Mathematics 180F (Fall 2013)
Contents
Day and Date
- Topics:
- Text:
- Tomorrow:
- Comments:
Day and Date
- Topics:
- Text:
- Tomorrow:
- Comments:
Friday December 6
- Topics:
- Text:
- Tomorrow:
- Comments: I will
have a review session for Math 180-F during reading period: Friday December
13 from 1:00-2:00 PM.
Thursday December 5
- Topics: FTC 2, integrals of rates as net change
- Text: 5.4, 5.5
- Tomorrow: Review for Exam 5 (Sections 4.9-5.5)
- Comments: After discussing several homework questions
from section 5.4 that involved using the Fundamental Theorem of Calculus,
Part 2 to take the derivatives of functions defined by integrals, we ended up
with a procedure that we can write as \[\frac{d}{dx} \left[\int_a^{g(x)} f(t)
dt \right] = f(g(x))\cdot g'(x) .\] We then discussed the material in Section
5.5 and noted it is the special case of using the FTC. Part 1, when the
function "f" is already known to be a derivative (a rate of change). Two
examples of this situation are integrating velocity to determine displacement
and integrating marginal cost to determine change in cost. We finished by
reviewing the definition of density functions (see below for more details)
and noting that density functions are really just rates of change. For
example, linear charge density at a point x on a wire is the limit that gives
the instantaneous rate of change of charge with respect to length of wire.
- Exam Five is now scheduled for Tuesday December 10.
- Here is a relatively detailed outline of the material we covered this
semester: SEMESTER REVIEW
Tuesday December 3
- Topics: fundamental theorem of calculus, part 2
- Text: 5.4
- Tomorrow: FTC 2, integrals of rates as net change
- Comments: we discussed various homework problems from
section 5.3 and then saw why part 2 of the fundamental theorem of calculus is
valid. Specifically, we expressed the signed area function \(A(x)\) of a
continuous function \(f(x)\) as an integral \(A(x) = \int_a^x f(t) \ dt\) and
then computed its derivative (in the special case where \(f\) is
decreasing)by "squeezing" \(\frac{A(x+h)-A(x)}{h}\) between \(f(x+h)\) and
\(f(x)\). This showed why \[A'(x) = \frac{d}{dx} \left[\int_a^x f(t) dt
\right] = f(x) \]. In the process we discussed why it is important to
distinguish the "t" and "x" variables.
- Here are the objectives for
Exam 5
Monday December 2
- Topics: fundamental theorem of calculus, part 1
- Text: 5.3
- Tomorrow: fundamental theorem of calculus, part 2
- Comments: After answering questions and reviewing the
process of computing a definite integral using Riemann sums, we proved that
if \(f(x) \) is continuous on [a,b] and \(F(x) \) is an antiderivative of
\(f(x)\), then \[\int_a^b f(x) \ dx = F(b)-F(a).\] This theorem allows us to
quickly compute the value of a definite integral of a function \(f(x) \) ---
provided we know an antiderivative of \(f(x) \). The down side is that there
are very many functions for which there is no "simple" antiderivative. For
example, \(f(x) = e^{x^2} \) has no antiderivative that can be expressed in
terms of any of the functions we have studied so far this semester. So
computing \(\int_0^5 e^{x^2} dx\) can only be done (or approximated) by using
Riemann sums.
Tuesday November 26
- Topics: definite integrals; importance of Riemann sums,
definition of charge density
- Text: 5.2; 5.3
- Tomorrow: fundamental theorem of calculus, part 1
- Comments: Today we looked at an example to see why
understanding Riemann sums is important in science. First we defined the
linear charge density of a point x on a wire immersed in an electromagentic
field to be the limit as the length goes to zero of the ratio of the charge
on a subinterval of the wire to the length of the wire. In symbols, if x is
in the interval \([x_{i-1}, x_i]\) and \(Q[x_{i-1}, x_i], \quad \Delta x\)
denote the total charge on the interval and the length of the interval
,respectively, then the linear charge density at the point x is \[\lambda(x)
= \lim_{\Delta x \rightarrow 0} \frac{Q[x_{i-1}, x_i]}{\Delta x} \] (provided
the limit exists). We then saw how we could use Riemann sums to develop a
formula for the total charge \(Q \) on a wire of length L: \[Q = \int_0^L
\lambda (x) dx.\] We then reviewed Left endpoint, Right endpoint and midpoint
methods for computing the signed area bounded by the graph of a continuous
function and the x-axis.
