§5.1 Constructing accurate graphs of antiderivatives

Introduction

Constructing the graph of an antiderivative

Multiple antiderivatives of a single function

Functions defined by integrals

Suppose that the following information is known about a function $f$: the graph of its derivative, $y = f'(x)$, is given in Figure 5.1. Further, assume that $f'$ is piecewise linear (as pictured) and that for $x \le 0$ and $x \ge 6$, $f'(x) = 0$. Finally, it is given that $f(0) = 1$.

On what interval(s) is $f$ an increasing function? On what intervals is $f$ decreasing?

On what interval(s) is $F$ an increasing function? On what intervals is $F$ decreasing?

On what interval(s) is $F$ concave up? concave down? neither?

At what point(s) does $F$ have a relative minimum? a relative maximum?

Use the given information to determine the exact value of $F(x)$ for $x = 1, 2, \ldots, 7$. In addition, what are the values of $F(-1)$ and $F(8)$?

Based on your responses to all of the preceding questions, sketch a complete and accurate graph of $y = F(x)$ on the axes provided, being sure to indicate the behavior of $F$ for $x \lt 0$ and $x \gt 7$. Clearly indicate the scale on the vertical and horizontal axes of your graph.

Suppose we change one key piece of information: in particular, say that $G$ is an antiderivative of $f$ and $G(0) = 0$. How (if at all) would your answers to the preceding questions change? Sketch a graph of $G$ on the same axes as the graph of $F$ you constructed in (e).

original function: $g(x) = \left| x \right| - 1$; initial condition: $G(-1) = 0$; interval for sketch: $[-2,2]$

original function: $h(x) = \sin(x)$; initial condition: $H(0) = 1$; 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 $[0, \pi]$).

original function:

; initial condition: $P(0) = 1$; interval for sketch: $[-1,3]$

Note: the area bounded by $x^2$ on $0 \lt x \lt 1$ is $\frac13$; the region bounded by the other quadratic piece of $p(x)$ is similar.

On what interval(s) is $A$ an increasing function? On what intervals is $A$ decreasing? Why?

On what interval(s) do you think $A$ is concave up? concave down? Why?

At what point(s) does $A$ have a relative minimum? a relative maximum?

Use the given information to determine the exact values of $A(0)$, $A(1)$, $A(2)$, $A(3)$, $A(4)$, $A(5)$, and $A(6)$.

Based on your responses to all of the preceding questions, sketch a complete and accurate graph of $y = A(x)$ on the axes provided, being sure to indicate the behavior of $A$ for $x \lt 0$ and $x \gt 6$.

How does the graph of $B$ compare to $A$ if $B$ is instead defined by $B(x) = \int_0^x g(t) \, dt$?