuburu
MaxEntropy_Man
Rishi
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Continuity question Empty Continuity question

Post by Rishi Mon May 13, 2013 7:11 am

Icould not go beyond the first two steps in solving this problem.

I saw the solution later.

Still not sure whether the MIT teacher is completely correct.

Problem.

Let f(x) = ax+b x < 1

x^4 +x +1 x >= 1


Find all a and b such that the function f(x) is differentiable.

From his solution, it appears that the derivative of a function is also continuous.

Is f'(x) is always continuous?

Rishi

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Post by MaxEntropy_Man Mon May 13, 2013 7:36 am

i haven't looked at your problem, but quick answer to your last question is no. consider the function f(x) = abs(x). the function is continuous everywhere, but its derivative at x=0 is not.
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Post by Rishi Mon May 13, 2013 7:56 am

MaxEntropy_Man wrote:i haven't looked at your problem, but quick answer to your last question is no. consider the function f(x) = abs(x). the function is continuous everywhere, but its derivative at x=0 is not.

Max,

Please look at Problem #5 on Exam #1 and the solution the teacher provided.

http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/exams/

I think the question is wrong.

He should have said.

"Find all a and b such that both f(x) and f'(x) is differentiable at x = 1."

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Post by Rishi Mon May 13, 2013 8:42 am

Max,

I got it.

The teacher is correct.

This goes to the very definition of continuity and derivatives.

f'(x) = limit h ---> 0 (f(x+h) -f(x)) /h

whether h < 0 or h > 0, f'(x) should be the same

f'(1) should be the same whether you compute it from left or right.

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Post by Jeremiah Mburuburu Mon May 13, 2013 12:13 pm

Rishi wrote:Icould not go beyond the first two steps in solving this problem.

I saw the solution later.

Still not sure whether the MIT teacher is completely correct.

Problem.

Let f(x) = ax+b x < 1

x^4 +x +1 x >= 1


Find all a and b such that the function f(x) is differentiable.

From his solution, it appears that the derivative of a function is also continuous.

Is f'(x) is always continuous?

rishi, you probably don't need this, but by posting this, i am trying to scrape off some rust. i didn't read any post in this thread except your first.

f ' is differentiable at all real numbers if a and b have such values that:

1. f is continuous, and

2. f ' is continuous

at all real numbers.

since linear and polynomial functios are continuous and differentiable at all real numbers, the piecewise-defined f(x) is differentiable if it is continuous and differentiable at x = 1.

for condition 1 above to be true, lim as x -> 1 of ax + b = 1^4 + 1 + 1, i.e. a + b = 3 -- A;

for condition 2 to be true, lim as x -> 1 of a (the derivative of ax + b) = 4(1)^3 + 1 (the derivative of f(x) at x =1), i.e. a = 5 -- B.

from A and B, a = 5 and b = -2.

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Post by Rishi Mon May 13, 2013 2:32 pm

Jeremiah Mburuburu wrote:
Rishi wrote:Icould not go beyond the first two steps in solving this problem.

I saw the solution later.

Still not sure whether the MIT teacher is completely correct.

Problem.

Let f(x) = ax+b x < 1

x^4 +x +1 x >= 1


Find all a and b such that the function f(x) is differentiable.

From his solution, it appears that the derivative of a function is also continuous.

Is f'(x) is always continuous?

rishi, you probably don't need this, but by posting this, i am trying to scrape off some rust. i didn't read any post in this thread except your first.

f ' is differentiable at all real numbers if a and b have such values that:

1. f is continuous, and

2. f ' is continuous

at all real numbers.

since linear and polynomial functios are continuous and differentiable at all real numbers, the piecewise-defined f(x) is differentiable if it is continuous and differentiable at x = 1.

for condition 1 above to be true, lim as x -> 1 of ax + b = 1^4 + 1 + 1, i.e. a + b = 3 -- A;

for condition 2 to be true, lim as x -> 1 of a (the derivative of ax + b) = 4(1)^3 + 1 (the derivative of f(x) at x =1), i.e. a = 5 -- B.

from A and B, a = 5 and b = -2.

