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Continuity question
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Jeremiah Mburuburu
MaxEntropy_Man
Rishi
7 posters
Page 1 of 1
Continuity question
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?
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- Posts : 5129
Join date : 2011-09-02
Re: Continuity question
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.
MaxEntropy_Man- Posts : 14702
Join date : 2011-04-28
Re: Continuity question
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."
Rishi- Posts : 5129
Join date : 2011-09-02
Re: Continuity question
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.
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.
Rishi- Posts : 5129
Join date : 2011-09-02
Re: Continuity question
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.
Jeremiah Mburuburu- Posts : 1251
Join date : 2011-09-09
Re: Continuity question
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- Posts : 5129
Join date : 2011-09-02
Re: Continuity question
i made a typing error. that passage should be as follows: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 ?
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 ')
Jeremiah Mburuburu- Posts : 1251
Join date : 2011-09-09
Re: Continuity question
To JM,
Does f being differentiable at some point imply f' will be continuous at that point?
Does f being differentiable at some point imply f' will be continuous at that point?
Rishi- Posts : 5129
Join date : 2011-09-02
Re: Continuity question
i think so, but i haven't proved that to myself.Rishi wrote:To JM,
Does f being differentiable at some point imply f' will be continuous at that point?
Jeremiah Mburuburu- Posts : 1251
Join date : 2011-09-09
Re: Continuity question
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:Jeremiah Mburuburu wrote:i think so, but i haven't proved that to myself.Rishi wrote:To JM,
Does f being differentiable at some point imply f' will be continuous at that point?
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!
Jeremiah Mburuburu- Posts : 1251
Join date : 2011-09-09
Re: Continuity question
correction: have you seen a graph of f after replacing a and b with the values required by the differentiability of f?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?
Jeremiah Mburuburu- Posts : 1251
Join date : 2011-09-09
Re: Continuity question
Jeremiah Mburuburu wrote:correction: have you seen a graph of f after replacing a and b with the values required by the differentiability of f?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?
Thanks.
I will do it when I get home.
Rishi- Posts : 5129
Join date : 2011-09-02
Re: Continuity question
Rishi wrote:Jeremiah Mburuburu wrote:correction: have you seen a graph of f after replacing a and b with the values required by the differentiability of f?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?
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.
Marathadi-Saamiyaar- Posts : 17675
Join date : 2011-04-30
Age : 110
Re: Continuity question
Marathadi-Saamiyaar wrote:Rishi wrote:Jeremiah Mburuburu wrote:correction: have you seen a graph of f after replacing a and b with the values required by the differentiability of f?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?
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.
Rishi- Posts : 5129
Join date : 2011-09-02
Re: Continuity question
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.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.
Idéfix- Posts : 8808
Join date : 2012-04-26
Location : Berkeley, CA
Re: Continuity question
Jeremiah Mburuburu wrote: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:Jeremiah Mburuburu wrote:i think so, but i haven't proved that to myself.Rishi wrote:To JM,
Does f being differentiable at some point imply f' will be continuous at that point?
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:
note the continuity and smoothness of the graph at x = 1.
Jeremiah Mburuburu- Posts : 1251
Join date : 2011-09-09
Re: Continuity question
>>> 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.Marathadi-Saamiyaar wrote:Rishi wrote:Jeremiah Mburuburu wrote:correction: have you seen a graph of f after replacing a and b with the values required by the differentiability of f?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?
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.
Kris- Posts : 5461
Join date : 2011-04-28
Re: Continuity question
I could solve this problem but GOOG is 900 so fuck this problem.
Bittu- Posts : 1151
Join date : 2011-08-19
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