Dido525
Find a and b such that
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Dido525
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the following function is differentiable for all values of x.
Dido525
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|dw:1350070251929:dw|
Dido525
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I figured out a= 1 but I can't figure out what b is equal to.
myininaya
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a=1 ?
Dido525
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Oops. I figured out b = 1.
Dido525
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I can`t solve for a.
myininaya
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Yep. Now that is right so far.
You just set the limx->0 for both of them and solve for a b.
Good job.
myininaya
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Because one this it must need to be is continuous
Dido525
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Yep . I set the both as lim x-> 0 and I got that b = 1. I can't solve for a however.
Dido525
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Should I derive both functions and solve for a?
Dido525
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But that just means it's continuous. Not differentiable.
myininaya
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Right that is one condition we must need though
myininaya
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We need it to run smoothly when they meet (no sharp corners)...
hmm...
thinking....
Dido525
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So I go:
|dw:1350070954216:dw|
Dido525
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right? O_o .
Dido525
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No. Thats not right.
Dido525
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Never mind.
myininaya
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Ok...how about this :
What do you think of both parts of the function having the same exact slope at x=0
This will give us that smooth transition at x=0 that i'm talking about
Dido525
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Yep. I knew that. But unless I know what a is equal to I can't solve for the derivative at x=0.
myininaya
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\[(\arctan(ax+1))'|_{x=0}=(\frac{\pi}{4} e^{\sin(x)})'|_{x=0}\]
Dido525
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Right...
Dido525
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I tried that.
myininaya
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Find derivative of both sides
Set x=0
And solve
you are done
Dido525
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Let me try again. Maybe I made a mistake.
Dido525
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The derivative of arctan(ax+1)
|dw:1350071299130:dw|
right?
myininaya
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Is that an a on top ?
Dido525
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It shoud say 1/(1+(ax+1)^2)
myininaya
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no it should say a on top
myininaya
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that is your issue
myininaya
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(ax+1)'=a
Dido525
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Ohh. Why?
Dido525
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Ohh....
Dido525
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haha... Chain rule.... oops.
myininaya
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\[(\arctan(u(x))'=\frac{u'(x)}{{1+(u(x))^2}}\]
Dido525
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so it would be \[\frac{ a }{ 1+(ax+1)^2 }\]
myininaya
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Yep and the other side you have?
Dido525
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|dw:1350071580411:dw|
Dido525
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right?
Dido525
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I really appreciate the help by the way. Thanks a lot.
myininaya
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I think you got it from here :)
Great job.
Dido525
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I got a = pi/2
Dido525
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@myininaya .
Dido525
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Thanks for making this so easy to understand! :) .
myininaya
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That is what I got :)
myininaya
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Gj again.