Monday November 25
- Topics: definite integrals
- Text: 5.2
- Tomorrow: definite integrals; fundamental theorem of
calculus part 1
- Comments: Today we introduced Riemann sums of a function
f(x) on domain [a,b] using a partition P and a sample C of points. The limit
as the norm of the partition goes to 0 is called the definite integral and
the fact that the limit of Riemann sums is the definite integral of f(x) over
the interval [a,b]. We also listed the basic properties of definite integrals
and worked some examples.
- The reworking of problem #1 on Exam 4 is due tomorrow (Tuesday
Nov 26) at the beginning of class.
Friday November 22
- Topics: approximating area, computing area, summation
notation
- Text: 5.1
- Tomorrow: definite integrals
- Comments: We discussed the importance of being able to
compute the area under the graph of a curve to science (lobsters on
treadmills) and then introduced the right endpoint, left endpoint, and
midpoint methods of approximating what we call the area under the curve
\(y=f(x), \quad [a,b] \). To do this well, we defined \(\Delta x = (b-a)/n \)
and introduced the summation notation \(\Sigma_{j=m}^n a_j \).
\[\begin{eqnarray*} R_n & = & \Delta x \Sigma_{j=1}^n f(a+ j\Delta x)
\\ L_n & = & \Delta x \Sigma_{j=0}^{n-1} f(a+ j\Delta x) \\ M_n &
= & \Delta x \Sigma_{j=1}^n f(a+ (j-\frac{1}{2}) \Delta x)
\end{eqnarray*} \] We then defined the area under a
continuous positive function \(y=f(x)\) on the interval \(a
\leq x \leq b\) to be \[A = \lim_{n \rightarrow \infty} = \lim_{n \rightarrow
\infty} L_n = \lim_{n \rightarrow \infty} M_n\]
Thursday November 21
- Topics: exam 4
- Text:
- Tomorrow: approximating area, computing area
- Comments:
Tuesday November 20
- Topics: exam 4 review
- Text: sections 4.1-4.7
- Tomorrow: exam 4
- Comments: Exam 4 is scheduled for Thursday
November 21.
Monday November 18
- Topics: antiderivatives
- Text: 4.9
- Tomorrow: exam 4 review
- Comments: antiderivatives are the result of doing a
"backwards derivative". More precisely, a function \(F(x)\) is an
antiderivative of the function \(f(x)\) if \(F'(x) = f(x).\) Every derivative
formula has a corresponding antiderivative formula. For example, since
\(\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)\), then the family of
antiderivatives of \(f(x)= -\csc(x)\cot(x)\) is the set of functions of the
form \(F(x) = -\csc(x)+C \) where \(C\) represents an arbitrary constant. The
specific notation for the antiderivatives of a function \(f(x)\) is called
the indefinite integral of \(f(x)\) and is written \\int f(x) \ dx. So for
our specific example, we have \[\int \csc(x) \ dx = -\csc(x)\cot(x). \]
Friday November 15
- Topics: applied optimization
- Text: 4.7
- Tomorrow: antiderivatives
- Comments: We used small group work to better understand
how to set up and solve optimization problems that are presented in paragraph
form.
- Announcement: Exam 4 is scheduled for Thursday November
21.
Thursday November 14
- Topics: applied optimization
- Text: 4.7
- Tomorrow: applied optimization
- Comments: We worked through more examples of applied
optimization problems in class.
Tuesday November 12
- Topics: applied optimization
- Text: 4.7
- Tomorrow: applied optimization
- Comments: This was the first of several days we will
spend on optimizing functions. We approached each example optimization
problem by phrasing it in the form: Optimize a function, Subject To some
constraints on that function. We then practiced at taking problems couched in
American English and converting them into mathematical problems for
maximizing of minimizing some function. Determining the maximum or minimum
then turned into a problem out of one of the previous sections. We also noted
that, at least in some cases, even though the problem does not satisfy the
hypotheses of the Extreme Value Theorem (and hence we do not know in advance
that a maximum or minimum exists), we can replace it with a problem that
is guaranteed to have that max or min. In one example, we
noted that solving the mathematical problem did not answer the given
question. Specifically, we asked to minimize a distance but the mathematical
problem we solved minimized the square of the distance. So, in solving these
problems, it is very important to record the answer to the question that is
posed.
Monday November 11
- Topics: sketching graphs
- Text: 4.6
- Tomorrow: applied optimization
- Comments: Today we saw how to use the signs of first and
second derivatives and asymptotes to determine the overall shape of the graph
of a function. By also plotting a few points on the graph, we were able to
draw a graph that illustrates where the function is increasing/decreasing or
concave up/down. This information will be useful in the next section where we
use it to determine maxima and minima of functions.
