# Jorge Basilio - PCC%md# Calculus II - Spring 2019## Instructor: *Dr. Basilio*# Lab 0## Introduction to Sage<span style="font-size:18pt; color:red">Due: Tuesday, April 9 by 11:59pm via Canvas</span>## Objectives1. Learn what `SageMath` is and why you are asked to learn it2. Become familiar with performing basic arithmetic calculations with `SageMath`3. Learn how to calculate: limits, derivatives, integrals with `SageMath`4. Learn how to make a PDF of your work to save and submit# 0. Introduction## Why use SageMath?- It's free!- Using it on web is free (hassle-free, no messy downloading needed)- Dowloading it is free (if you want to use it without internet)- It's open source!- No hidden algorithms (you can look at source code if you wish--and look "under the hood" so to speak)- It will prepare you for the future!    - You will likely need to learn some basic programming no matter what you study in college    - In Pyschology, for example, it's common to learn Python (and the R package) when learning statistics# 1. Some reading assignments:## - Read: [Why use SageMath?](https://github.com/sagemathinc/cocalc/wiki/SageInCalculus)> *An excerpt:*> > #### Why not just use graphing calculators?> > Back in their day, graphing calculators were rather popular. I used one in high school in the 1990s.> > - Let's say you're working with a large data set. Using a system like Sage, a professor can upload data into a project, and distribute it to all of his students with a click. With a graphing calculator, the data would have to be entered by each student, by hand.> > - Students can take the images and outputs of their computations in Sage and easily add them to any document for their classes, or undergraduate research papers suitable for publication. A picture on a graphing calculator is idle and immovable.> > - The appearance of graphs and 3D plots on a computer is vastly more realistic and comprehensible than the display of a graphing calculator.> > - The "online help" systems available (such as web-pages) such as Sage's wiki can be a tremendous boon to the student who is new to Sage. A graphing calculator just has a manual.> > - Many faculty working with Sage have made online videos, to help new students learn Sage.> > - If a student learns Sage, then the student learns Python "along the way." Python is an extremely popular computer programming language, used in industry. (Ranked #3 in this article published in IEEE Spectrum.)## - Read: [Getting Started with SageMath](https://mosullivan.sdsu.edu/sagetutorial/about.html#getting-started)> - Read: **About Sage**> - Read: **Sage as a Calculator**> - Read: Section: Arithmetic and Functions>      1. Basic Arithmetic>      2. Integer Division and Factoring>      3. Standard Functions and Constants## - Reference: [Sage for Undergraduates](http://www.gregorybard.com/Sage.html)> - Reference: this site has an entire book you can download for more information# 2. Getting Started1. Make an account at the [CoCalc website](https://cocalc.com/) page so you can **login** to the free Sage server. No nosy questions, just make up a username and set a password. Just be sure to use a modern web browser (Google Chrome, Mozilla Firefox, etc).2. Create a new **Sage Worksheet** and begin experimenting, ORUpload worksheets using the "Upload" link in the upper-left corner, ORIf you are making a worksheet to submit an assignment, give the worksheet a title like **"Math5B-Lab_0– YourLastName_YourFirstName-S19"**.3. At the beginning of your worksheet include course info and lab info by using **comments** (see the template you can follow)    - By looking at this document, you are encouraged to copy and paste lines of code and modify them :-)4. Have some fun and make a few calculations# 3. Handy Short Cuts- `Command+Enter` (on Mac) or `Shift+Enter` (on PC) runs the code and gives you an output# 4. Quick Examples

# this a SageMath worksheet# comments are designated by a hashtag and are not read by the program# how to add; hit `Run` when you are ready to tell SageMath to compute2+2

# evaluating a functionf(x) = x^2 # defines f; note the notation is similar to math notation!f(-2)

# graphing a function# this will graph f(x)=x^2 with x in (-2,2)plot(x^2,x,-2,2)

# limitsf(x)=x*e^(-x) # note: must use * to multiplylimit(f,x=ln(2))

# another limitf(x)=1/xlimit(f,x=0)

# notice we have to be careful in the above limit!# the above example doesn't give us the expected 'DNE' (Recall: LHL is -Infinity whereas RHL is +Infinity)# to do a one sided limit:f(x)=1/xlimit(f, x=0, dir="left") # this means `approaches from the left`

# and the RHLf(x)=1/xlimit(f,x=0, dir="right")

# derivativef(x)=x^3*e^(sqrt(x))derivative(f,x) # 'd(f,x)' is basically 'df/dx'

# the above is correct but not easy to look at# we can use show() function to make the output look prettierf(x)=x^3*e^(sqrt(x))show(derivative(f,x))

# better way to use show()f(x)=x^3*e^(sqrt(x))g=derivative(f,x)show(g)

# anti-derivatives or indefinite integralsintegral(x^3,x) # integral(f,x) the ",x" is like our "dx"

# definite integralintegral(x^3*sin(x),x,-1,1)

# a hard integralI=integrate(x^4*exp(x),x)show(I)

# variables have to be declared (x is predefined for convenience)var('t')show(plot(sin(t),(t,-1,4*pi)), axes_labels = ['t','P']) # t for time and P for population

# Assignment for Lab_0%md# 5. Assignment1. Evaluate: $\cos(\pi/8)$, $e^8$, $\sqrt{8}$, $\ln(8)$2. Now, have Sage compute numerical approximations for the expression in \#1. *(Consult the reading from above "Getting Started with Sage")*3. Compute $e$ to one hundred decimal places.4. Let $f(x)=x^3+7^x-\ln(e^{x^2}\cdot \sqrt{x})$.    a. Compute: $\lim_{x \to 1} f(x)$.    b. Compute: $f'(x)$.    c. Compute: $\int f(x) dx$.    d. Verify the Fundamental Theorem of Calculus by defining $F(x)=\int f(x)dx$ and checking that $F'(x)=f(x)$.    e. Compute: $\int_2^8 f(x) dx$.    f. Plot $f$ for $x \in (2,8)$.    g. Make your plot prettier by labeling the $x$ and $f(x)$ axes appropriately.5. Download your Sage worksheet *(extension .sagews)* and submit it via Canvas.6. Submit a pdf of your lab via Canvas.**Note:** Use comments to indicate which problem you are working on (see template)

# printing PDF%md# 6. How to create a PDFWhen you have completed the assignment, download the worksheet and also a PDF version of your file and submit both files via Canvas. To accomplish this, select `Print` (*You'll see a prompt that says "Conver to HTML") then select `Download`. A pop-up will give you a PDF of your worksheet. You may now downnload and save it to your computer. As before, give your file the name **"Math5B-Lab_0– YourLastName_YourFirstName-S19.pdf"**.**Note \#1:** Do **not** use `Convert to PDF` *(try it if your curious but the formatting is not as nice looking as when it is converted to html then printed)***Note \#2:** Do **not** use `Ctrl+P`. It doesn't seem to print the entire document.

# Bonus%md# 7. Bonus Section (Optional): Python`SageMath` is built using the `Python` programming language.In fact, you can type `Python` code directly on Sage Worksheets without any special instructions.Below, I'll give a few simple examples for those of you who are  curious ;-)

# example of python codeprint("Hello World!")

# example of python code# defining a more complicated functiondef f(x):    if -1 <= x <=0:        return -x    elif 0 < x < 0.5:        return x    else:        return 0.5f(-0.1)

f(0)

f(0.1)

#mix and match; use sage to plot the function f defined with pythonplot(f)