Activity 5.1.3.
For each of the following functions, sketch an accurate graph of the continuous antiderivative that satisfies the given initial condition. In addition, sketch the graph of two additional antiderivatives of the given function, and state the corresponding initial conditions that each of them satisfy.
(a)
original function: \(g(x) = \left| x \right| - 1\text{;}\) initial condition: \(G(-1) = 0\text{;}\) interval for sketch: \([-2,2]\)
(b)
original function: \(h(x) = \sin(x)\text{;}\) initial condition: \(H(0) = 1\text{;}\) interval for sketch: \([0,4\pi]\)
Note: the area bounded by one hump of the sine function is exactly \(2\) (e.g., on the interval \([0, \pi]\)).
(c)
original function:
\begin{equation*}
p(x) = \begin{cases}x^2, \amp \text{ if } 0 \lt x \lt 1 \\ -(x-2)^2, \amp \text{ if } 1 \lt x \lt 2 \\ 0 \amp \text{ if } x \le 0 \text{ or } x \ge 2 \end{cases}\text{;}
\end{equation*}
initial condition: \(P(0) = 1\text{;}\) interval for sketch: \([-1,3]\)
Note: the area bounded by \(x^2\) on \(0 \lt x \lt 1\) is \(\frac13\text{;}\) the region bounded by the other quadratic piece of \(p(x)\) is similar.