Friday November 8
- Topics: L'Hospital's Rule
- Text: 4.5
- Tomorrow: sketching graphs
- Comments: Today we looked at L'Hospital's Rule and did
several examples where f(x), f'(x), g(x), g'(x) were all continuous
functions. We also reviewed the seven indeterminate forms for limits (
\(\frac{0}{0}, \frac{\infty}{\infty}, 0 \cdot \infty, \infty - \infty,
1^{\infty}, 0^0, \infty^{0}\)) and noted that using algebra allows us to
exploit L'Hospital's Rule in all of the last five. We also talked about
comparing the growth of functions and introduced the notation \(f(x) \ll g(x)
\) to mean \[\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \infty. \]
We briefly mentioned that a limit of the form "\(0^{\infty}\)" was not
indeterminate. Here is why. Let f(x) and g(x) be functions for which
\[\lim_{x \rightarrow \infty}f(x) =0 \text{ and } \lim_{x \rightarrow
\infty}g(x) = \infty.\] Then,
\[\lim_{x \rightarrow \infty}g(x) \ln(f(x))=\infty \cdot -\infty = -\infty\]
so
\[\begin{eqnarray} \lim_{x \rightarrow \infty}f(x)^{g(x)} & = &
e^{\ln(f(x)^{g(x)})} \\ & = & e^{g(x) \ln(f(x))} \\ & = &
e^{-\infty} \\& = & 0.\end{eqnarray}\]
Thursday November 7
- Topics: problem day
- Text: 4.4
- Tomorrow: L'Hospital's Rule
- Comments:
Tuesday November 5
- Topics: shapes of graphs
- Text: 4.4
- Tomorrow: problem day
- Comments: We discussed and looked at examples of how to
use first and second derivatives to determine intervals on which a function
is increasing, decreasing, concave up, and convave down. This completes the
second of three steps in learning how to extract enough information from a
function to present an accurate graph of that function. Knowing how the
different shapes of a graph relate to the derivatives gives us a better sense
of the meaning of many comments we hear on the news (or might make as part of
a future job). For example, every week there are comments on the news along
the lines of "the unemployment rose but is slowing down". In other words, the
function U(t) that tracks how many people are unemployed at time t now has a
positive first derivative but a negative second derivative. So the graph of
U(t) is increasing but is concave down which doesn't bode as well for the
future as a would a concave up graph. Can you explain in your own
words why, in this situation for the unemployment function (positive
derivative, negative second derivative) it would be better to have a negative
third derivative than a positive one?
Monday November 4
- Topics: Mean Value Theorem; monotonicity
- Text: 4.3
- Tomorrow: shapes of graphs
- Comments: We looked at the Mean Value Theorem and then
used it to verify that the sign of the first derivative of a function \(f\)
can tell us on which intervals \(f\) is increasing or decreasing.
Specifically, if \(f'(x) \lt 0\) for all \(x \text{ in } (a,b)\) then \(f(x)
\) is strictly decreasing on the interval \((a,b)\). Similarly, if the
derivative is positive on an interval, then the function is increasing.
Friday November 1
- Topics: Extreme Values
- Text: 4.2
- Tomorrow: Mean Value Theorem; monotonicity
- Comments: There were lots of definitions and terminology
presented in class today as well as four theorems that underpin the way
calculus is used in "optimization". The mathematical approach is to find the
largest or smallest outputs of a function. The terms we defined were absolute
and local extrema (both maxima and minima) and critical points. The Extreme
Value Theorem tells us when we are guaranteed a function has
both absolute extrema (when the function is continuous on a closed, bounded
domain). Fermat's Theorem on critical points tells us that local extrema of a
function can only occur at critical points, and Theorem 3 tells us that
absolute extrema of a function can only occur at end points of the domain, at
points where the derivative of the function fails to exist, and at points
where the derivative of the function is zero. We then worked examples at
finding the critical points and absolute extrema of functions.
- The fourth theorem was Rolle's Theorem that gives a three-part scenario
where we are guaranteed the existence of a point where the
derivative equals zero (f is continuous on [a,b], differentiable on (a,b) and
f(a)= f(b)). We will do more with Rolle's Theorem on Monday.
