Dido525 3 years ago Find a and b such that

1. Dido525

the following function is differentiable for all values of x.

2. Dido525

|dw:1350070251929:dw|

3. Dido525

I figured out a= 1 but I can't figure out what b is equal to.

4. myininaya

a=1 ?

5. Dido525

Oops. I figured out b = 1.

6. Dido525

I can`t solve for a.

7. myininaya

Yep. Now that is right so far. You just set the limx->0 for both of them and solve for a b. Good job.

8. myininaya

Because one this it must need to be is continuous

9. Dido525

Yep . I set the both as lim x-> 0 and I got that b = 1. I can't solve for a however.

10. Dido525

Should I derive both functions and solve for a?

11. Dido525

But that just means it's continuous. Not differentiable.

12. myininaya

Right that is one condition we must need though

13. myininaya

We need it to run smoothly when they meet (no sharp corners)... hmm... thinking....

14. Dido525

So I go: |dw:1350070954216:dw|

15. Dido525

right? O_o .

16. Dido525

No. Thats not right.

17. Dido525

Never mind.

18. myininaya

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

19. Dido525

Yep. I knew that. But unless I know what a is equal to I can't solve for the derivative at x=0.

20. myininaya

$(\arctan(ax+1))'|_{x=0}=(\frac{\pi}{4} e^{\sin(x)})'|_{x=0}$

21. Dido525

Right...

22. Dido525

I tried that.

23. myininaya

Find derivative of both sides Set x=0 And solve you are done

24. Dido525

Let me try again. Maybe I made a mistake.

25. Dido525

The derivative of arctan(ax+1) |dw:1350071299130:dw| right?

26. myininaya

Is that an a on top ?

27. Dido525

It shoud say 1/(1+(ax+1)^2)

28. myininaya

no it should say a on top

29. myininaya

30. myininaya

(ax+1)'=a

31. Dido525

Ohh. Why?

32. Dido525

Ohh....

33. Dido525

haha... Chain rule.... oops.

34. myininaya

$(\arctan(u(x))'=\frac{u'(x)}{{1+(u(x))^2}}$

35. Dido525

so it would be $\frac{ a }{ 1+(ax+1)^2 }$

36. myininaya

Yep and the other side you have?

37. Dido525

|dw:1350071580411:dw|

38. Dido525

right?

39. Dido525

I really appreciate the help by the way. Thanks a lot.

40. myininaya

I think you got it from here :) Great job.

41. Dido525

I got a = pi/2

42. Dido525

@myininaya .

43. Dido525

Thanks for making this so easy to understand! :) .

44. myininaya

That is what I got :)

45. myininaya

Gj again.