Activity 5.2.2.
Suppose that \(f\) is the function given in Figure 5.2.2 and that \(f\) is a piecewise function whose parts are either portions of lines or portions of circles, as pictured.
Two sets of adjacent coordinate axes are provided. The left axes have \(x\) ranging horizontally from \(-4\) to \(4\) and \(y\) ranging vertically from \(-2\) to \(2\text{.}\) The grid is \(1 \times 1\text{.}\) The righthand grid and axes are the same size with the same horizontal and vertical scale.
On the left axes, the graph of \(y = f(x)\) is provided (which is the same function considered in Activity 5.1.2). This piecewise linear function is \(0\) for \(x \lt 0\) and \(x \gt 7\text{.}\) On the interval \(0 \lt x \lt 1\text{,}\) \(f(x) = x\text{.}\) On the interval \(1 \lt x \lt 2\text{,}\) \(f(x)\) is a quarter circle that connects points \((1,1)\) and \((2,0)\text{.}\) On the interval \(3 \lt x \lt 4\text{,}\) \(f(x) = -1\text{.}\) On the interval \(4 \lt x \lt 5\text{,}\) \(f(x) = x-5\text{.}\) And on the interval \(5 \lt x \lt 7\text{,}\) \(f(x)\) is the top half of of the circle of radius \(1\) centered at \((6,0)\text{.}\)
At right, blank coordinate axes for plotting \(y = A(x) = \int_2^x f(t) \, dt\text{.}\)
(a)
(b)
(c)
Sketch a precise graph of \(y = A(x)\) on the axes at right that accurately reflects where \(A\) is increasing and decreasing, where \(A\) is concave up and concave down, and the exact values of \(A\) at \(x = 0, 1, \ldots, 7\text{.}\)
(d)
(e)
With as little additional work as possible, sketch precise graphs of the functions \(B(x) = \int_3^x f(t) \, dt\) and \(C(x) = \int_1^x f(t) \, dt\text{.}\) Justify your results with at least one sentence of explanation.
