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Friday, January 27, 2006:
• What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph?
• What are the domain and the range of the inverse tangent function? What happens to arctan(x) when x is very large (close to infinity)?
• If x is an angle between -$pi$/2 and $pi$/2 and tan(x)=a, explain how do you find sin(x).
Monday, January 30, 2006:
• Explain, using your own words, what an indeterminate form is.
• What are the seven types of indeterminate forms?
• How do you calculate a limit involving an indeterminate power?
Wednesday, February 1, 2006:
• What is the differentiation rule corresponding to the method of integration by parts?
• True or false: every integral involving the product of two functions can/should be computed using integration by parts. Briefly explain your answer.
• In example 3 (p. 513), the author chose $u=t^2$ and $dv=e^t dt$. What happens if we choose $u=e^t$ and $dv = t^2 dt$?
Friday, February 3, 2006:
• What substitution(s) can you use to find an antiderivative of $(\sin x)^3*(\cos x)^7$? If more than one substitution works, which one would make the computation easier?
• In example 6 (p. 521), the author tries to separate either a $\sec^2 \theta$ factor or a $\sec \theta$ $\tan \theta$ factor? Why?
• How do the trig identities at the bottom of page 523 help you in computing an antiderivative of $\sin(2x) \cos(3x)$?
Wednesday, February 8, 2006:
• Identify one similarity and one difference between the substitution method described in section 8.3 and the substitution rule introduced in section 5.5.
• Why do you use a trigonometric substitution when you compute integrals involving square roots of polynomials of degree 2?
• Write $3x-x^2$ as a sum or a difference of two squares, one being a constant and the other one an expression involving x.
Friday, February 10, 2006:
• In some cases, when we compute an integral of a rational function, we have to start with a long division of polynomials. When is that necessary, and why?
• What (kind of) functions can appear in the answer when you compute the integral of a rational function?
• Why does it suffice to consider only linear and quadratic factors (distinct or repeated) in the denominator of a rational fraction? That is, why don't we have to consider, for example, factors of degree three?
Wednesday, February 15, 2006:
• What are the two types of improper integrals? For each of them, briefly explain, in your own words, how do they differ from the definite integrals studied so far.
• Is it possible to have an unbounded region with a finite area? Give an example, or explain why that's not possible.
• What happens with the comparison theorem if we do not impose the condition $g(x) \ge 0$? Is the new statement always true, sometimes true, sometimes false, always false?
Friday, February 17, 2006:
• Why is it more difficult to compute integrals than to compute derivatives?
• The author states that there are basically only two methods to compute integrals. What are those methods? Do you agree with his statement, or not? Justify!
• What is an elementary function? Do all elementary functions have elementary functions as antiderivatives? If yes, explain, and if no, give an example.
Friday, February 24, 2006:
• What is a differential equation, and why do we study differential equations?
• What is a solution of a differential equation? How many solutions does a differential equation have?
• What is an initial value problem? What is a solution of an initial value problem?
Monday, February 27, 2006:
• What is a direction field, and how can you use one to visualize a solution of a differential equation?
• Briefly describe, using YOUR OWN WORDS, the main idea behind Euler's method.
• What is an autonomous differential equation? How can you recognize an autonomous differential equation from its direction field?
Wednesday, March 1, 2006:
• Why is a separable equation called ''separable''? Give an example of a differential equation that is not separable.
• Explain, IN YOUR OWN WORDS, the general idea behind the method of solving a separable differential equation.
• The solutions of separable differential equations are most of the times given implicitly. What does that mean, and why do you think that happens?
• What is the relative growth rate? Which one is more intuitive: the growth rate or the relative growth rate?
• Give three examples of models involving exponential growth/decay. Explain one of it in detail (USING YOUR OWN WORDS).
• Best Bank has recently introduced a new CD (that is ''certificate of deposit,'' not ''compact disk''!) product: for a mere \$1 fee, it compounds the 6% anual interest continuously, instead of monthly. If you have \$1000 to keep for a year, would you consider the new CD product?
Friday, March 3, 2006:
• How is the length of a curve defined?
• On page 584, the author states: "We recognize this expression as being equal to ..." Why?
• How is the arclength of a curve different from the arclength function?
Monday, March 6, 2006:
• Describe, in your own words, the method used to compute the surface area of a surface of revolution.
• Why does the formula for the surface area of a surface of revolution involve a definite integral?
