Mathematics Finding Local Minimum and Maximum of A Function
Data 100, Fall 2023 Homework 1B Due Date: Thursday, August 31, 11:59 PM Submission Instructions You must submit this assignment to Gradescope by the on-time deadline, Thursday, August 31, 11:59 PM. Please read the syllabus for the grace period policy. No late submissions beyond the grace period will be accepted. While course staff is happy to help you if you encounter difficulties with submission, we may not be able to respond to lastminute requests for assistance (TAs need to sleep, after all!). We strongly encourage you to plan to submit your work to Gradescope several hours before the stated deadline. This way, you will have ample time to contact staff for submission support. There are three parts to this assignment listed on Gradescope: • Homework 01 Coding: Submit your Jupyter Notebook zip file for Homework 1A, which can be generated and downloaded from DataHub using the grader.export() cell provided. • Homework 01 Written: Submit a single PDF to Gradescope that contains both (1) your answers to all manually graded questions from the Homework 1A Jupyter Notebook and (2) your answers to all questions in this Homework 1B document. • Syllabus Quiz: The assignment is multiple-choice style on Gradescope. You may change or update your answers anytime before the deadline. To receive credit on this assignment, you must submit both your coding and written portions to their respective Gradescope portals as well as the syllabus quiz. Your written submission (a single PDF) can be generated as follows: 1. Access your answers to manually graded Homework 1A questions in one of three ways: • Automatically create PDF (recommended): We have provided a cell to generate your written response in the Homework 1A notebook for you. Run the cell and click to download the generated PDF. This function will extract your response to the manually graded questions and put them on separate pages. This process may fail if your answer is not properly formatted; if this is the case, check out common errors and solution described on Ed or follow either of the two ways described below. • Manually download PDF : If there are issues with automatically generating the PDF, on DataHub, you can try downloading the pdf by clicking on File->Save andExportNotebookAs…->PDF. If you choose this route, you must take special care to ensure all appropriate pages are chosen for each question on Gradescope. 1 Homework 1B 2 • Take screenshots: If that doesn’t work either, you can take screenshots of your answers (and your code if present) to manually graded questions and include them as images in a PDF. The manually graded questions are listed at the top of the Homework 1A notebook. 2. Answer the below Homework 1B written questions in one of many ways: • Type your answer. We recommend LaTeX, the math typesetting language. Overleaf is a great tool to type in LaTeX. • Download this PDF, print it out, and write directly on these pages. If you have a tablet, you may save this PDF and write directly on it. • Write your answers on a blank sheet of physical or digital paper. • Note: If you write your answers on physical paper, use a scanning application (e.g., CamScanner, Apple Notes) to generate a PDF. 3. Combine these two sets of answers together into the same PDF and submit to the appropriate Gradescope written portal. You can use PDF merging tools, e.g., Adobe Reader, Smallpdf (https://smallpdf.com/merge-pdf) or Apple Preview (https: //support.apple.com/en-us/HT202945). 4. Important: When submitting on Gradescope, you must tag pages to each question correctly (it prompts you to do this after submitting your work). This significantly streamlines the grading process for our readers. Failure to do this may result in a score of 0 for untagged questions. You are responsible for ensuring your submission follows our requirements. We will not be granting regrade requests nor extensions to submissions that don’t follow instructions. If you encounter any difficulties with submission, please don’t hesitate to contact staff before the deadline. Collaboration Policy Data science is a collaborative activity. While you may talk with others about the homework, we ask that you write your solutions individually. If you discuss the assignments with others, please include their names at the top of your submission. Homework 1B 3 Homework 1A Manually Graded Questions 0. This is not a question. This is a reminder to include your Homework 1A manually graded questions (automatically generated into a PDF) in your single written PDF submission to Gradescope. Calculus and Algebra x2 + x + 1 . Find the local minimum and maximum point(s) of x f (x) and write them in the form (a, b). 