Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes. The piecewise function describing the boundary of one such track is given as follows:
\[
f(x) = \begin{cases}
x^2 + 4, & 0 \le x \\
4x^2 - 4x + 40, & x \ge 3
\end{cases}
\]
Based on the given information, answer the following question:
Question: 1
Find \(f'(x)\) for \(0 < x < 3\).
Show Hint
When differentiating a piecewise function on an open interval, completely disregard the other branches! Simply pinpoint which interval matches your constraint, isolate that basic polynomial expression, and apply standard differentiation rules directly. For \(0 < x < 3\), the second branch \(4x^2 - 4x + 40\) plays absolutely no role in determining the local slope.
Always double-check which interval your target point belongs to! A common error is mistakenly using the first branch formula. Since \(4 > 3\), only the branch \(4x^2 - 4x + 40\) is valid. Differentiate first, then plug in the value.
To test continuity at a boundary point of a piecewise function, always evaluate the limits from both sides. Find the value coming from the left branch, find the value coming from the right branch, and see if they match up perfectly! If they don't, the graph has a "jump" break at that point.
Test for differentiability of \(f(x)\) at \(x = 3\).
Show Hint
Continuity is a necessary prerequisite for differentiability! Always check continuity first. If a graph is broken or has a jump gap at a point, it can never be differentiated at that point, saving you the time of calculating derivatives entirely!
