Reading Questions
Hass, Weir, and Thomas
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- What is the midpoint rule for estimating areas bounded by the graphs of
functions?
- What is an antiderivative of a function (see Section 4.9).
- What is true about the limits of \(R_N, L_N, \text{ and } M_N\) if the
function f(x) is continuous on the interval [a,b]?
- In a definite integral, what is the integrand?
- Do all functions have definite integrals? If not, what functions are we
sure are integrable?
- What is a Riemann sum for a function f on the interval [a,b]?
- What does the notation \( \left. F(x) \right|_a^b \) mean?
- What does \(\int_a^b f(x) / dx\) mean in terms of area? Do not assume
that f(x) is positive for all x.
- Why doesn't it matter if we use \(F(x)=x^2 \text{ or } G(x)=x^2 -5\) as
antiderivatives when we compute \(\int_a^b 2x / dx\)?
- What is the geometric meaning of the function A(x) in this section?
- Is there a function whose derivative is \(\sin(x^2)\)? If so, give
one.
- Do you like Part I or Part II of the Fundamental Theorem of Calculus
better? Why?
- What is the difference between displacment and distance travelled?
- What is marginal cost?
- What does it mean if the velocity of a particle is negative?
- The Substitution Method is the reverse of what derivative rule.
- Why are the limits of integration different in the two integrals in the
Change of Variables Formula for Definite Integrals.
- In the examples, how does the author choose the functions he sets equal
to \(u\)?
- What is the definition of a transcendental function (use the index)?
- Why did the author use \(u = 2x\) in Example 2?
- The author uses the inverse trigonometric functions in this section. What
exactly does \(y = sin^{-1}(x)\) mean?
- What does it mean to say that the function \(P_0 e^{kt}\) satisfies the
differential equation \(y' = ky\)?
- Explain what the doubling time of an exponential function is. Do not just
give the formula.
- How does carbon dating work?
- Does the formula for the area between two curves work if the graphs of
both functions are completely below the x-axis?
- Why is it important to find any points of intersection of the two
functions?
- Name a Figure in this section where integrating along the \(y\)-axis
would give fewer integrals than integrating along the \(x\)-axis.
- What is Cavalieri's principle?
- Do you understand how Riemann sums were used to derive the integral for
flow rate in Equation 5?
- What is the average value of a function \(f\) on an interval [a,b]?
- What is the difference between the disk method and the washer method?
- Is there anything special about the solids whose volume can be found
using the disk or washer methods?
- Using a method from this section, would you integrate with respect to x
or y if a solid of revolution was obtained by revolving a region around a
vertical line?
- Does the Shell Method only apply to volumes of solids of revolution or
can it be used to find the volumes of other types of solids?
- When using the method of Cylindrical Shells on a solid obtained by
rotating about the \(x\)-axis, is the "thickness" variable \(x \text{ or }
y\)?
- Is it possible that finding the volume of a solid of revolution will be
easier using the shell method than using the disk method?
Section 6.5 (Not Assigned)
- What derivative formula is reversed by the Integration by Parts
formula?
- What is the goal of using integration by parts?
- Do you understand the method used in Example 5?
- Which trigonometric identity mentioned in this section is most unfamiliar
to you?
- In the Products of Powers of Sines and Cosines subsection, why does
having m an odd integer lead to a useful substitution?
- How many integrals are in the largest table of integrals in our
textbook?
- If \(x=a \tan(\theta)\) what does \(a \sec(\theta)\) equal?
- In example 3, why did the authors factor the 4 out of the square
root?
- What trig substitution should be considered if \(\root{x^2 - a^2}\)
occurs in the integral?
- No reading questions for this section.
- What is the difference between a proper rational function and an improper
rational function?
- What is the first step in integrating an improper rational function?
- Give an example that is not in the book of an integral whose partial
fraction decomposition would require using repeated quadratic factors.
- What is a p-integral over [0,a]?
- What is a p=integral over \([a, \infty) \)?
- What is the purpose of the Comparison Test for Improper Integrals?
