Question

Find the value of c and d so that f(x) is both continuous and differentiable.
e^x+x^2+C, x(less than or equal to)0
dx+2, x>0

Answer 1

In order for f(x) to be continuous and differentiable, then there cannot be any holes, gaps or sharp edges in the function. So the first segment and second segment must have equal values at x=0, and the slopes must also match up.

Answer 2

so by solving the first equation for c, and the second for d, i can substitute x=0 to find the values for c & d?

Answer 3

To find the value of C, we set x=0, which will make the segments connect without a hole or gap. However to make it not have a sharp corner, we have to make sure the slope a x=0 is equal for both segments.

Answer 4

\[e^0+0^2+C\]
1+0+C?

Answer 5

how do you find the slope of these two equations?

Answer 6

For the first part, you have to equate that to the other segment:
\[e^0+0^2 + C = 1+C = d(0) + 2\]
To find the slope of the two segments, you have to take the derivatives of both segments at x=0

Answer 7

so would C=1 then?

what do you mean derivatives at x=0?

Answer 8

I'm assuming you know what a derivative is(or at least the definition of a derivative/finding the tangent line)

Answer 9

yes! or, basically. the derivative of \[e^x+x^2+C\] would be \[e^{x}+2x+C\] ?

Answer 10

If C is a constant, the the derivative of C = 0. What would be the derivative of the second segment/function?

Answer 11

\[dx+2\] would just be \[d\]

Answer 12

So just evaluate both of those derivatives at x=0 to find the slopes for each. For it to be smooth, and therefore differentiable, the slopes at x=0 must be equal.

Answer 13

okay, but for the first one, if x=0, then will slope=1?
and the second... where do i sub in x=0?

Answer 14

You don't have an x to sub(because the slope is always equal to d).

Answer 15

so, how do i know the two slopes match up?

Answer 16

The slopes match up if the slopes are equal.
\[e^0+2(0)=d\]