1. (3 points) Let f (x) = Homework 1B 4 3 2. (3 points) Let g(x, y, z) = 4×3 y 2 − e2x ln(z) + yz 2 + x−ln(y) . x−z (a) Holding all other variables constant, take the partial derivative of g(x, y, z) with ∂ respect to x, ∂x g(x, y, z). (b) Holding all other variables constant, take the partial derivative of g(x, y, z) with ∂ g(x, y, z). respect to y, ∂y Homework 1B 5 Probability and Statistics 3. (3 points) Much of data analysis involves interpreting proportions – lots and lots of related proportions. So let’s recall the basics. It might help to start by reviewing the main rules from Data 8 (Chapter 9.5), with particular attention to what’s being multiplied in the multiplication rule. For this question, assume we have a bag filled with 14 marbles: 6 blue marbles, 3 red marbles, and 5 yellow marbles. (a) Yash selects 2 marbles without replacement. What’s the probability that the selected marbles have the same color? You can leave your solution in equation form. (b) Yash selects 4 marbles with replacement. What’s the probability that he sees at least 1 blue marble? Homework 1B 6 Content Warning: This question includes discussion about cancer. If you feel uncomfortable with this topic, please contact your GSI or the instructors. 4. (3 points) Consider the following scenario: Only 1% of 40-year-old women who participate in a routine mammography test have breast cancer. 80% of women who have breast cancer will test positive, but 9.6% of women who don’t have breast cancer will also get positive tests. Suppose we know that a woman of this age tested positive in a routine screening. What is the probability that she actually has breast cancer? (Note: You must show all of your work, and also simplify your final answer to 3 decimal places.) Hint: Data 8 Chapter 18.2. Homework 1B 7 Linear Algebra 5. (6 points) A common representation of data uses matrices and vectors, so it is helpful to familiarize ourselves with linear algebra notation, as well as some simple operations. Define a vector ⃗v to be a column vector. Then, the following properties hold: • c⃗v with c some constant, is equal to a new vector where every element in c⃗v is equal 1 2 to the corresponding element in ⃗v multiplied by c. For example, 2 = . 2 4 • ⃗v1 + ⃗v2 is equal to a new vector with elements equal to the elementwise addition of 1 −3 −2 ⃗v1 and ⃗v2 . For example, + = . 2 4 6 The above properties form our definition for a linear combination of vectors. ⃗v3 is a linear combination of ⃗v1 and ⃗v2 if ⃗v3 = a⃗v1 + b⃗v2 , where a and b are some constants. Oftentimes, we stack column vectors to form a matrix. Define the column rank of a matrix A to be equal to the maximal number of linearly independent columns in A. A set of columns is linearly independent if no column can be written as a linear combination of any other column(s) within the set. If all columns in a matrix are linearly independent, it means that the matrix is full column rank. For example, let A be a matrix with 4 columns. If three of these columns are linearly independent, but the fourth can be written as a linear combination of the other three, then rank(A) = 3. Alternatively, if all four columns of A were linearly independent, rank(A) = 4., and A would be full column rank. For each of the following matrices, state the rank of the matrix and whether or not the matrix is full column rank. If the matrix is not full column rank, state that it is not full column rank and give a linear relationship among the vectors—for example: ⃗v1 = ⃗v2 . | | 1 1 (a) ⃗v1 = , ⃗v2 = , A = ⃗v1 ⃗v2 0 1 | | Homework 1B 8 | | 3 0 (b) ⃗v1 = , ⃗v2 = , B = ⃗v1 ⃗v2 −4 0 | | | | | 0 5 10 (c) ⃗v1 = , ⃗v2 = , ⃗v3 = , C = ⃗v1 ⃗v2 ⃗v3 1 0 10 | | | 0 −2 2 | | | (d) ⃗v1 = 2, ⃗v2 = −2, ⃗v3 = 4 , D = ⃗v1 ⃗v2 ⃗v3 3 5 −2 | | |
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