Final Exam Information

The Final Exam is cumulative and will take place Friday, May 2 from 1:30pm-4:30pm.  Room assignments are as follows:

Last Names A - R in 114 DKH

Last Names S - Z in 119 DKH

General Instructions/Advice

• The final exam may contain a few multiple choice or short answer question, but most of the questions will be free response requiring you to show all of your work.  The logical steps getting you to an answer are worth much more than the final answer itself.
• Your objective on the exam is to demonstrate that you understand the concepts and can carry out essential calculations.  Therefore, you should add enough detail to your answers to convince the grader that you understand what you are doing.
• No calculators will be allowed on the exam.
• Studying and practicing homework problems is important for the exam, but not all questions on the exam are drawn from the homework. Theory is also important. You could be asked to state simple definitions, describe concepts, or even state simple theorem
• Work problems.  Re-work homework questions, mid-term exam questions, and examples done in lecture and in discussion.  If you have trouble with one of then, go back and read that section from the text and your notes.  Then try another problem.  Studying and practicing problems we've already done is very important.  However, not all questions will be drawn from the homework or examples.  Thus, it is important to focus on understanding the concepts involved in those problems and to also practice other problems.  You can find more problems in the textbook and in the Self-study tab on MathZone.
• Practice under exam conditions.  It is very easy to fool yourself into believing you understand the material better than you really do, especially if you work problems with the solutions in front of you.  Thus, you should work entirely on your own before looking at the solutions. 
• Remember good test-taking strategies the day of the exam.  Read all of the problems and then do the easier problems first.  The easiest ones may not be at the beginning.  Don't leave early.  If you have extra time, read over the entire exam to check for small arithmetic errors and rework any problems you were not sure of the first time through.

Final Exam Content

Chapter 0

Chapter 0 covers precalculus material and as such will not be covered on the exam, except where the material in this chapter is important for concepts and calculations in later chapters.

Chapter 1, Section 2:

• know the conceptual definition of right and left handed limits of a function f(x) at a point x = a
• know several ways in which limits are undefined (i.e typical graphs)

Chapter 1, Section 3:

• generally you should be able to evaluate limits by applying limit rules (sums, products, quotients, compositions, etc.)

Chapter 1, Section 4:

• know the precise definition of continuity
• have three or four graphical examples in your head of how a function can be discontinuous at a point
• understand what a removable discontinuity is
• know which standard functions are continuous: polynomials, rational functions where denominator does not vanish, roots, sine, cosine, tangent (where defined), etc.
• be able to determine where a function is continuous using Theorem 4.2's limit properties

Chapter 1, Section 5:

• know what is meant by a vertical asymptote of a function and how to express this in terms of left/right handed limits being infinity or -infinity
• know what is meant by an horizontal asymptote of a function and how to express this in terms of limits of the function as x approaches infinity or -infinity
• be able to find vertical and horizontal asymptotes for simple functions

Chapter 1, Section 6:

• know the precise definition of the limit and be able to draw a graph to demonstrate that definition
• be able to find a delta given an epsilon for the limit of a simple function (graphically, numerically, or algebraically)

Chapter 2, Section 2:

• know the limit definition of the derivative
• know the different interpretations of the derivative
• be able to compute the derivative directly from the definition
• know the relationship between differentiable and continuous functions ( Theorem 2.1)
• be able to draw graphs of the different ways in which a function fails to be differentiable at a point
• be able to answer questions about the derivative of a function given the graph of the function

Chapter 2, Section 3:

• generally be able to differentiate simple sums and scalar multiples of functions
• know how differentiation relates to velocity and acceleration problems

Chapter 2, Section 4:

• know the product and quotient derivative rules
• be able to apply these rules efficiently

Chapter 2, Section 5:

• know the Chain Rule for differentiating composite functions
• be able to apply this rule efficiently

Chapter 2, Section 6:

• know the special limits used in finding the derivatives of sine and cosine (Lemma 6.3 and Lemma 6.4)
• know the formulas for the derivatives of the six primary trig functions (Table on page 200)
• be able to use these derivatives in more complex differentiations involving products, quotients, compositions, etc.

Chapter 2, Section 7:

• know the derivatives of e ^x, a ^x ,ln x
• be able to use these efficiently in more complex derivative calculations

Chapter 2, Section 8

• know what it means for a function to be defined implicitly (have a simple example such as x²+ y² = 4 in your head to illustrate)
• know the technique for finding derivatives of functions defined implicitly
• know the formulas for the derivatives of sin^-1x, tan^-1x
• be able to use these formulas in complex derivative calculations

Chapter 2, Section 9:

• be able to give a precise statement of the Mean Value Theorem (Theorem9.4) and be able to illustrate what it says with a simple graphical argument

Chapter 3, Section 1:

• be familiar with how to compute the linear approximation L(x) of a function at a point
• be familiar with the type of calculation using linear approximations that appears in Example 1.3

Chapter 3, Section 2:

• know how to identify the basic indeterminate forms 0/0 and infinity/infinity when they arise in limits
• know the statement of L'Hopital's Rule, how to use it in basic cases, and when NOT to use it
• be able to recognize the indeterminate forms other than the two basic ones and how to compute limits in these cases

