1. If f is a function and P_3 is the third order Taylor Polynomial for f, what do f and P_3 have in common?

2. If f is a function and P_3 is the third order Taylor Polynomial for f, what determines the error with which P_3 approximates f ? (That is, what property of f does the error depend on?)

3. What is the n th order Taylor Polynomial of a function f used for?

4. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. Explain why f(x)=|x| cannot be equal to any Maclaurin series.

2. If a function f has a Taylor (power) series representation around x_0, what can you say about the coefficients of that series?

3. Section 11.7 says that any function f that has infinitely many derivatives at x = x_0 has a Taylor series expansion based at x_0 . In this context, what is the relevance of Taylor's theorem?

4. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. In example 1, the book gives a power series representation for e^x. How do they justify the assertion that the given series is equal to the exponential function?

2. Can a power series be differentiated term by term? How about integration? What happens to the radius of convergence after these operations?

3. At the bottom of page 593, the authors find a power series representation for arctan(x). Then they set x=1 in the expression to obtain a series expression for Pi/4. However, as they remark, this move isn't justifiable without further proof. Why doesn't the previous information justify the move?

4. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. What is the radius of convergence of a power series? What is the interval of convergence? What is the relationship between the two? What are possible values of radius of convergence?
2. Suppose you have a power series that is based at x=2 and that you know the power series converges at x=5. What, if anything, can you conclude about the radius and interval of convergence?
3 . What test is usually used to determine the radius of convergence of a power series?
4. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. Explain the difference between absolute and conditional convergence. What does Theorem 10 say about the relationship between the two?

2. What does the alternating series test (AST) say ?

3. Can you conclude divergence based on AST?

4. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered

1. What is the "comparison test"? Give an intuitive explanation for why it works.

2. What is a p-series? How can you tell if a p-series converges?

3. What is the ratio test?

4. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1) Decide if the following statements are True or False

i) If a series converges, its general term goes to 0

ii) If the terms of a series go to 0, then series converges

2) What, if any, is the difference between a "sequence" and "series"? What is the relationship between convergence of sequences and convergence of series?

3) When does a geometric series converge?

4) Was there anything in this section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. What is the formal definition of a sequence?

2. Give at least two alternative ways (other than the formal definition) of viewing a sequence.

3. Do you think the converse of the Fact on page 550 is correct?

4. What is a monotone sequence?

5. Was there anything in this section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1) How does integration figure into calculating physical work? What is it that makes the use of integration necessary in computing work?

2) What is Hooke's law?

3)What is Newton's law of gravity?

4) Was there anything specific in this section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1) In finding volumes by cross sectioning, what do we assume about cross sectional area when Delta x (thickness) is small? What is the volume of such a section, approximately?

2) What are the cross sections of solids of revolutions like?

3) What is the area of the washer? (Your answer should include the terms "inner radius" and "outer radius")

4) Was there anything specific in this section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. What does the word "error" mean in connection with approximating sums for integrals?

2. If a function f is decreasing over an interval, what can you say about the relationship between Rn and Ln on that interval? Why does this relationship hold?

3. If a function f is concave up over an interval, what can you say about the relationship between Tn and Mn on that interval? Why does this relationship hold?

4. The Simpson's rule can be thought of as a "weighted average" of two other sums. What are they? Which one gets more weight and why?

5. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. What are the two important advantages of symbolic methods of solving of differential equations over numerical and graphical methods? What is the main disadvantage of symbolic methods?

2. Give an example of a separable differential equation and an example of a non-separable differential equation.

3.Briefly what does the book suggest as a general approach for solving the separable DE?

4. Was there anything in this section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. What makes an integral improper?

2. What is an absolutely convergent improper integral?

3. What does it mean to say that "convergent integrals have skinny tails?"

4.Was there anything in this section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. What is the method given in the book to find the limit of a rational function as x goes to infinity (or negative infinity)?

2. Briefly explain what an indeterminate form means. Give at least 3 types of indeterminate forms. There are actually more indeterminate forms than those described in the book. Do you know any additional indeterminate forms?

3. With what types of indeterminate forms does the l'Hopital's rule deal with?

4. Was there anything in this section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1) Complete the following expressions to a square: i) x^2+3x+1 ii) -x^2-x (Just give the final answer, don't show intermediate steps)

2) State what trig substitution is used for integrals involving each of thefollowing: i) sqrt(4-x^2) ii) sqrt(x^2-3) iii) sqrt(3x^2+5)

3) Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. Why do we bother calculating antiderivatives by hand, instead of just asking Maple do it for us?

2. Integration by parts is related to which differentiation formula?

3. Which of the following functions is a good candidate for antidifferentiation by parts:

f(x) = x exp(x^2 ) or f(x) = x^2 exp(x)? Explain?

4. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

Notes for Students (xviii)

and

Section 5.4

i) What is the authors' answer to "Why do we study calculus"?

ii) What are the 3 steps in using the method of substitution?

iii) What rule of differentiation justifies the method of substitution?

iv) Was there anything specific that was confusing to you? Is there any specific (or general) question you would like to have answered?