Question

Find the value of c and d so that f(x) is both continuous and differentiable.

Answer 1

Please help, anyone

Answer 2

First, can you find \(f'(x)\)?

Answer 3

Of which of the two functions?

Answer 4

Well, there is really only one function, \(f(x)\).

Answer 5

Well yeah, I mean e^x+x^2+c or dx+2?

Answer 6

Or both?

Answer 7

Both.

Answer 8

You want \[
\lim_{x\to0^+}f'(x) =\lim_{x\to0^-}f'(x)
\]

Answer 9

The derivative of the first line is 2x+e^x but I'm not sure about the second one

Answer 10

@wio Would it be f'(x)=2x+e^x and d

Answer 11

\[\lim_{x \rightarrow 0^+}2x+e^x=\lim_{x \rightarrow 0^-}d\]
So \[d=2x+e^x\]

Answer 12

?

Answer 13

Hold on one second...

Answer 14

Okay, so first of all... \[
\lim_{x\to0^+}2x+e^x = L
\]And \(L\) should be constant with respect to \(x\).

Answer 15

We want \(L=d\), so you have to find \(L\).

Answer 16

Is it 1? L=1

Answer 17

Yeah

Answer 18

So d=1. What about c?

Answer 19

Also, you mixed up \(0^+\) and \(0^-\), though it doesn't affect the problem at all.

Answer 20

Oh ok, I'll fix that

Answer 21

Now you need to do: \[
\lim_{x\to0^+}f(x) =\lim_{x\to0^-}f(x)
\]

Answer 22

\[\lim_{x \rightarrow 0^+}x+2=\lim_{x \rightarrow 0^-}e^x+x^2+c\]

Answer 23

What do I do with this?

Answer 24

@wio

Answer 25

Do you know how to evaluate limits?

Answer 26

\[\lim_{x \rightarrow 0^+}2=\lim_{x \rightarrow 0^-}c+1\]

Answer 27

Is that right? Do I solve for c now?

Answer 28

yes.

Answer 29

@wio Ok then I get c=1. I think that's all.

Answer 30

however, when you plugged in \(1\), you should have gotten rid of the \(\lim\) parts.

Answer 31

Is your avatar wearing some sort of doggy collar?

Answer 32

when you plugged in \(x=0\)^ I meant to say

Answer 33

Ok, I'll fix that too. Thank you for your time!
And haha no its a jacket