\( \newcommand{\lt}{ < } \newcommand{\gt}{ > } \newcommand{\amp}{ & } \)

Section9Written Assignments

As the term progresses, more written assignements will be added to the list here. These must be turned in no later than their posted due dates. Written assignments are each worth 1 % of your final grade, and there will be nine in total. These are where you will get to show me your mathematical writing skills, and improve them if necessary.

The most important thing to understand about your written homework is that you are not merely finding and giving answers. You are explaining to someone what the problem was, how you handled it, and summarizing the results. In general, your write-ups should respect these “five C's”:

  1. Clarity. Am I able to read your work? Is your writing legible? Have you written complete sentences that make sense, where appropriate? If you have charts, graphs, and mathematical expressions, have you clearly indicated what they represent? Generally, any graphs need their axes labeled with regularly spaced tick marks, and they need some kind of overall title. If something about your write-up causes me to pause and wonder what you mean, you will lose clarity points.

  2. Correctness. Are your numerical answers and conclusions accurate? Often you can check that your numbers are correct by substituting numbers back into earlier equations. Also, by asking yourself if the numbers that you find make sense in the context of the problem. It's not a bad idea to include these checks with your work.

  3. Conciseness. Have you rambled on with extra content that is not relevant to solving the problem? This is distracting and hurts the overall ability of your write-up to communicate effectively.

  4. self-Containment. The response to the question that you submit should make it clear what the original question was. Put yourself in the shoes of another student from the class who does not have the original homework assignment in front of them. Would they be able to understand what you are talking about? At a minimum, this means that you must give an introduction of some kind that lets your reader know what you are about to investigate. Also, include any and all charts, graphs, and mathematical expressions that were given in the problem, and explain their meaning, even if you do so merely with labels.

  5. Conclusions. Some problems come with context (“word problems”). When writing your final answer to such a question, you need to put that answer in its full context with a conclusion statement that is a complete English sentence. As an example, writing “\(x=70\)” or “He needs $70” will not be good enough if there is more context to the problem. Instead, something like “Dmitri needs $70 in order to purchase a new lawnmower” is a conclusion statement with all of the context in it. For problems without context, conclusion statements like “So the domain of \(f\) is \([0,\infty)\).” are acceptable.

  • Written HW 1

    Due Monday April 4 or Tuesday April 5

    1. Define a function \(z\) by \(z(x)=2x^2+2x+4\). Find and simplify the expression

      \begin{equation*}\frac{z(3+x)-z(3)}{x}\text{.}\end{equation*}

    2. Find the domain of the function \(g\) where \(g(x)=\sqrt{5+6x}\).

  • Written HW 2

    Due Monday April 11 or Tuesday April 12

    1. Let the function \(f\) be defined by

      \begin{equation*}f(x)=\frac{1}{x},\quad x\text{ in }[-3,0)\cup(0,2]\text{.}\end{equation*}
      What is the range of this function? It will help if you make a graph of this function with a restricted domain.

    2. Determine whether each of the following rational functions is even, odd, or neither. Provide a complete explanation.

      1. \(f(x)=\frac{4x^2+7}{7x^8+3x^2}\)

      2. \(g(x)=\frac{x^3+4x^2}{7x^2+1}\)

      3. \(h(x)=\frac{4x^3+2x}{7x^4}\)

  • Written HW 3

    Due Monday April 18 or Tuesday April 19

    1. Here is a graph of a function \(f\).

      <<SVG image is unavailable, or your browser cannot render it>>

      Write a formula for \(f\), using the notation for piecewise-defined functions.

    2. Suppose that

      \begin{equation*}f(x)=\begin{cases}\left\lvert x\right\rvert \amp -\infty\lt x\lt 2\\-x^2+4x+3\amp x\geq2\end{cases}\end{equation*}
      Solve the equation \(f(x)=3\).

