Worksheet for Section 6.1

Secton 6.1 introduces antiderivatives, in this section looking at them graphically and numerically. For example, suppose you know that the function $F(x)$ has derivative $F'(x)=f(x)=3x^2$ -- then what is $F(x)$? In this case, $f(x)=3x^2$ is in a form that suggests the result of applying the Power Rule for derivatives, and in fact, with $F(x)=x^3$, $F'(x)=3x^2=f(x)$ as needed. However, note that also $F(x)=x^3+4$, $F(x)=x^3-12$, and any other function of the form $F(x)=x^3+C$ (where $C$ is a constant) will also satisfy $F'(x)=3x^2=f(x)$. In other words, the antiderivative $F(x)$ of the function $f(x)$ is not unique -- there are many possible antiderivatives, differing only by a constant term. In fact, if $F(x)$ is an antiderivative of $f(x)$, then another function $G(x)$ is also an antiderivative of $f(x)$ just in case $G(x)=F(x)+C$ for some constant $C$. (You will see more about this later in the chapter.) The constant $C$ is an important part of this expression, making sure that you get a general solution for the antiderivative. In some cases, extra information is given in a problem which can help you determine a value for the constant $C$ -- in such cases, the extra information allows you to go from the general solution to a particular solution. For example, suppose $F'(x)=-16x$. First, find a general solution for $F(x)$. Now suppose $F(0)=10$ (this extra information is called an initial condition). Use this to find a value for $C$ in the general solution.

Since the derivative $f'(x)$ for $f(x)$ gives information about slopes of tangent lines to the graph of $f(x)$, this information can be used to generate graphs of $f(x)$ when you are given a graph of $f'(x)$ -- however, again there will be more than one possibility. Consider, for example, if $f'(x)=\vert x\vert$. Graph $f'(x)$ and consider what your graph says about slopes of tangent lines to $f(x)$. Use this to generate graphs of possible functions $f(x)$, with initial conditions $f(0)=0$, $f(0)=1$, or $f(0)=-1$. To make the problem a bit more interesting, suppose $g'(x)=\vert x\vert-2$ and $g(0)=2$. Sketch a graph of $g(x)$ showing the coordinates of all critical points and inflection points. In order to find the coordinates of points on the graph, you will need to use the given initial condition along with the Fundamental Theorem of Calculus.