Functions, derivatives, and antiderivatives have many entangled properties. For example, over intervals where the first derivative of a function is always positive, we know that the function itself is always increasing. (Do you understand why?) Many of these relationships can be expressed graphically. Consequently, it is imperative that you fully understand the meaning of some commonly used graphical expressions. These expressions are loosely defined in Definition 4.1.1.

Definition4.1.1Some Common Graphical Phrases

These are loose, informal definitions. Technical, precise definitions can be found elsewhere.

Positive function. The vertical coordinates of points on the function are positive. A function is positive whenever its graph lies above the horizontal axis.

Negative function. The vertical coordinates of points on the function are negative. A function is negative whenever its graph lies below the horizontal axis.

Increasing function. The vertical coordinates of the function increase as you move along the curve from left to right. Linear functions with positive slope are always increasing.

Decreasing function. The vertical coordinates of the function decrease as you move along the curve from left to right. Linear functions with negative slope are always decreasing.

Concave up function. A function is concave up at \(a\) if the tangent line to the function at \(a\) lies below the curve. The bottom half of a circle is concave up.

Concave down function. A function is concave down at \(a\) if the tangent line to the function at \(a\) lies above the curve. The top half of a circle is concave down.

Subsection4.1.1Exercises

Answer each of the following questions in reference to the function shown in Figure 4.1.2. Each answer is an interval (or intervals) along the \(x\)-axis.

Use interval notation when expressing your answers. Make each interval as wide as possible; that is, do not break an interval into pieces if the interval does not need to be broken. Assume that the slope of the function is constant on \(\ointerval{-\infty}{-5}\text{,}\) \(\ointerval{3}{4}\text{,}\) and \(\ointerval{4}{\infty}\text{.}\)