Prereq Review for Calc I

Here are a few problems you should try in order to reacquaint yourself with a bit of precalculus math and familiarize yourself with our notation in Calculus I.

The problems

  1. Define two intervals in set theoretic notation as \[A = \{x\in\mathbb R: -1<x\leq 2\} \text{ and } B = \{x\in\mathbb R: -3<x\leq 1\}.\] Draw the set \(S = A\cap B\) on the set of axes shown in Figure 1. Note that \(S\) should be an interval; be sure to indicate whether the endpoints are half-open or half closed.

  2. Write down the half open interval shown in Figure 2 in set theoretic notation.

  3. Let \(I=\{x\in\mathbb R: 0 \leq x < 4\}\) and define \(f:I\to\mathbb R\) by \(f(x) = \frac{1}{2} (x^2-3x)\). Sketch the graph of \(f\) on the empty axes in Figure 3.

  4. Find an equation for the line containing the points \((-1,-1)\) and \((5,2)\).

  5. Simplify the rational expression \[\frac{x^3 - 2x^2 + 2x - 1}{x-1}\] to a polynomial. Be sure to explain your work, indicating:

    1. How the numerator factors and
    2. Any assumptions that must be made for the original rational expression to equal a polynomial

Figures

Figure 1: Empty axes
Figure 2: A half-open interval
Figure 3: More empty axes