Chapter 3, Section 3:

• there are many definitions here with subtle differences, so be sure to be able to distinguish between them
• know the precise statement of the Extreme Value Theorem
• know what the critical numbers of a function are, how to find them, and why they are important (see Theorem 3.2)
• be familiar with Theorem 3.3 and Remark 3.3 and how this helps in finding absolute extrema such as in Example 3.11

Chapter 3, Section 4:

• be able to determine, using the derivative of a function, the intervals on which the function is increasing and decreasing
• know the "First Derivative Test" for local extrema, and be able to apply it

Chapter 3, Section 5:

• be able to determine, using the second derivative of a function, the intervals on which the function is concave up and concave down
• know what inflection points are and how to find them
• know the "Second Derivative Test" for local extrema and be able to apply it

Chapter 3, Section 6:

• know how to sketch a fairly simple curve, indicating regions where the function is increasing/decreasing and concave up/down, the local extrema, the inflection points, and any horizontal and vertical asypmtotes and intercepts

Chapter 3, Section 7:

• be able to solve simple word problems involving optimization
  • know the steps involved at the top of p. 308 or the ones presented in class
  • know how to show that your answer really is a max or min using the 1st Derivative Test or the 2nd Derivative Test
  • practice writing your work up clearly so others can understand it

Chapter 3, Section 8:

• be able to solve word problems involving related rates (know how to work the steps listed near the bottom of p. 321 or the ones from class)

Chapter 4, Section 1:

• know what is meant by an antiderivative of a function
• be able to compute antiderivatives for power functions and all the functions in the table on page 347
• be familiar with antiderivatives for 1/x and f'(x)/f(x)
• be familiar with the sum and scalar multiplication rule (Theorem 1.3) for finding complicated antiderivatives from simpler ones

Chapter 4, Section 2:

• know how to use the sum formulas like those given in Theorem 2.1
• know how to use the sum rule, Theorem 2.2, in computing complicated sums from simpler ones

Chapter 4, Section 3:

• know the technical language, notation, and concepts related to Riemann sums
• be able to set up the approximation to the area under a curve with n rectangles as a Riemann sum (using sigma notation and right-hand evaluation points)
• be able to compute areas under simple curves approximately using n rectangles for a general n
• be able to compute areas exactly by limits of n rectangle approximations

Chapter 4, Section 4:

• be familiar with the distinctions between signed area and total area and how to compute each
• know how to use the addition rule, Theorem 4.2, and property (4.2) in computing complicated integrals from simpler ones
• review Theorem 4.3 and explain why it is true in terms of areas
• know how to compute the average value of a function over an interval
• know the precise statement of the Integral Mean Value Theorem

Chapter 4, Section 5:

• know the precise statements of The Fundamental Theorem of Calculus, Part I and Part II
• know how to compute definite integrals using antiderivatives via FTC Part I
• know how to compute derivatives of area functions (definite integrals with an x as the upper limit of integration) via FTC Part II
• know how to compute derivatives of definite integrals in which the upper limit of integration involves a function of x

Chapter 4, Section 6:

• know how to find antiderivatives by substitution - the method is described at the top of page 395
• be aware of how to change the limits of integration in a definite integral when making a substitution

Chapter 5, Section 1:

• be able to compute the area between two curves using both integrations along the x axis and along the y axis

Chapter 5, Section 2:

• be familiar with the fundamental formula for volume (2.1) and how to use it
• be familiar with how to set up formulas for volumes of revolution using partitions, areas of disks, and Riemann sums
• know how to compute volumes of this kind using disks and washers.

Chapter 5, Section 4:

• be able calculate the arc length of a curve for "simple" functions

Chapter 5, Section 6:

• be able to calculate the work done in "simple" situations

Textbook Problems

A list of other practice problems already exists for each of the 4 exams - refer to their individual web pages for those lists.  The following are problems from the Chapter Reviews related to the topics listed above.  Please note - these problems are simply given as another resource for practice problems.  Remember I do not guarantee that all questions will be drawn from the homework, examples, or these practice problems.

Chapter 1 Review p. 141:  True/False #3, 4, 6, 7.  Exercises #11-18, 20-23, 27-28, 31-34, 39-41, 43-49, 54

Chapter 2 Review p. 236:  Exercises #2-20, 23-47, 49, 53-57, 65-67, 71-74

Chapter 3 Review p. 339:  True/False #4-7, 9.  Exercises #1-4, 9-12, 17-30, 35-42, 47, 51

Chapter 4 Review p. 426:  Exercises #1-24, 27-30, 31-34, 35a, 35b, 36a, 36b, 39-40, 45-60

Chapter 5 Review p. 506:  Exercises #1-4, 6-8, 11, 14-15, 19-22 (set up only), 33-34

Definitions/Theorems

The following are definitions/theorems that you should be able to state:

• Definition of continuity
• Precise definition of the limit
• Limit definition of the derivative
• Mean Value Theorem
• L'Hopital's Rule
• Extreme Value Theorem
• Integral Mean Value Theorem
• Fundamental Theorem of Calculus, Parts I and II