Thursday October 31
- Topics: Exam 3
- Text: Sections 3.1-3.11
- Tomorrow: Extreme values
- Comments:
Tuesday October 29
- Topics: exam review
- Text: Sections 3.1-3.11
- Tomorrow: Exam 3
- Comments:
Monday October 28
- Topics: linear approximations
- Text: 4.1
- Tomorrow: exam review
- Comments: We looked at linear approximations and the
linearization of functions. The basic linear approximation formula is \(f(a+
\Delta x) -f(a) \approx f'(a) \Delta x\). This formula is most useful when
one does not know the function \(f(x)\) but does know both \(f(a) \text{ and
} f'(a).\) The formula is more accurate the smaller the value of \(\Delta
x\).
The linearization of the function \(f\) at the input \(a\) is the function
\( L(x)=f(a) + f'(a)(x-a)\). This is the function whose graph is the tangent
line to the graph of \(f(x)\) at the point \((a,f(a))\). If we set \(\Delta x =
x-a\), the linear approximation formula \(f(a+ \Delta x) -f(a) \approx f'(a)
\Delta x\) can be re-written in the form \(f(x) \approx L(x) \) for numbers
\(x\) that are close to \(a\)
We also discussed the error and percentage error when using the linear
approximation formula. The textbook also give a formula that provides a bound
on that error: \(\text{Error } = |\Delta f - f'(a) \Delta x| \leq K \frac{1}{2}
(\Delta x)^2 \) where \(K\) is the largest value of the absolute value of
\(f''(x)\) that occurs on the interval from \(a\) to \( a+\Delta x\).
Exam three is scheduled for next Thursday (October 31) and will cover all of
Chapter 3. Here is the objectives
list.
Friday October 25
- Topics: Related rates
- Text: 3.11
- Tomorrow: linear approximations
- Comments: We carefully worked three examples of related
rates problems by following a procedure designed to help transform a "word
problem" into a mathematics problem. The basics of that procedure are:
- Carefully read the problem all the way through.
- Find, label and describe all quantities relevant to the problem ---
especially those that have rates of change specified or implied in the
problem statement.
- If possible, draw a picture and label the appropriate parts of the
picture.
- Find an equation relating the above quantities.
- Use the Chain Rule to differentiate both sides of the equation.
- Answer the question that was asked.
Exam three is scheduled for next Thursday (October 31) and will cover all of
Chapter 3. Here is the objectives
list.
Thursday
October 24
- Topics: implicit differentiation
- Text: 3.10
- Tomorrow: Related rates
- Comments: Today we focussed on distinguishing explicitly
defined functions and implicitly defined functions. Although the equations
defining the latter can sometimes be hard (or even impossible) to rewrite in
an explicit manner, it is always possible to find the
derivative of the implicit function. The method requires that we pay
close attention to when we need to use the chain rule. For example, if we are
given the equation \(x^2 +y^3 = y\) which implicitly defines a function
\(y=f(x) \) for points \((x,y)\) near the point \((0,-1)\), then by rewriting
the equation as \(x^2 +(f(x))^3 = f(x)\) and taking the derivative of both
sides we obtain \(2x +3(f(x))^2 f'(x) = f'(x)\). This last equation can then
be solved for \(f'(x)=\frac{-2x}{3y^2-1} \).
The homework for tomorrow includes reading Section 3.11 on related rates
before class. Note for homework : Use the derivative at P to write the
equation of the line perpendicular to the tangent at P. Then use that line to
determine R.
Friday October 18
- Topics: Derivatives of inverse functions, using the
chain rule with exponential and logarithm functions
- Text: 3.8, 3.9
- Tomorrow: derivatives of hyperbolic trigonometric and
inverse hyperbolic trigonometric functions, implicit differentiation
- Comments: We continued adding to our toolbox of
differentiation formulas by including a method (and formula) for finding the
derivative of an invertible differentiable function. After the next class
period, we will have all of the basic formulas and rules for derivatives
(except the generalized power rule). You can find a list of all of these
(except implicit differentiation) in numbers 1-5 of this
web page.
Next week we will start using derivatives to analyze the behavior of
functions.
Thursday Oct 17
- Topics: Problem Day
- Text:
- Tomorrow: Derivatives of inverse functions
- Comments: We spent the day working example problems from
the homework and problems from this
handout.
Tuesday Oct 15
- Topics: chain rule
- Text: 3.7
- Tomorrow: Problem Day
- Comments: We derived the formula for the chain rule and
looked at a simple example that illustrated that the rate of change of the
composition \(F(x)=f(g(x))\) is expressed as the product of the rates of
change of f and g: \[F'(x)=f'(g(x))g'(x).\]
We then worked a number of examples.
