CPM Homework Help : APCALC Problem 7-123

Determine the values of $a$ and $b$ so that the function defined below is continuous and differentiable.

$f ( x ) = \left\{ \begin{array} { l l } { a e ^ { x } } & { \text { for } x \leq 1 } \\ { \operatorname { ln } ( x ) + b } & { \text { for } x > 1 } \end{array} \right.$

Hint:

$\text{Continuous means }\lim \limits_{x\rightarrow 1^{-}}f(x)=\lim \limits_{x\rightarrow 1^{+}}f(x)=f(1).$

Differential means $\lim\limits_{x\rightarrow 1^{-}}f'(x)=\lim\limits_{x\rightarrow 1^{+}}f'(x)=f(1)$ , but only if it's continuous as well.

More Help:

Write and solve a system of equations, each evaluated at $x = 1$ . The first equation will equate the two pieces of $f(x)$ .
The second equation will equate the two pieces of $f^\prime(x)$ .