*** Mac users using the Remote Desktop to access Scientific Notebook need to bring an Ethernet cord to the eam and use it to connect to the internet. That is, you should not connect to the internet using a wireless connection. *** Topics for Eam 2 You will be allowed to use a new blank SN file ONLY during the eam so bring your laptop. All other programs should be closed and your wireless should be turned off. CALCULATORS ARE NOT ALLOWED. The eam will cover sections 2.6 (Rates of Change) and.1-4.1. Format The eam will be out of 100 points and will consist of two parts. The first part will be done with paper and pencil only, and the second part will be completed using Scientific Notebook. For Part I, I will ask you to compute some derivatives (including higher-order) using the rules (or shortcuts) to differentiation from sections.1,.2,., and.5. Part I will roughly be worth 0 out of 100 points and you will have 0 minutes to complete it. Note: you can start Part II of the eam as soon as you finish Part I. There will be questions on the eam that "require" you to use SN (in other words to do them by hand would be hard and/or long) so make sure you know how to use the program! In general, you may use any feature/tool of SN when answering the questions for Part II unless otherwise stated. Suggestions for study Look over all of your class notes and make sure you understand everything we have talked about. Look over this review sheet and make sure you understand everything listed here. Review the suggested homework problems from each section, your Web Assign problems, in-lab and take-home quizzes, and your labs. There are additional practice problems at the end of this review sheet. Section 2.6: Derivative Be able to compute an average rate of change between two points and understand the relationship to the slope of a secant line. Understand the difference between average rate of change and instantaneous rate of change. Know how to solve real-world problems relating to rate of change. Understand velocity and acceleration as interpretations of a derivative. In general, understand the derivative as an instantaneous rate of change. Section.1: Basic Rules of Differentiation Be able to find the derivative of a function f by using the constant multiple, sum/difference, constant, or power rules. You will be doing this by hand on Part 1 of the eam. Section.2: Product and Quotient Rules Be able to find the derivative of a function f by using the product and quotient rules.you will be doing this by hand on Part 1 of the eam. Section. - The Chain Rule. Although this section covers the Chain Rule, you will also need to know the Basic Rules, Product Rule and Quotient Rule as some of the Chain Rule problems will incorporate other rules. Be able to find the derivative of a function f by using the chain rule. You will be doing this by hand on Part 1 of the eam. **Be able to apply to finding equations of tangent lines, where tangent lines have certain slopes, and various instantaneous rates of change.** I will ask you to find the equation of the tangent line again. Want More Practice?. Self Check Eercises (pp. 192): 1-2 AND Chapter Review Eercises 1
(pp. 24-245): 1-0, 47-50, 56-59 Section.4 - Marginal Functions in Economics Given a demand function, be able to find a revenue function. In addition, if given a cost function be able to find the profit function. Understand "marginal" cost, profit, or revenue as interpretations of the derivative. Know the difference between total cost for 150 units and the cost for the 150th unit. Be able to interpret a marginal cost (or profit or revenue) as the APPROXIMATE cost for the NEXT unit. We did not cover Elasticity of Demand. Want More Practice?.4 Self Check Eercises (pp. 208): 1 AND Chapter Review Eercises (pp. 24-245): 60-62 Section.5 - Higher Order Derivatives Be able to find higher order derivatives. Make sure you can do this using SN as well as by hand. Know about velocity and acceleration and how they relate to the derivative of a position function. Want More Practice?.5 Self Check Eercises (pp. 216): 1&2 AND Chapter Review Eercises (pp. 24-245): 1-6 Section 4.1 - Applications of the First Derivative Know the definition of relative (or local) etrema (maimums and minimums). Note that when we refer to the "maimum" (or minimum) that refers to the y value. Be able to find the domain of a function. Be able to find critical numbers. **Remember, a critical number must be in the domain of f.** Know how to use f to tell where f is increasing/decreasing. That is be able to give the sign chart for f and what it tells you about f. Remember we have used SN to solve equations like f 0 and f 0. Graphically be able to tell where f is increasing/decreasing and has relative etrema. Be careful using SN with functions involving radicals other than square roots, e. f 2. It is easiest to do those by hand. Know how to use the first derivative test for relative etrema. Want More Practice? 4.1 Self Check Eercises (pp. 258): 1,2 AND Chapter 4 Review Eercises (pp. 28-0): 1-10, 11-18 (but do not sketch) 2
Practice Problems **Note this list of problems is by no means complete and to focus solely on these problems would be unwise.** 1. Without using SN, find the derivatives of each of the following. You do not need to simplify your answers. a. f 7 5 1 5 4 b. f 72 7 2 5 4 10 c. f 100 2 8 10 7 d. f 2 99 5 20 5 e. Find f for f 100 5 2 1 9 f. Find f 4 for f 5 12 4
2. Suppose that a consumer receives a certain amount of pleasure or utility from a certain product when units are received. The utility of the product is given by the function: U 5 2 2 64 a. Graph the utility function for 50 50 and an appropriate y window. b. Find U 0, U 5, U 10, and U 15. c. Find lim U and interpret the results. d. Find the marginal utility for this product. Give your answer in simplified form. e. What is the marginal utility when 0, 5, 10, and 15 units are received? f. For what (positive) values is marginal utility equal to zero?. Let f 5 0 4 100 120 2 64. Be sure to use calculus to answer the following: a. Give a sketch of f for the following window. y 80 60 40 20 b. Find the domain for f. -1 1 2 4 5-20 -40-60 -80 c. Find all the critical numbers for f. d. Find the intervals where f is increasing and where f is decreasing. e. Classify all the critical points of f as relative maima, relative minima, or neither. Be sure to give the y values of all etrema. f. Find the values of all points where the tangent line has slope equal to 45. g. Find the equation for the tangent line at 1 and sketch your line on the graph 2 above. 4
4. Let f 2 2. Be sure to use calculus to answer the following: 2 2 a. Give a sketch of f for the following window. y 10 5-8 -6-4 -2 2 4 6 8 10 12-5 b. Find the domain for f. c. Find all the critical numbers for f. d. Find the intervals where f is increasing and where f is decreasing. -10 e. Find all relative etrema of f, if any. Remember that the relative etreme values are the y values. 5. You are told that the domain of a function f is all real numbers ecept 0. In addition, you are given the sign chart for f. f undefined, 0 0 0, 1 1 1, f undefined 0 a. Complete the sign chart for f using and to indicate where f is increasing and where it is decreasing. b. On what intervals is f increasing? On what intervals is f decreasing? Give your answer using interval notation. c. Which value(s) are critical numbers for f? d. Which value(s) yield a relative min? What is the rel min in each case? e. Which value(s) yield a relative ma? What is the rel ma in each case? f. Sketch a possible graph of y f. 5
y 6. The graph of the derivative f appears below. Answer the following questions about f. f '() 6 4 2-2 -1 1 2-2 -4 a. On what interval(s) is f increasing? b. On what interval(s) is f decreasing? 7. The weekly demand for the LectroCopy photocopying machine is given by the demand equation p 2000 0. 04 on 0 50000 where p denotes the wholesale unit price in dollars and denotes the quantity demanded. The weekly total cost function for manufacturing these copiers is given by C 0. 000002 0. 02 2 1000 120000 where C denotes the total cost incurred in producing units. a. Find the revenue function, the profit function. b. Find the marginal revenue function, the marginal profit function. c. What is the total profit for 5000 copiers? What is the profit from the 5000th copier? d. Compute marginal profit 4000, 10000, and 12000. e. Carefully interpret the results from part (d). 8. The demand function for the Luminar desk lamp is given by p d 0. 1 2 0. 4 5 where is the quantity demanded (measured in thousands) and p is the unit price in dollars. a. What is the average rate of change of the unit price as the quantity demanded ranges from 2500 to 000? Be sure to include the units. b. What is the average rate of change of the unit price as the quantity demanded ranges from 2700 to 000? Be sure to include the units. c. What is the average rate of change of the unit price as the quantity demanded ranges from 2900 to 000? Be sure to include the units. 6
d. What is the rate of change of the unit price when the quantity demanded is 000? Be sure to include the units. e. What is the unit price at that level of demand? f. As the interval from 1 to 2 gets smaller, what do you notice about the relationship between the average rate of change over the interval 1, 2 and the instantaneous rate of change of f? 9. The number of bacteria N t in a certain culture t minutes after an eperimental bactericide is introduced obeys the rule N t 10000 1 t 2 2000 a. What is the population in the culture 1 minute and 2 minutes after the bactericide is introduced? b. How fast is the population growing after 1 minute? 2 minutes? Be sure to include the units. 7