General Instructions:

• The final exam will consist of 10 questions. Some may be multiple choice..
• Remember that 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 those grading the exam that you understand what you are doing.
• Studying and practicing homework problems is important for the exam, but not all questions on an exam are drawn from the homework. Theory is also important. You could be asked to state simple definitions, describe concepts, or even state simple theorems.

Calculator Use Instructions:

• DUE TO THE LOGISTICAL PROBLEMS THAT GO ALONG WITH CHECKING ALL CALCULATORS FOR DISALLOWED STORED ITEMS, CALCULATORS WILL NOT BE PERMITTED ON THE FINAL EXAM!
• The exam will be carefully designed in a way that a calculator will not be needed to answer any of the questions.

Final Examination 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

Sec 2:

• know the conceptual definition of right and left handed limits of a function f(x) at a point x = a and the definition of "limit" in terms of one sided limits
• know several ways in which limits are undefined (i.e typical graphs), and in particular an example involving a straight line with a hole in it

Sec 3:

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

Sec 4:

• be able to state accurately the definition of continuity
• have three or four graphical example in your head of how a function can be discontinuous at a point
• what is a removeable discontinuity?
• know what standard functions are continuous: polynomials, rational functions where denominator does not vanish, roots, sin, cos, tan (where defined), etc.
• be able to determine where a function is continuous using Theorem 4.2's limit properties and known continuous functions

Sec 5:

• be able to find horizontal and vertical asymptotes for simple functions using infinite limits and limits at infinity

Chapter 2

Sec 2:

• know the definition of the derivative of a function as a limit of the change in f(x) over the change in x
• be able to compute the derivative directly from the definition for

• simple polynomial
• simple rational functions
• simple square root functions

• know the statement and meaning of Theorem 2.1
• be able to draw graphs of the different ways in which a function fails to be differentiable at a point (see Figure 2.19a, b, c, d)

Sec 3:

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

Sec 4:

• know the form of the product, reciprocal and quotient derivative rules
• be able to apply these rules efficiently in simple examples

Sec 5:

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

Sec 6:

• know by heart the formulas for the derivatives of the six primary trig functions (Theorems 6.1, 6.2, 6.3 and Table on page 200)
• be able to use these derivatives in more complex differentiations involving products, quotients, compositions, etc.

Sec 7:

• know the derivatives of e ^x, a ^x , ln x, log_a x
• be able to use these efficiently in more complex derivative calculations
• know the technique referred to as logarithmic differentiation and in particular how to differentiate functions of the form f(x) = x ^g(x), f(x) = h(x) ^x, and f(x) = h(x) ^g(x)

Sec 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, cos^-1x, tan^-1x
• be able to use these formulas in complex derivative calculations

Chapter 3

Sec 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 approximations that appears in Example 1.3

Sec 2:

• know how to identify the basic undeterminate forms 0/0 and infinity/infinity when they arise in limits
• know the statement of L'Hopital's Rule and how to use it in basic cases
• be able to recognize the indeterminate forms other than the two basic ones and how to compute limits in these cases. In this regard, review Examples 2.6 through 2.11

Sec 3:

• there are a lot of definitions here with subtle differences, so be sure to be able to distinguish between them
• know the precise statement of the Extreme Value Theorem
• what does one mean by the critical numbers of a function, know how to find them in simple examples, and know 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

Sec 4:

• be able to determine, using the derivative of a function, the intervals on which a function is increasing and decreasing
• what is the "First Derivative Test" for local extrema, and can you apply it in simple cases?
• know how to handle examples with fractional exponents such as in Example 4.4

Sec 5:

• be able to determine, using the second derivative of a function, the intervals on which the function is concave up and concave down
• review Definition 5.2 and know how to find inflection points in simple cases
• what is the second derivative test for local extrema and how do you apply it?

Sec 7:

• be able to solve simple word problems involving optimization
  • review the steps involved at the top of page 308
  • then review Examples 7.1, 7.3, and 7.3 as examples of what to do

Sec 8:

• be able to solve simple word problems involving related rates
  • review the steps involved near the bottom of page 321
  • then review Examples 8.1, 8.2, 8.3 and 8.5 as examples of what to do

Chapter 4

Sec 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
• know Theorem 1.1 and what it means for antiderivatives
• be familiar with antiderivatives for 1/x and also f'(x)/f(x) (and especially Theorem 1.4)
• be familiar with the sum and scalar multiplication rule (Theorem 1.3) for finding complicated antiderivatives from simpler ones

Sec 2

• know how to write a sum of many terms with a "pattern" in sigma notation, and also how to take a sum in sigma notation and write it out term by term
• know Theorem 2.1 and how to use it
• know how to use the sum rule, Theorem 2.2, in computing complicated sums from simpler ones
• kow how to compute the sum of function values, such as in Examples 2.6 and 2.7

Sec 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 evalustion points) (Example 3.4 is an example of this type of calculation)
• 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

Sec 4

• be familiar with the distinctions between signed area and total area and how to compute each
• be familiar with the distinctions between distance traveled and displacement (and position) 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 (Example 4.6 is a good example)
• review Theorem 4.3 and explain why it is true in terms of areas
• know how to compute the avergae value of a function over an interval

Sec 5

• know the precise statement of Theorem 5.1, FTC Part I
• know how to compute definite integrals using antiderivatives via FTC Part I
• know the precise statement of Theorem 5.2, FTC Part II
• know how to compute derivatives of area functions (definite integrals with a x as the upper limit of integration) via FTC Part I
• review Examples 5.8 and 5.9 as examples in which the upper (and possibly lower) limits of integration involve functions of x

Sec 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 (i.e. review Exercise 6.10)

Chapter 5

Sec 1

• be able to compute the area between two curves using both integrations along the x axis and along the y axis. Review each of the examples in this section

Sec 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 diasks and washers. Review the examples in this section

Sec 4

• be able to set up integrals that represent the arc length of a curve and the surface area of a surface of revolution

Sec 5

• be able to solve simple projectile problems of the type shown in Examples 5.1, 5.2, 5.3, and 5.4