Mathematics 265 Introduction to Calculus I

Study Guide :: Unit 2

Functions

Finishing This Unit

  1. Review the objectives of this unit and make sure you are able to meet all of them.
  2. If there is a concept, definition, example or exercise that is not yet clear to you, go back and reread it, then contact your tutor for help.
  3. Definitions are important, for each definition you should be able to give an example (something that is covered by the definition) and a counterexample something that is not so covered.
  4. Do the exercises in the “Learning from Mistakes” section for this unit.
  5. You may want to do Exercises 1(a)-(d), 2, 5 (domain only), 6 (domain only), 9(a)-(d), 10 and 11-15 from the “Review” (page 42 of the textbook).

Learning from Mistakes

There are mistakes in each of the following solutions. Identify the errors, and give the correct answer.

  1. Define each of the functions below as a set of pairs.

    1. The velocity $v$ depends on the time $t$.
    2. The bacteria population $B$ depends on the amount of oxygen $o$.

    Erroneous Solutions

    1. $\{ (v,t)|v\;{\text{is}}\;{\text{the}}\;{\text{velocity}}\;{\text{at}}\;{\text{time}}\;t\}$

    2. $\{ (o,B)|o\;{\text{is}}\;{\text{the}}\;{\text{amount}}\;{\text{of}}\;{\text{oxygen}}\;{\text{for}}\;{\text{the}}\;{\text{population}}\;B\}$

  2. Let $S(t) = 3.75{t^2} + 500$ be a function, where $S$ is the salary for the number of units sold $t$. Find the value of $S$ at $t = 4 - \sqrt {14} $.

    Erroneous Solution

    \begin{align*} S(4 - \sqrt {14} ) & = 3.75{(4 - \sqrt {14} )^2} + 500 \\ & = 3.75(16 - 14) + 500 \\ & = 507.50 \end{align*}

  3. Find the domain of each of the following functions.

    1. $g(t) = \displaystyle{\frac{4 - t}{6 - \sqrt{t^2 - 9}}}$
    2. $h(x) = \tan(2x)$.

    Erroneous Solutions

    1. We need $t^2 -9 \gt 0$; $t^2 \gt 9$; so $t \gt 3$ and $t \lt -3$. We also want $4 - t \neq 0$, so $t \neq 4$; therefore, the domain in interval notation is the union of the intervals, $(-\infty,-3)$, $(3,4)$ and $(4,\infty)$.

    2. Since $\tan(u)$ is undefined for

      $u = \dfrac{k\pi}{2}$,

      we need

      $2x \neq \dfrac{k\pi}{2}$, so $x \neq \dfrac{k\pi}{4}$,

      the domain is all numbers except $\displaystyle{\frac{k\pi}{4}}$ for any integer $k$.

  4. Sketch the graph of a single function that satisfies all of the conditions listed below.

    • Domain is $[3,12]$.
    • The pairs $(4,5)$ and $(5,4)$ are in the function.
    • The function is constant and equal to $1$ over the interval $(8,12]$.

    Erroneous Solution

    Figure 2.38. Erroneous solution to “Learning from Mistakes” Question 4.

  5. If the graph of the function $f$ is shown below, give a sketch of the following functions.

    Figure 2.39. Graph for “Learning from Mistakes” Question 5.

    1. $F(x) = 2f(x) - 1$
    2. $F(x) = f(x + 2) +3$.

    Erroneous Solutions

    1. The function $F$ is a vertical stretch by a factor of $2$ and a shift down of one unit.

      Figure 2.40. Erroneous solution to “Learning from Mistakes” Question 5(a).

    2. The function $F$ is shift to the right by $2$ units and a shift up by $3$ units.

      Figure 2.41. Erroneous solution to “Learning from Mistakes” Question 5(b).