Why should be we concerned about f' being differentiable ?

Rishi

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Post by Jeremiah Mburuburu Mon May 13, 2013 5:40 pm

Rishi wrote:
Jeremiah Mburuburu wrote:
Rishi wrote:Icould not go beyond the first two steps in solving this problem.

I saw the solution later.

Still not sure whether the MIT teacher is completely correct.

Problem.

Let f(x) = ax+b x < 1

x^4 +x +1 x >= 1


Find all a and b such that the function f(x) is differentiable.

From his solution, it appears that the derivative of a function is also continuous.

Is f'(x) is always continuous?

rishi, you probably don't need this, but by posting this, i am trying to scrape off some rust. i didn't read any post in this thread except your first.

f ' is differentiable at all real numbers if a and b have such values that:

1. f is continuous, and

2. f ' is continuous

at all real numbers.

since linear and polynomial functios are continuous and differentiable at all real numbers, the piecewise-defined f(x) is differentiable if it is continuous and differentiable at x = 1.

for condition 1 above to be true, lim as x -> 1 of ax + b = 1^4 + 1 + 1, i.e. a + b = 3 -- A;

for condition 2 to be true, lim as x -> 1 of a (the derivative of ax + b) = 4(1)^3 + 1 (the derivative of f(x) at x =1), i.e. a = 5 -- B.

from A and B, a = 5 and b = -2.

Why should be we concerned about f' being differentiable ?
i made a typing error. that passage should be as follows:

f is differentiable at all real numbers if a and b have such values that:

1. f is continuous, and

2. f ' is continuous


(i changed the first f ' to f; the second remains f ')

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Post by Rishi Mon May 13, 2013 5:49 pm

To JM,

Does f being differentiable at some point imply f' will be continuous at that point?

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Post by Jeremiah Mburuburu Tue May 14, 2013 2:17 am

Rishi wrote:To JM,

Does f being differentiable at some point imply f' will be continuous at that point?
i think so, but i haven't proved that to myself.

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Post by Jeremiah Mburuburu Tue May 14, 2013 1:14 pm

Jeremiah Mburuburu wrote:
Rishi wrote:To JM,

Does f being differentiable at some point imply f' will be continuous at that point?
i think so, but i haven't proved that to myself.
rishi, have you seen a graph of f after replacing the values of a and b with those required by the differentiability of f? it's educational to do so. it's easily done if you have a graphing calculator. on the Y= screen of my ti-84, i entered the equation:

Y1 = (5x - 2)(x < 1) + (x^4 + x + 1)(x >= 1)

which graphed y = 5x - 2 for x < 1, and x^4 + x + 1 for x >= 1.

the (x < 1) that you see in the equation is a logical expression or test that returns a 1 if x < 1, and a 0 otherwise. the (x >= 1) is similar.

perhaps someone will post here a similar graph produced by wolfram alpha or some such software.

observe the continuity and smoothness of the graph, and bask in its warmth. tTotT'ly cool, yaar!

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Post by Jeremiah Mburuburu Tue May 14, 2013 1:29 pm

Jeremiah Mburuburu wrote:have you seen a graph of f after replacing the values of a and b with those required by the differentiability of f?
correction: have you seen a graph of f after replacing a and b with the values required by the differentiability of f?

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Post by Rishi Tue May 14, 2013 4:23 pm

Jeremiah Mburuburu wrote:
Jeremiah Mburuburu wrote:have you seen a graph of f after replacing the values of a and b with those required by the differentiability of f?
correction: have you seen a graph of f after replacing a and b with the values required by the differentiability of f?



Thanks.

I will do it when I get home.

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Post by Marathadi-Saamiyaar Tue May 14, 2013 4:34 pm

Rishi wrote:
Jeremiah Mburuburu wrote:
Jeremiah Mburuburu wrote:have you seen a graph of f after replacing the values of a and b with those required by the differentiability of f?
correction: have you seen a graph of f after replacing a and b with the values required by the differentiability of f?

Thanks.

I will do it when I get home.

