continuous functions and differentiability (g(x) piecewise)

[Math wiz ya rite 09]

g(x)=

x+b, x<0
cosx, x greater than or equal to 0

Is there a value for b that will make g(x) continuous at x=0? Is there a value that will make g(x) differentiable at zero? If there is, what is it? If not, why not? Give reasons for your answeres.

Thanks

 

[soroban]

Re: continuous functions and differentiability

Hello, Math wiz ya rite 09!

\(\displaystyle g(x) \;= \;\left\{\begin{array}{ccc}x\,+\,b& \quad & x\,<\,0 \\
\cos x & \quad & x\,\geq\,0 \end{array}\)

Is there a value for $\displaystyle b$ that will make $\displaystyle g(x)$ continuous at $\displaystyle x=0$? . . . . yes

To be continuous at $\displaystyle x=0$, the curves must "meet" there.
If you make a sketch, the answer is obvious.

                  |
                  o b
                * |
              *   |
            *   1 *
          *       |   *
        *         |     *
      *           |      *
    *             |
  - - - - - - - - + - - - -*- - - - - -
                  |
                  |          *
                  |           *
                  |             *
                  |                *
                  |

And we see that $\displaystyle b$ must be $\displaystyle 1.$

Is there a value that will make $\displaystyle g(x)$ differentiable at $\displaystyle x=0$? . . . . no

At $\displaystyle x=0$, the line has slope $\displaystyle 1$
. . and the cosine curve has slope $\displaystyle \,-\sin(0)\,=\,0$.