• What is the meaning of the term ''ds'' in the formulas for the surface area (Formulas 7 and 8)?
Wednesday, March 8, 2006:
• What is a parametric curve, and why do we use this terminology ("parametric")?
• How do we recognize what a curve given by parametric equations looks like?
• Explain why the graph of a y=f(x) function can be thought of a parametric curve. Is every parametric curve the graph of a function y = f(x)?
• How do you recognize a point on a parametric curve where the tangent is vertical?
• The parametric curve discussed in Example 1 has two tangents at the point (3,0). How do you explain that?
• What happens to the length of the parametric curve in Example 4 if we allow the parameter t to vary from 0 to $4\pi$? How do you explain that?
Monday, March 20, 2006:
• What are the geometric meanings of the polar coordinates of a point?
• What is the curve described by the polar equation $r=2$? What is the curve described by the polar equation $\theta= \pi/3$?
• Explain, in your own words, how would you graph a curve given in polar coordinates (without using a graphing device!)
Wednesday, March 22, 2006:
• What is the meaning of the term $1/2 r^2 d \theta$ in formula 4, and where does it come from?
• In Example 1, why are the limits of integration from $-\pi/4$ to $\pi/4$ and not from $-\pi/2$ to $\pi/2$?
• What is the basic principle behind the formula for the arclength of a curve given in polar coordinates?
Wednesday, April 5, 2006:
• Name a few differences between sequences and series.
• Explain, IN YOUR OWN WORDS, what is a convergent series. When is a geometric series convergent?
• Consider the following two statements:
• The sequence $(a_n)_n$ converges to zero
• The series $\sum_{n=1}^{\infty} a_n$ is convergent
What is the relationship between these two statements? (Does the first imply the second? Does the second imply the first? Do each of them imply the other?)
Friday, April 7, 2006:
• There are two major questions about series:
1. Is a given series convergent or not?
2. If the series is convergent, what is the sum of the series?
Which of these questions can be answered using the integral test?
• The sequence $(a_n)_n$ is given by $a_n = f(n)$, where $f$ is a function defined for all real values greater than or equal to 1. What conditions on the function $f$ should we check if we want to apply the integral test?
• What is a p-series, and for what values of p is the series convergent? How do you relate this to what you know about the convergence of improper integrals?
Monday, April 10, 2006:
• What are the basic ideas behind the comparison and limit comparison tests? Which of the two tests appears to be easier to use?
• What are some frequent types of series we compare a series with when using one of the comparison tests? Why do you think we look for such test series?
• What series would you compare the series$\sum_{n \ge 1} \frac{n+2}{2n^2+3}$ with? Is the given series convergent or divergent?
Wednesday, April 12, 2006:
• How do you recognize an alternating series? Give an example of an alternating series, and an example of a series that is not alternating.
• Give an example of an alternating series for which the alternating series test does not apply. Can an alternating series be convergent even if it doesn't satisfy all the conditions of the alternating series test?
• For the series in example 4, how many terms should you add if you want your estimate to be correct to 4 decimal places? Justify your answer.
Friday, April 14, 2006:
• Can a series with positive terms be conditionally convergent? If yes, give an example. If no, justify your answer.
• Explain, in your own words, the main idea behind the Ratio Test. Why is the ratio test inconclusive if the limit is 1?
• Explain, in your own words, the main idea behind the Root Test. Why is the root test inconclusive if the limit is 1?
Wednesday, April 19, 2006:
• Give an example of a series for which you would use the limit comparison test to determine whether it converges/diverges. Explain how would you recognize that the limit comparison test may be useful for such a series.
• Give an example of a series for which you would use the alternating series test to determine whether it converges. Explain how would you recognize that the alternating series test may be useful for such a series.
• Give an example of a series for which you would use the ratio test to determine whether it converges/diverges. Explain how would you recognize that the ratio test may be useful for such a series.
• Give an example of a series that is a power series, and an example of a series that is not a power series.
• What is the radius of convergence of a power series, and how would you find it?
• What is the interval of convergence of a power series? How would you determine whether a given power series converges or diverges at an endpoint of the interval?
Friday, April 21, 2006:
• In Example 2 (page 791), why do we have to divide by 2 and change the sign of x/2 in the denominator?
• Why is it useful to have power series representation of a function?
• True or false: the interval of convergence of the series obtained by differentiating a power series term by term is the same as the interval of convergence of the original series. Justify your answer!
Wednesday, April 26, 2006: 