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.

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?

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.

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 ?

Rishi wrote:To JM,

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

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.

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 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?

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.

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.

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.

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!

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.