Take-home Assessment 3

MAT 135 – Calculus Take-Home Assessment 3 30 points; due at 6:00 PM on Tuesday, 23 June 2009 Reminders: • This and all Take-Home Assessments are to be completely individually. As stated in the syllabus policy on academic honesty, any collaboration with others on this assignment must take place only in the initial stages of the work. All substantive work on solutions must be your own, and you may be subject to a random oral examination on your work if it doesnt appear that you have done the work to the point that you understand it. • You may use technology on this assignment; indeed some questions below require the use of technology. But you must always show a completed solution for each problem unless it says to state an answer. • You may hand-write your mathematical work, but a typed-up version is preferred. If you hand-write your work, make it neat, organized, and easy to read, and leave all margins empty for comments. You may lose points if you do not do these things. • For all work requiring technology (i.e. Winplot or Excel work), please hand in a printout of your work AND email the corresponding file to the professor.

√ 1. [6 points] Use the limit-based definition of the derivative to calculate the derivative of y = x. Do NOT use any derivative rules developed in Chapter 3. (Hint: The answer, as indicated 1 in an example in class, is y 0 = √ .) 2 x 2. [10 points] A home builder began construction of the first house in a new subdivision 15 months ago. Since then, the number of houses built in the subdivision has grown according to this graph: 350 320 300 280 260 Number of homes

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This is an example of what’s called logistic growth, characterized by slow growth in the beginning that reaches a very rapid pace and then slows down. Logistic growth models the 1

real-life behavior of many things, particularly populations in restricted environments – such as homeowners in a subdivision with only a limited number of home sites available. Using this graph, particularly the grid markings, and some hand-drawn tangent lines, make an accurate sketch of the derivative of this function. Show all work you use in making the graph. Then, write a 3-5 page informative paragraph aimed at a typical home-buying family (who do not know calculus) that uses the derivative to address the following questions: • How has the builder’s activity changed over the last 15 months? • When was the builder the busiest? And how do you know that? • Can the family expect a lot of construction traffic through their neighborhood in the coming months after they move in? 3. In Take-Home Assessment 1, you looked at some data for an airplane’s altitude (a, in feet) as a function of time (t, in minutes) during the first 30 minutes of its flight. Please refer to the table found on the handout for that Assessment (posted on Angel if you don’t have a paper copy any more). (a) [4 points] What was the average rate of change in the airplane’s altitude from 10 minutes into the flight to 20 minutes into the flight? What was the average rate of change in the airplane’s altitude from 7 minutes to 10 minutes? (b) [6 points] Suppose we wanted to find the value of a0 (10), the derivative of the altitude function at time t = 10. State the units of measurement of this quantity. Then, explain why we cannot compute this value exactly; and then devise a plan for how you might use the previous part of this question to approximate the value of a0 (10). (c) [4 points] Use your answer from the previous part of this question and an Excel spreadsheet to construct a table for a0 (t), the derivative of the altitude function. Make sure to explain how you got your table entries.

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