- If X is a continuous random variable, why is there zero probability of X
taking on the specific value 3?
- What is the name of the Greek letter \(\mu \)?
- Following the Conceptual Insight on page 451, give a list of 8 numbers in
which \(N(7)= 5\).
- What is the definition of the error when approximating a
definite integral using Simpson's Rule?
- Is the Trapezoid or Simpson's Rule likely to give a better approximation
when estimating a definite integral?
- What is "\(K_4\)" in the eror bound for Simpson's Rule?
- In developing the arclength formula, why does the author assume that
\(f'(x)\) exists and is continuous?
- Can a polygonal approximation of the arclength of a curve ever be larger
than the arclength? Why?
- What is a truncated cone?
- There are no reading questions for this section.
- There are no reading questions for this section
- What is the formula for the Taylor polynomial that agrees with the
function \(f(x)\) to order \(n\)?
- What is the "zeroeth" derivative of \(f(x)\)?
- What does the \(R_n (x)\) in Taylor's Theorem refer to?
- There are no reading assignments for this chapter.
- What does it mean for a sequence to diverge to infinity?
- Give an example of a recursively defined sequence that is not in this
section of the text.
- What dies it mean for the sequence \(\{a_n\}\) to be monotonic
decreasing?
- What is an infinite series and how does it relate to its sequence of
partial sums?
- What does the Divergence Test say about ain infinite series if the
\(n\)th term \(a_n\) does converge to zero?
- What does Theorem 1 say about the sum of two divergent infinite
series?
- Can we apply the integral test (Theorem 2) to the infinite series that
sums all of the terms in the sequence {a_n} where a_n=(cos(n)?
- Give two values of p for which the corresponding p-series diverge and one
value of p for which the corresponding p-series converges.Give two values
of p for which the corresponding p-series diverge and one value of p for
which the corresponding p-series converges.What is the advantage of
comparing (or limit comparing) a series to a geometric series or
p-series?
- Does the harmonic series converge or diverge?
- Why can't you use a comparison test on an alternating series?
- What is the difference between absolute and conditional convergence or an
infinite series ?
- Does the Alternating Harmonic series diverge, converge conditionally, or
converge absolutely?
- Explain why, in the proof of the Ratio Test, \( \|a_{M+2}\| < r^2
\|a_M\|\).
- What does \(\frac{n!}{(n+1)!}\) equal?
- Why wouldn't the root test work well with an infinite series involving
\(n!\)?
- What does the radius of convergence of a power series tell us?
- How do you take the derivative of a power series?
- Do you understand Example 8?
- What is the difference between a Taylor Series for a function \(f(x)\)
and a Maclaurin series for the same function?
- Is it possible for a function to have a convergent Taylor Series that
does not converge to the function?
- What are the first three terms of the Binomial Series if \(a=1/2\)?
- What is a parametrization of the line through the points \(P=(a,b) \text{
and } Q=(c,d)\)?
- How many parametrizations are there for a single curve?
- What is What is the standard parametrization of a circle of radius \(R\)
centered at the point \(P=(a,b)\)?
- Do you understand why the formula given in Theorem 1 gives the arclength
of the parametrized curve?
- What is the angular velocity of of a particle moving with constant speed
along a circle of radius \(R\)?
- What is the formula for the speed of a particle moving along a
paramatrized path \(c(t) = (x(t),y(t))\)?
- What are the Cartesian coordinates [the \((x,y)\) coordinates] of a point
with polar coordinates \((3,5\pi/3)\)?
- What is the difference in graphing \(r\) as a function of \(\theta\) in
rectangular coordinates and and graphing \(r\) as a function of \(\theta\)
in polar coordinates?
- Do you understand the technique for graphing polar equations illustrated
in Example 8?
- What is the formula for arc length in polar coordinates given in this
section?
- What is the equation in polar coordinates of the "rose" graphed in this
section?
- Does formula 2 in Theorem 1 compute the area under the curve
\(r=f(\theta)\)?