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.

Macy enjoys spending time with her friends. They talk about a lot of different things, from hobbies to current events and more. However, she feels like her friends don’t really know the real her, and she feels like the friendships are somewhat superficial. According to Social Penetration Theory, Macy’s relationships likely have [depth] depth of penetration and [breadth] breadth of penetration.

Jasmine dislikes going to the theater, but her partner, Cecelia, loves it. Cecelia tries to convince Jasmine to like the theater by saying “It’s the best thing ever, you will love it!” According to Social Judgement Theory, this message is likely to fall in the latitude of [latitude] and result in [contrast].

9. A retail company uses a perpetual inventory system that continuously updates its inventory records after purchases and sales. Management is deciding whether the company should still perform a physical inventory count at year-end. What is the primary reason for performing the physical count? 1. To determine the selling price of every inventory item 2. To replace the perpetual inventory records 3. To identify shrinkage, damage, and recording errors 4. To determine which inventory method is legally required 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.

5. Martin Supply uses a perpetual inventory system and the moving weighted average method. Martin has 40 units that cost $10 each. Martin then purchases 60 units that cost $14 each. After the purchase, Martin calculates a new average cost per unit by dividing the total cost of the 100 units by 100 units. Martin then sells 50 units. What is Martin Supply’s cost of goods sold? 1. $500 2. $600 3. $620 4. $700 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.