CPM Homework Help : APCALC Problem 4-152

Determine the value of $a$ such that $f$ is differentiable at $x = 1$ .

$f ( x ) = \left\{ \begin{array} { l l } { ( x + 2 ) ^ { 2 } - 3 } & { \text { for } x < 1 } \\ { a \operatorname { sin } ( x - 1 ) + 6 } & { \text { for } x \geq 1 } \end{array} \right.$

Step 1:

In order to prove that $f\left(x\right)$ is differentiable at $x = 1$ , you must demonstrate that the derivatives agree from the left and right.

Step 2:

But do not forget that 'differentiability implies continuity'. In other words, if the function is not continuous at $x = 1$ , then it CANNOT be differentiable (even if the derivatives agree). So use the Three Conditions of Continuity to demonstrate that $f\left(x\right)$ is continuous at $x = 1$ .

Use the eTool below to visualize the problem.
Click the link at right for the full version of the eTool: Calc 4-152 HW eTool