  • Written HW 4

    Due Wednesday April 27 or Thursday April 28

    1. Relative to the graph of \(y=\sqrt{x}\), the graphs of the following equations have been changed in what way?

      1. \(y=\sqrt{\frac{x}{5}}\)

      2. \(y=\sqrt{x}+5\)

      3. \(y=\frac{\sqrt{x}}{5}\)

      4. \(y=\sqrt{x+5}\)

    2. Using only what you know about the graph of \(y=\sqrt{x}\) and graph transformations, plot a graph of \(f\), where \(f(x)=4\sqrt{2x-6}+3\). It would be good to check your result by computing outputs from \(f\) in two ways: using this formula and using your graph. Note: simply plotting points using the formula will not be considered acceptable; you must demonstrate that you understand the graph transformations suggested by the formula.

  • Written HW 5

    Due Monday May 2 or Tuesday May 3

    1. By hand, on graph paper, graph each of the functions defined below. Each graph should have all of the following identified and labeled, assuming the graph has these in the first place:

      • zeros (also known as horizontal intercepts)

      • the \(y\)-intercept

      • the local behavior of the curve near any horizontal intercept

      • the long term behavior of the curve should be clear

      1. \(f(x)=(x-3)(x-5)(x+1)\)

      2. \(g(x)=(2x+3)^3(x-2)\)

  • Written HW 6

    Due Monday May 9 or Tuesday May 10

    1. By hand, on graph paper, graph each of the functions defined below. Each graph should have all of the following identified and labeled, assuming the graph has these in the first place:

      • zeros (also known as horizontal intercepts)

      • the \(y\)-intercept

      • vertical asymptotes

      • the local behavior of the curve near any horizontal intercept

      • the local behavior near any vertical asymptote should be clear (does the curve approach the asymptote curving upward/downward from the left/right?)

      • the long term behavior of the curve should be clear; if it is a horizontal or slanted asymptote, it should be labeled with its equation

      1. \(h(x)=\frac{x^2-3x-4}{x-2}\)

      2. \(k(x)=\frac{3(x+2)(x-1)^2}{(x+3)(x-4)^2}\)

  • Written HW 7

    Due Wednesday May 18 or Tuesday May 24

    1. Let \(f(x)=\frac{4x+7}{9x+5}\). Find a formula for \(f^{-1}\).

    2. Some sociologists feel that if \(x\)is a person's age, then \(f(x)=\frac{1}{2}x+7\) gives the youngest age for which it is socially acceptable to date someone of that age. 1 

      1. What does it mean in context to say that \(f(20)=17\)?

      2. In context, what should be the domain of this \(f\)? (According to this model anyway.) Hint: how young must a person be so that it is only “acceptable” for them to be dating people of their own age?

      3. In words, \(f\) cuts its input in half and then adds \(7\). Therefore, in words, what should \(f^{-1}\) do? Use this to write a formula for \(f^{-1}\).

      4. Write a sentence that explains what \(f^{-1}(x)\) means for a person who is \(x\) years old.

      5. So what does it mean to say \(f^{-1}(40)=66\)?

  • Written HW 8

    Due Wednesday May 25 or Tuesday May 31

    1. The amount of a certain drug (in mg) in the body \(t\) hours after taking a pill is given by \(A(t) = 22 (0.87)^t\).

      1. What is the dose in the pill?

      2. What proportion of the drug leaves the body each hour?

      3. What is the amount left in the body after 10 h

      4. How long will it be until there less than 1 mg left in the body? (you may use technology to assist you)

    2. Solve the equation by using logarithms.

      \begin{equation*}96(1.215)^x = 36(1.555)^x\end{equation*}

  • Written HW 9

    Due the day of your Final Exam

    1. Convert this exponential expression to the form \(Q = a\,b^t\).

      \begin{equation*}Q = 0.56e^{−0.2t}\end{equation*}

    2. A radioactive substance decays at a continuous rate of -21.5 % per year, and 72 mg of the substance is present at the start of 2016.

      1. Find a formula for \(A(t)\), the amount present \(t\) years after 2016.

      2. How much will be present in the year 2020?

      3. When will the amount drop below 7.2 mg? (Show steps with logarithms)