AiyerE:

Great that you have interest in learning. But, you been there dont that. So pick some other new topic to study and you will feel good about it. Of course, if you are learning to help your HS children, it is good but not good. Bcz very rarely they like to learn from parents.

If I were to study I will pick Accounting - bcz I hate it and think it is a totally fake field. Hence, that is the one I want to know something about.

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Post by Rishi Tue May 14, 2013 6:19 pm

Marathadi-Saamiyaar wrote:
Rishi wrote:
Jeremiah Mburuburu wrote:
Jeremiah Mburuburu wrote:have you seen a graph of f after replacing the values of a and b with those required by the differentiability of f?
correction: have you seen a graph of f after replacing a and b with the values required by the differentiability of f?

Thanks.

I will do it when I get home.

AiyerE:

Great that you have interest in learning. But, you been there dont that. So pick some other new topic to study and you will feel good about it. Of course, if you are learning to help your HS children, it is good but not good. Bcz very rarely they like to learn from parents.

If I were to study I will pick Accounting - bcz I hate it and think it is a totally fake field. Hence, that is the one I want to know something about.

I am thinking of learning biology.

I never had an opportunity to learn it.

I do not know the difference between a virus and a bacteria.

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Post by Idéfix Tue May 14, 2013 6:33 pm

Rishi wrote:I am thinking of learning biology.

I never had an opportunity to learn it.

I do not know the difference between a virus and a bacteria.
Most of what I know of biology, I learned after college. I didn't learn it well in school. I now find it more fascinating than math. I think that is because I learned in school more math than I need to use in my life, while biology is mostly new to me.
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Post by Jeremiah Mburuburu Wed May 15, 2013 11:54 am

Jeremiah Mburuburu wrote:
Jeremiah Mburuburu wrote:
Rishi wrote:To JM,

Does f being differentiable at some point imply f' will be continuous at that point?
i think so, but i haven't proved that to myself.
rishi, have you seen a graph of f after replacing the values of a and b with those required by the differentiability of f? it's educational to do so. it's easily done if you have a graphing calculator. on the Y= screen of my ti-84, i entered the equation:

Y1 = (5x - 2)(x < 1) + (x^4 + x + 1)(x >= 1)

which graphed y = 5x - 2 for x < 1, and x^4 + x + 1 for x >= 1.

the (x < 1) that you see in the equation is a logical expression or test that returns a 1 if x < 1, and a 0 otherwise. the (x >= 1) is similar.

perhaps someone will post here a similar graph produced by wolfram alpha or some such software.

observe the continuity and smoothness of the graph, and bask in its warmth. tTotT'ly cool, yaar!

rishi, here's wolfram alpha's plot of your piecewise function f, with a and b replaced with values that make f differentiable:

Continuity question Gif&s=3&w=312.&h=55




Continuity question Gif&s=4&w=300.&h=185

note the continuity and smoothness of the graph at x = 1.

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Post by Rishi Wed May 15, 2013 12:39 pm

Looks great.

Thanks.

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Post by Kris Wed May 15, 2013 2:19 pm

Marathadi-Saamiyaar wrote:
Rishi wrote:
Jeremiah Mburuburu wrote:
Jeremiah Mburuburu wrote:have you seen a graph of f after replacing the values of a and b with those required by the differentiability of f?
correction: have you seen a graph of f after replacing a and b with the values required by the differentiability of f?

Thanks.

I will do it when I get home.

AiyerE:

Great that you have interest in learning. But, you been there dont that. So pick some other new topic to study and you will feel good about it. Of course, if you are learning to help your HS children, it is good but not good. Bcz very rarely they like to learn from parents.

If I were to study I will pick Accounting - bcz I hate it and think it is a totally fake field. Hence, that is the one I want to know something about.
>>> I would like to study linguistics or German if I can get the bandwidth. Would also like to take do different programming languages, but this one will be like a hobby because it was like playing with puzzles the last time I did this.

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Post by Bittu Wed May 15, 2013 2:25 pm

I could solve this problem but GOOG is 900 so fuck this problem.

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