Monday October 14
- Topics: higher order derivatives; derivatives of
trigonometric functions
- Text: 3.5, 3.6
- Tomorrow: chain rule
- Comments: Higher order derivatives are derivatives of
derivatives. The second derivative is the derivative of the first derivative,
the third derivative is the derivative of the second, etc. For notation we
use
function (0th derivative) |
\(f(x)\) |
\(f^{(0)}(x)\) |
none |
first derivative |
\(f'(x)\) |
\(f^{(1)}(x)\) |
\(\frac{df}{dx}\) |
second derivative |
\(f''(x)\) |
\(f^{(2)}(x)\) |
\(\frac{d^2f}{dx^2}\) |
third derivative |
\(f'''(x)\) |
\(f^{(3)}(x)\) |
\(\frac{d^3f}{dx^3}\) |
fourth derivative |
none |
\(f^{(4)}(x)\) |
\(\frac{d^4f}{dx^4}\) |
\(n\) th derivative |
none |
\(f^{(n)}(x)\) |
\(\frac{d^nf}{dx^n}\) |
We also derived the derivative of the cosine and used the quotient rule to
find the derivative of the tangent. Then we listed the derivatives of all six
trigonometric functions.
function |
derivative |
cofunction |
derivative of cofunction |
\(\sin(x)\) |
\(\cos(x)\) |
\(\cos(x)\) |
\(-\sin(x)\) |
\(\tan(x) \) |
\(\sec^2 (x)\) |
\(\cot(x)\) |
\(- \csc^2(x) \) |
\(\sec(x) \) |
\(\sec(x)\tan(x) \) |
\(\csc(x)\) |
\(-\csc(x)\cot(x)\) |
Friday October 11
- Topics: Rates of change
- Text: 3.4
- Tomorrow: higher order derivatives; derivatives of
trigonometric functions
- Comments: We looked at how derivatives are useful for
computing the rate of change of functions with respect to their input
variables. For a function \(y=f(t)\) that gives the position of an object at
time \(t\), the rate of change of position with respect to time \(f'(t)\) is
called the velocity of the object at time \(t\). Velocity
can be negative if the object is moving opposite to the positive \(y\) axis
and speed is the absolute value of velocity. We also looked
at the rate of change of an inflating sphere with respect to its radius and
it was pointed out that the rate of change of volume of a sphere with respect
to its radius is exactly equal to the surface area of a sphere of that
radius. One intuitive way to think about this is to consider an inflating
spherical onion where the inflation is being done by "adding" new, very thin,
onion skins to the current sphere. This idea will be made precise in second
semester calculus. We also looked at two applications of rates of change:
Newton's Law of cooling which gives a mathematical model of the rate of
change of the temperature of an object with respect to time and marginal
cost, which is the change in cost, after producting x items, of producing one
more item. Since this can be mathematically modeled by
\(\frac{C(x+1)-C(x)}{1}\) where \(C(x)\) is the cost function, then we see
that marginal cost is approximated by the derivative of the cost function at
\(x\) since we can use \(h=1\) in the approximation \[C'(x) \approx
\frac{C(x+h)-C(x)}{h}.\]
Thursday October 10
- Topics: Sections 2.1-3.1
- Text: Exam Two
- Tomorrow: Rates of change
- Comments:
Tuesday October 8
- Topics: Exam Two Review
- Text: objectives for exam 2
- Tomorrow: Exam 2
- Comments:
Monday October 7
- Topics: product and quotient rules for derivatives
- Text: 3.3
- Tomorrow: Exam Two Review
- Comments: We carefully developed the product rule for
derivatives and noted the quotient rule and then worked a number of
examples.
The rules are: \[\begin{eqnarray}\frac{d}{dx}\left[ f(x)g(x) \right] & =
& f'(x)g(x)+f(x)g'(x) \\ \frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] &
= & \frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)}. \end{eqnarray}\]
Friday October 4
- Topics: derivatives as functions
- Text: 3.2
- Tomorrow: product and quotient rules for
derivatives
- Comments: Today we defined the derivative
function as distinct from evaluating the derivative of a
function at the number \(x=a\). We also proved the power rule and noted the
two linearity rules and the derivative of the natural exponential function.
We ended by noting the relationship between the graph of a function \(f(x)\)
and the graph of its derivative function \(f'(x)\).
Exam 2 is scheduled for Thursday October 10 and will cover the material in
Sections 2.1 to 3.2. As usual, you may stay until 1:50. Here is a link to the objectives for exam 2.
Thursday October 3
- Topics: Problem session
- Text: 2.7, 2.8, 2.9, 3.1
- Tomorrow: derivatives as functions
- Comments: We practiced evaluating limits with
indeterminate forms. This included using the definition to compute the
derivatives of functions.
Exam 2 is scheduled for Thursday October 10 and will cover the material in
Sections 2.1 to 3.2. As usual, you may stay until 1:50. Here is a link to the objectives for exam 2.
Tuesday October 1
- Topics: derivatives
- Text: 3.1
- Tomorrow: Problem Session
- Comments: The rates at which the outputs of functions
change with respect to changes in their inputs is at the heart of the
differential calculus. We introduced this idea by looking at graphs of a
function \(f(x) \) near a point \((c,f(c)) \). For certain "nice" functions,
the slopes of the (secant) lines that pass through the points \((c,f(c))
\text{ and } (c+h,f(c+h)) \) appear to be limiting to a single value as \( h
\text{ limits to } 0 \). When this happens, we say that the function \(f\) is
differentiable at \(c\) or that \(f\) has a derivative at \(c\). For
notation, we use \(f'(c)\) to designate this slope. That is, if the following
limits exist, then we have \[\begin{eqnarray} f'(c) & = & \lim_{h
\rightarrow 0} \frac{f(c+h)-f(c)}{h} \\ & = & \lim_{x \rightarrow c}
\frac{f(x)-f(c)}{x-c} \end{eqnarray}\]
We ended by working several examples.
Monday September 30
- Topics: formal limits
- Text: 2.9
- Tomorrow: derivatives
- Comments:
We finished off Chapter 2 by giving the formal definition of a
limit. Specifically,
- \(\lim_{x \rightarrow c}f(x) = L\) means
- Given any positive number \(\epsilon\), there is a positive number
\(\delta \) satisfying: if \(0 \lt |x-c| \lt \delta, \) then \(|f(x)-L|
\lt \epsilon\).
We then worked several examples by, first, doing some scratchwork to see how
the \( \epsilon \text{ and } \delta \) are related and second,
carefully presenting each step of a logical argument that
starts with \(0 \lt |x-c| \lt \delta, \) and ends with \(|f(x)-L| \lt
\epsilon\).
The ability to clearly present a carefully crafted logical argument is one
of the most important skills that an undergraduate can develop and writing
clear and logically correct formal proofs of the existence of limits is
excellent pratice.
Friday September 27
- Topics: trigonometric limits, Intermediate Value
Theorem
- Text: 2.6, 2.8
- Tomorrow: formal limits
- Comments: We proved two essential trigonometric limits
that will be used in Chapter 3 to develop formulas for the derivatives of the
trig functions. Specifically, we showed that \[\lim_{\theta \rightarrow 0}
\frac{\sin(\theta)}{\theta}= 1\] and then used that fact and trigonometric
identities to show \[\lim_{\theta \rightarrow 0}
\frac{1-\cos(\theta)}{\theta}= 0.\] We then talked about one of the most
important theorems in calculus.
Intermediate Value Theorem(IVT): If \(f\) is a continuous
function on the interval \( [a,b] \) and if \(L\) is any number between
\(f(a) \text{ and } f(b)\), then there is a number \(c\) between \(a\) and \(
b\) for which \(f(c)=L\).
We then used the IVT three times to see that there is a solution, \(x_0
\),to the equation \(x^3 +2x+1=0\) and that \(x_0 = -\frac{3}{8} \pm
\frac{1}{8}\). This technique is called the Bisection Method and is one of
the very first algorithms used in computers to approximate solutions to
equations. The algorithm that is used by your calculator to approximate the
solution probably uses the Bisection Method to obtain a rough estimate but
then shifts over to a more sophisticated, and computationally much faster,
method to refine that estimate to 12 (or more) significant figures of
accuracy.
I updated the course schedule earlier this week so be sure to use the
current one when looking at what we will be covering. I also made some
changes to homework due dates for this week so check the homework page to see
what is due next.
Thursday September 26
- Topics: problem session; trigonometric limits;
- Text: Problem session
- Tomorrow: trigonemetric limits, Intermediate Value
Theorem
- Comments: Rather than collect homework, we looked at a
number of example problems from section 2.5 (indeterminate forms) and a
single example from section 2.6 that used the Squeeze Theorem. We noted: (1)
it is very useful to notice occurrences of the difference of two squares \(
A^2 - B^2\) when we need to factor expressions and (2) when computing limits
of elementary functions, it is a very good idea to first check the "form" of
the limit. Limits that are not in one of our indeterminate forms do not
require algabraic simplification for us to compute them.
Tuesday September 24
- Topics: continuity; indeterminate forms; limits of
noncontinuous functions; trigonometric limits.
- Text: 2.6
- Tomorrow: trigonometric limits; problem session
- Comments: We started off looking at questions from the
homework. We then listed the indeterminate forms for limits
\[\frac{0}{0}, \quad \frac{\infty}{\infty}, \quad \infty \cdot 0 \quad \infty
-\infty, \quad 1^{\infty}, \quad \infty^0\] and worked a number of examples.
We finished off by stating and proving the Squeeze Theorem. This is one of
the few careful proofs we will see this semester but it nicely illustrates
the fact that mathematics is about what we actually know.
The is the only reason that people ue mathematics to answer computational
questions.
Monday September 23
- Topics: Continuity
- Text: 2.4; 2.5
- Tomorrow: continuity; indeterminate forms; limits of
noncontinuous functions; trigonometric limits.
- Comments: Today's topic is continuity of functions. We
looked at the three part defition of a function \( f\) being continuous at
the number \( x=c \):
- The number c must be in the domain of the function f (i.e., f(c) must be
a number.)
- The limit \(\lim_{x\rightarrow c}f(x)\) must exist.
- The two numbers above must be equal \(\lim_{x\rightarrow c}f(x) =
f(c)\).
We also looked at the rules for combining known continuous functions to
obtain other continuous functions and began to explore how to evaluate limits
when the functions involved are not continuous. Specifically when the limits
take on one of the indeterminate forms \(\frac{0}{0},
\frac{\infty}{\infty}, \infty \cdot 0, \infty - \infty, 1^{\infty}, \infty^{0}
\).
Friday September 20
- Topics: Basic limit laws; limits at infinity
- Text: 2.3; 2.7
- Tomorrow: Continuity
- Comments: Today we looked at the basic theorem for
manipulating limits (Theorem 1 on page 77). This theorem applies
only to functions whose limits are known to exist. In words,
it tells us that the limit of a sum, difference, product, quotient, or root
of functions is the sum, difference, product, quotient or root of the limits,
respectively. We used very basic examples to illustrate the use of the
theorem as a precursor to the more sophisticated uses we will see later in
the semester.
- We also began looking at "limits to infinity" and noted that if such a
limit exists, say \(\lim_{x\rightarrow \infty} f(x) = L \), then the graph of
\(y=f(x) \) will have the line \( y=L \) as a horizontal asymptote to the
right. The book also discusses limits as x goes to negative infinity which
behave in a similar fashion.
- We finished with a brief discussion of Hilbert's Hotel as an example
illustrating that "infinity" does not always behave the way we expect. This
class does not address how to think carefully about infinity and we only use
the symbol "\( \infty \)" as a notational aid to depict the concept of
functions "growing without bound". If you are intrigued by Hilbert's Hotel,
you might consider reading the book "Infinity and the Mind" by Rudy Rucker.
It is written with calculus students in mind as an audience and discusses
Hilbert's Hotel along with many other fascinating aspects of "infinity".
Thursday, September 19
- Topics:
- Text: Exam 1
- Tomorrow: Basic limit laws; limits at infinity
- Comments: Tutoring Hours are now posted on-line at this link.
Tuesday September 17
- Topics: Exam 1 review; exam 1 objectives
- Text: Exam 1 review
- Tomorrow: Exam 1
- Comments: Tutoring Hours are now posted on-line at this link.
Monday September 16
- Topics: Limits, numerically and graphically
- Text: 2.2
- Tomorrow: Exam 1 review; exam 1 objectives
- Comments: Although the limit of a function \(f\) at
\(x=c\)is what the function "ought to be", it is not always clear what the
actual value of the limit is. In these cases it is useful to apply numerical
or graphical techniques to obtain evidence that the limit does not exist or,
if it deoes, evidence of the value of the limit. Once one obtains a
reasonable estimate for the value \(L\), then it is possible to "prove" that
\(\lim_{x\rightarrow c}f(x)=L \) by making a valid argument that \(|f(x)-L|
\) can be made arbitrarily small provided that \(x\) is
sufficiently close to \(c\) but we do not care what happens when
\(\mathbf{x=c}\).
