MAT 125: Calculus 2

Fall 2021

hw03

  1. Read textbook sections 6.1 and 6.2. Always pre-read the section(s) to be covered the next class. I will not explicitly list this starting next homework.

  2. Read the guidelines for written homework. Follow it for hw03a.

  3. hw03a: Chapter 5 review (p.310) # 3, 4, 19, 34, 41ac*, 54.

    Note for 41c: set up an integral to answer the question, then evaluate it on Mathematica (Mma) and find an estimate with one decimal place of accuracy to answer the question. Give units with the answer.

    General note on Mathematica (Mma) portion in written homework: Unless otherwise specified, from now on, Mma portion of written homework is to be saved as PDF (not left as .nb files) and uploaded to Moodle. (You will be told when you should upload .nb files; otherwise, convert to PDF.) You will also be told when you can use Mma (otherwise do problems by hand). So for example, in this HW, you must use Mma on #41, and should convert to PDF, but you should do the other problems by hand.

  4. hw03b: Mathematica review. Before you begin this part, make sure you have completed hw02 (Mathematica installation, Intro Lab, and Hands-on Start Video) and the in-class Crash Course.

    Open a new notebook, save it as yourlastnameHW03b.nb. Put your name, PO number, and homework number at the top. Use "subtitle" format for this cell.

    Complete the following in your notebook. Label each problem with the corresponding letter (sections or subsections are nice formatting for this) and include any necessary explanation in a text cell with each problem.

    1. Enter the function, $h(y) = \sin^3(y^2 + e) – 0.3$.

      Yes, let $y$ be the independent variable: $x$ doesn't have to be the independent variable! Use good Mathematica functional notation: the cubic part may take some thought; think about a different way to notate the cubic part. Hint: it won't go right next to the "$\sin$" part.

    2. Produce a graph of the function defined in part (a), for $0 \leq y \leq 2\pi/3$.

    3. It should look like a minimum occurs around $y = 1.5$. Find the approximate $y$ and $h(y)$ for this minimum. Give $y$ to at least 6 decimal places.

      Then, using Mathematica, prove that this indeed is a minimum (as opposed to a maximum). Use a Calculus test, don't just look at the plot.

      Include some text at the end of this section to explain your work and give the coordinates of the point. Do not use the built-in FindMinimum or FindMinValue commands, as we want to see that you understand Calculus. Instead, use Solve, NSolve, or FindRoot.

      If you use FindRoot in this homework, use the WorkingPrecision option with 12 decimal places. See the command sheet found in the MmaIntro folder.

      Our goal for the final parts is to find the magnitude of the area between the graph of $h(y)$, the $y$-axis, the line where $y=0$, and the place where the curve crosses the $y$-axis around $2$.

    4. Find where the function crosses the $y$-axis for $2 \leq y \leq 2.25$. Find a numerical value good to at least 5 decimal places for the solution. Don't just "zoom in" with a plot; use a Mathematica command.

    5. Find the area under the curve described above, for $y$ between zero and the value you found in part (d).

      Give a picture of the region we are considering (no "fill" necessary, you can just plot the function on the interval), and discuss whether or not your answer for the area makes sense by estimating the area:

      Click on your plot, then press Ctrl-D to bring up drawing tools. Draw geometric region(s) on top of your plot to show the geometric figures you are using to estimate the area. (This estimate can be very rough: just use a few regions like rectangles and triangles.)

      Write some text to talk about your exact value and estimate.

    Save your notebook to your personal folder on the network or to your computer and then upload the .nb file to Moodle assignment hw03 with the correct file name as instructed above.

  5. Jamboard. In class on Day 3 we are going to do some drawing. We'll use Google's Jamboard for this. Please install the Jamboard app on your smartphone if you can. (The app is easier to use than the web interface.) I'll share a Jamboard with you that you can use as a test.
To submit electronically on Moodle before coming to class on Day 3: