General Rules

• You are allowed one (and only one) 3"-by-5" note card with anything you wish written on both sides for use during the exam.
• No other notes or books are allowed at the exam.
• You must have your graphing calculator with you, with angular measure set to radian mode.

Differences between an in-class skills test and an exam

• Exam problems will not necessarily be just like a homework problem or an example done in class. We will expect you to be able to apply what you have learned about derivatives and demonstrate that you understand the concepts.
• Because we are asking you to think on an exam, you will have more time than the 50-minute class period. You will have one hour and 55 minutes.
• An exam will require you to explain your answers carefully and show your work even more than a skills test.

This exam covers Chapter 2--limits, the derivative, and continuity--and Section 1.7--continuity. There will be no questions whose only purpose is to test your knowledge of Chapter 1, but you must be familiar with that material since much of this newer material depends upon it.

This sheet is meant to highlight the important topics on the exam. You should not assume that a topic will not be on the exam just because it is not mentioned on this sheet. All of the material in the sections mentioned above is fair game.

General Topics and Skills

• You should understand the concept of a rate of change in its different forms, and in particular know the difference between average and instantaneous rates of change. You should be familiar with the term derivative and understand that it measures the instantaneous rate of change of a function. You should be able to compute and represent rates of change algebraically (using formulas), numerically, and graphically. You should understand and be able to discuss the practical meaning and applications of a rate of change.
• You should understand the information that the signs of the first and second derivatives give you.
• You should understand the difference between an approximate answer and an exact answer and know which methods give which.

• 2.1   You should know the difference between average and instantaneous velocity. You should be able to compute or estimate average and instantaneous velocity if you know position as a function of time. You should be able to represent average and instantaneous velocity as slopes of lines. You should understand that instantaneous velocity is a limit of average velocities. You should understand the meaning of the limit notation and be able to work with it.
• 2.2  Limits:  You should know the definition of limit both in words and with symbols. Using your graphing calculator, you should be able to find an appropriate delta given a specific epsilon. You should be able to show that a limit exists using the delta-epsilon definition for a linear function. You should be familiar with the properties of limits, when limits exist, and the definition of continuity.
• 2.3   You should know the meaning of derivative, tangent line, and instantaneous rate of change. You should be able to estimate derivatives numerically using a table or graphically by approximating the slope of a tangent line. You should be able to compute the derivative at a point using the limit definition. You should understand how to find the equation of a tangent line if you know the derivative.
• 2.4   Given the graph of a function f, you should be able to sketch the graph of f'. You should understand how the sign of the derivative relates to whether or not the function is increasing or decreasing. Given two or more graphs you should be able to tell whether a given graph could be the graph of the derivative of any other graph. You should be able to find the derivative of a function algebraically using the limit definition of derivative. You should know the derivatives of f(x)=c, f(x)=mx+b and f(x)=xⁿ.
• 2.5   You should be able to explain in ordinary English the practical meaning of a derivative of a function that models a real-world situation. You should understand what the correct units for a derivative are. You should be familiar with the dy/dx notation for derivatives. You should understand and be able to use the linear/tangent line approximation.
• 2.6   You should know what is meant by a second derivative and how it is related to concavity. You should be able to explain and interpret the practical meaning of a real-world function.
• 1.7/2.7   Continuity and Differentiability:  You should know the definition of "continuous at x=c" and "continuous on an interval" as applied to functions, and you should be able to recognize where a function presented graphically is not continuous. You should be able to recognize where a function presented graphically is not differentiable. You should know the logical relationship between continuity and differentiability at a point.

Suggested review problems

• p. 61: 1
• p. 69: 31, 33, 35
• p. 93: 9, 11, 13
• p. 99-102: 1-12, 22-25, 38, 39, 31ab, 32, 33, 34ab, 36
• p. 103: 3, 4, 6, 10, 12, 16, 17, 27-29, 34, 36