I have read and understand the course’s Generative AI policy. I understand that generative AI is a professional tool that may be used only as permitted by the course policy. I also understand that using AI in unauthorized ways, including during quizzes or exams, submitting AI-generated work as my own, or using AI to avoid engaging in the learning process, violates the course’s academic integrity expectations.  

Set up the integral that represents the length of the curve \(x=e^{t}-t, \qquad y=4e^{t/2}, \qquad 0\le t\le 2.\)

Final Submission Before picking up your phone, make sure that: You have completed every multiple-choice question. You have completed all free-response questions. You have finished all scratch work. You have written your name on each page at the top right corner. You will not make any further changes to any answer. Picking up your phone begins the final submission process. Once you pick it up, you may not return to the exam questions, revise any answer, or add anything to your handwritten work. 1. Show your work to the webcam Put all pages you wrote on in the correct order, including scratch work. Your free response solutions should be first, with any scratch work afterwards. Before picking up your phone: State aloud: “I have finished the exam and am now beginning the scanning and submission process.” Hold each completed page clearly in front of the webcam, one page at a time. Make sure the entire page is visible. 2. Scan your written work to a PDF You may now use your phone only to scan, transmit, and upload your completed work. iPhone or iPad: Open the built-in Files app, tap the three-dot menu, and select Scan Documents. Android: Open Files by Google and select Scan. You may alternatively use Google Drive or another PDF-scanning app. Scan every page into one PDF. Make sure that: Every page is included. The pages are in the correct order. Every page is readable. No page is sideways or upside down. You may not use your phone to access Canvas, view exam questions, search for information, communicate with another person, access course materials, or use any calculator, AI tool, or other unauthorized resource. 3. Email the PDF to yourself Using your UC email account on your phone, email the PDF to your own UC email address, and CC me at gruberan@ucmail.uc.edu. Use the subject line “Midterm 3 – Last Name, First Name”. The emailed copy provides a timestamped backup, but emailing the PDF does not complete your submission. The PDF uploaded to this Canvas question is your official submission. 4. Retrieve the PDF on your exam computer On the computer running Honorlock, navigate to UC webmail. Open only the message containing the PDF you just sent. Download the attached PDF to your computer. Return to this Canvas question. Do not read, search, or open any other emails or attachments. 5. Upload and submit Upload the PDF to this question, and submit the quiz.

Which of the graphs of the following parametric equations are circles? I. \(x(t)=2+3\sin(t)\) \(y(t)=-1+3\cos(t)\) II. \(x(t)=\cos(2t)\) \(y(t)=\sin(2t)\) III. \(x(t)=-1+1\cos(t)\) \(y(t)=2+3\sin(t)\) IV. \(x(t)=9+\cos(t)\) \(y(t)=6+\cos(t)\)

Compute the 15th derivative \(f^{(15)}(0)\) of the function \(f(x)=e^{-x^5}\).

Use a known Maclaurin series to find the sum of the following series. \(\displaystyle 3-\frac{\pi^2}{2!\cdot 3}+\frac{\pi^4}{4!\cdot 3^3}-\frac{\pi^6}{6!\cdot 3^5}+\frac{\pi^8}{8!\cdot 3^7}+\cdots\)

Free Response  Work all four questions below on your own paper, showing all steps. Label each question and part clearly. Show all work, including all key steps. Answers without supporting work receive no credit. Your work must be legible enough for me to read and follow your reasoning. Use arrows or other notation if needed to make the order of your work clear. FR1. (20 points) For each series below, name the convergence test you will use, show your work, and conclude whether the series diverges, converges absolutely, or converges conditionally. (a) (5 points) \(\displaystyle\sum_{n=1}^{\infty}\frac{3^n}{n!}\) (b) (7 points) \(\displaystyle\sum_{n=1}^{\infty}\frac{n+1}{n^3+2}\) (c) (8 points) \(\displaystyle\sum_{n=1}^{\infty}(-1)^n\,\frac{n}{n^2+1}\) FR2. (8 points) (a) (5 points) Find the third degree Taylor polynomial for \(f(x)=\sin x\) centered at \(a=\dfrac{\pi}{6}\). (b) (3 points) Explain why you cannot use the Alternating Series Estimation Theorem to find the maximum possible error of using your answer from part (a) to approximate \(\sin 1\). FR3. (12 points) (a) (5 points) Find the power series representation for \(f(x)=\dfrac{1}{6-x}\) centered at \(a=0\). (b) (5 points) Use your answer from part (a) to find the power series representation for \(g(x)=\dfrac{1}{(6-x)^2}\) centered at \(a=0\). (c) (2 points) Use your answer from part (b) to find the power series representation for \(h(x)=\dfrac{x^2}{(6-x)^2}\) centered at \(a=0\). FR4. (10 points) Consider the curve given by the parametric equations \(x(t)=t^2-4t, \qquad y(t)=t^3-3t.\) (a) (5 points) Find all values of \(t\) at which the curve has a vertical tangent line, and give an equation for each such line. (b) (5 points) Find all values of \(t\) at which the curve has a horizontal tangent line, and give an equation for each such line.

Find the sum of the series \(\displaystyle -\frac12+\frac13-\frac14+\cdots=\sum_{n=2}^{\infty}\frac{(-1)^{n+1}}{n}.\)

April was happily surprised by fancy latte art when she ordered at a cafe. However, she found the barista who made the art a little weird and unlikable. According to Expectancy Violation Theory, April likely has [violation] violation valence and [communicator] communicator reward valence.

41. Franklin Company reports the following amounts: Cash: $20,000Short-term investments: $10,000Accounts Receivable: $30,000Inventory: $50,000Current liabilities: $40,000 For this question, quick assets include cash, short-term investments, and Accounts Receivable. Inventory is excluded. Use the following formula: Quick ratio = Quick assets ÷ Current liabilities What is Franklin Company’s quick ratio? 1. 1.00 2. 1.25 3. 1.50 4. 2.75 Instructions to students: Type in the number of the answer of your choice (type in either 1, 2, 3, or 4). Do not type in a decimal after inputting the number.