Question

Find all values of the constants a and b for which the following function is differentiable:

Answer 1

\[f(x)= \begin{cases}
x^2+4x+1, & x\geq 1 \\
ax+b, & x < 1 \\
\end{cases}\]

I understand (I hope not to have misunderstood) that when a function is differentiable at one point, it means that it is continuous at that point as well. And for a function to be a differentiable one at a certain point, its derivative coming from the left must be the same to that coming from the right. So in this case:

\[\large{\lim_{x\to 0^-}f’(x) = \lim_{x\rightarrow 0^+} f’(x)}\]
\[a= 2x + 4\]
\[a= 2(1) + 4\]
\[a= 6\]

Since the derivative of a constant is zero, does it mean that \(b\) can be any number?

Can a function be differentiable but not continuous?

Sorry for the silly questions.

Answer 2

You are correct a = 6 and but b must be equal to zero since the value of the function f(x) must be continous

Answer 3

why do you have as x approaches 0?

Answer 4

it should be as x approaches 1

Answer 5

you should have two equations
\[\lim_{x \rightarrow 1^-}f(x)=\lim_{x \rightarrow 1^+}f(x) \\ \lim_{x \rightarrow 1^-}f'(x)=\lim_{x \rightarrow 1^+}f'(x) \]

Answer 6

Yep. I made a mistake there |:

Answer 7

the function has to be continuous and smooth to be differentiable

Answer 8

the first equation i wrote was continuity assuming f(1) exists and is equal that limit
the second equation i wrote is for smooth

Answer 9

Ah okey, then in order to find the values for which the function is differentiable at that point I must find:

1) Its slope at that point.
2) Go back to find the value of \(b\).

In other words, make the slopes equal to each other, right?

Answer 10

@freckles

Answer 11

yep slopes must be equal and both functions must meet

Answer 12

you got "a" correct from insisting the slopes be equal at 1 independent of which side you approach it from.
b = 0 comes from making sure the value of f(x) must be the same also as I approach form the left or right

Answer 13

Aaah okey. \(b=0\). That makes sense.

That answers my 2nd question too. Thank you both.

Answer 14

np

Answer 15

you are welcome...have a great day!

Answer 16

cool video

Answer 17

:) you may create it using geogebra if you're interested

Answer 18

http://web.geogebra.org/app/#

Answer 19

The gif makes it super clear why it must be 0 (:

Thanks @ganeshie8 !!!

Answer 20

np :)

Answer 21

let me give u another gif by fixing b=0 and varying a

Answer 22

Great

Answer 23

note that the value of \(\large a\) affects both continuity and differentiability

Answer 24

I can see it. That's pretty cool.

Answer 25

Was it hard to program geoboard to do that?

Answer 26

or whatever it is called

Answer 27

I mean do you think geo'whatever' is programmer friendly

Answer 28

not at all, its pretty easy freckles... its not even code.. just add a slider and set the range, thats all.. .

Answer 29

let me see if the online version allows exporting gif

Answer 30

looks fun
i might have to look more into that
and check out the pricing
also i have no programming skills

Answer 31

its 100% free

Answer 32

1) click this link
http://web.geogebra.org/app/#

2) click "Algebra"

3) enter below command in input bar :
`y=If[x<1,6x+b,x^2+4x+1]`

4) click somewhere on the grid on right side

Answer 33

you should see something like this :
http://prntscr.com/52d0pe

Answer 34

yea

Answer 35

that's neatness
thanks

Answer 36

but the online version doesn't seem to have Export->gif option, you will have to download it to create gifs i think...

do u use windows ?

Answer 37

Yes

Answer 38

http://www.geogebra.org/download

Answer 39

here is the path to direct exe file
http://download.geogebra.org/installers/?os=win

Answer 40

i got it

Answer 41

installing the virus now

Answer 42

i mean geo'whatever' thing

Answer 43

lol i have been using it like forever its very user friendly and you will love it once u get use to it

Answer 44

Seems like you can do a lot with it

Answer 45

indeed, there is a 3D version also where u can plot surfaces/planes etc...

Answer 46

the CAS allows to work integrals numerically and many other things, but i use it mostly for graphing equations

Answer 47

for example, you can enter below two commands in input bar :

f(x) = x^2 + x
g(x) = f'(x)

Answer 48

g(x) shows the graph of derivative of f(x)

Answer 49

Enjoy... !

Answer 50

I will definitely have to play with it to get the hang of it

Answer 51

and thanks ganeshie