Reading Assignments
:
- Read 7.5
- Answer the following questions:
- 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).
:
- Read 7.7
- Answer the following questions:
- 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?
:
- Read 8.1
- Answer the following questions:
- 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 $dv = t^2 dt$?
:
- Read 8.2
- Answer the following questions:
- 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 $\sin(2x) \cos(3x)$?
:
- Read 8.3
- Answer the following questions:
- 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.
:
- Read 8.4
- Answer the following questions:
- 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?
:
- Read 8.8
- Answer the following questions:
- 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?
:
- Read 8.5
- Answer the following questions:
- 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.
:
- Read 10.1
- Answer the following questions:
- 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?
:
- Read 10.2
- Answer the following questions:
- 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?
:
- Read 10.3 and 10.4
- Answer the following questions:
- 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?
:
- Read 9.1
- Answer the following questions:
- 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?
:
- Read 9.2
- Answer the following questions:
- 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)?
:
- Read 11.1 and 11.2
- Answer the following questions:
- 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?
:
- Read 11.3
- Answer the following questions:
- 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!)
:
- Read 11.4
- Answer the following questions:
- 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?
:
- Read 12.2
- Answer the following questions:
- 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?)
:
- Read 12.3
- Answer the following questions:
- There are two major questions about series:
- Is a given series convergent or not?
- 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?
:
- Read 12.4
- Answer the following questions:
- 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?
:
- Read 12.5
- Answer the following questions:
- 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.
:
- Read 12.6
- Answer the following questions:
- 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?
:
- Read 12.7 and 12.8
- Answer the following questions:
- 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?
:
- Read 12.9
- Answer the following questions:
- 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!
:
- Read 12.10
- Answer the following questions:
- What "ingredients" do you need to construct a Taylor series? What
is the "recipe"?
- What is the difference between a Taylor series and a Maclaurin
series? What is the difference between a Taylor series and a Taylor
polynomial?
- Why is it very important to be able to evaluate the remainder of
the Taylor series? How can we do that?
Last modified:
November 2, 2007
Copyright ©2007
Catalin Zara