We did a few examples of this process and also noted that infinite limits
indicate the existence of asymptotes in the graph of $f$ and that we can also
compute the "one-sided" limits \(x\rightarrow c-}f(x) \) and \(x\rightarrow
c+}f(x) \). We ended by generating the graph of a function that satisfied
several different restrictions associated with limits.
Exam 1 will take place this coming Thursday September 19.
It will cover: logic, sets, Chapter 1, and Section 2.1. Here is the Exam Objectives Sheet.
Friday September 13
- Topics: Limits; rates of change, tangent lines
- Text: 2.1
- Tomorrow: Limits, numerically and graphically
- Comments: Conceptually, calculus can be divided into the
differential calculus and the integral calculus. For the
next several weeks we will be focusing on the former but both heavily rely on
the concept of limits. One, intuitive way of describing a limit of the
function \(f\) at \( x=a \), is to say that, regardless of whether or not the
number \(a\) is in the domain of \(f\), the limit of \(f\) as \(x\)
approaches \(a\) is the number that \(f(a)\) "ought" to be.
We noted that not all functions have this property but that calculus is
designed to work with those that do. We then presented an example of such a
function. Specifically, we showed that \(f(x) = \frac{9x^2 -1}{3x-1} \)
"ought" to take on the value \(2\) when \(x=1/3\) by showing that it is
always possible to force \(f(x) = 2 \pm \text{ (any error
bound) }\) by only using values of \(x\) that satisfy \(x=1/3 \pm \text{
(tolerance) } \) for some tolerance (we used the tolerance
that is one-third of the desired error bound). That is
- \[ \text{if } x=1/3 \pm \frac{\text{error bound}}{3}, \text{ then } f(x)
= 2 \pm \text{ (error bound).}\]
Exam 1 will take place this coming Thursday September 19.
It will cover: logic, sets, Chapter 1, and Section 2.1. I will post an Exam
Objectives sheet before Monday evening.
Thursday September 12
- Topics: Exponentials; logarithms
- Text: 1.6
- Tomorrow: Limits; rates of change, tangent lines
- Comments: Continuing the theme of chapter one, we
reviewed exponential and logarithmic functions. Logarithm functions are the
inverses of exponential functions so
- \( \log_b(b^x) =x \text{ for each } x \in \mathbb{R}, \text{ the domain
of } b^x \)
- \( (b^{\log_b(x)}) =x \text{ for each } x \in (0,\infty), \text{ the
domain of } \log_b(x) \)
- We also introduced the hyperbolic trigonometric functions:
- \( \cosh(x) = \frac{1}{2} \left( e^x + e^{-x} \right) \)
- \( \sinh(x) = \frac{1}{2} \left( e^x - e^{-x} \right) \)
- \( \cosh^2 (x) - \sinh^2 (x) =1 \).
The other hyperbolic trigonometric functions are defined in a similar manner
to the circular trigonometric functions. For example, the hyperbolic tangent,
\(\tanh(x) \), is the result of dividing the hyperbolic sine by the hyperbolic
cosine.
Tuesday September 10
- Topics: Inverse functions
- Text: 1.5
- Tomorrow: Exponentials; logarithms
- Comments: Today was another review day. We pointed out a
number of useful trigonometric identities that can be found on pages 29 and
30 of our textbook. and noted the easily remembered values of the
trigonometric functions:
|
0 |
π/6 |
π/4 |
π/3 |
π/2 |
|
|
|
|
|
|
|
|
|
|
|
|
- We then discussed inverse functions and noted that only one-to-one
functions can have an inverse. Inverse functions "unwind" the effects of the
original function in the following sense. If \(f\) is a function with domain
D and range R, and \(f^{-1}\) is the inverse function to \(f\) having domain
R and range D, then \(\text{for each } x \in D, \ f^{-1}(f(x)) = x \) and
\(\text{for each } y \in R, \ f(f^{-1}(y)) = y \).
We finished by using \(tan(x)\) and it's inverse function \(\arctan(x)\) to
illustrate that the graph of \(f^{-1}\) is the reflection across the line
\(y=x\) of the graph of \(f\).
Monday September 9
- Topics: Polynomials, rational functions, algebraic
functions, trigonometric functions
- Text: 1.3; 1.4
- Tomorrow: Inverse functions
- Comments: Today we reviewed the definitions and
terminology for power functions, polynomials, rational functions, algebraic
functions, and trigonometric functions. We defined one radian to be the
measure of an angle from the center of a unit circle which subtends an
arclength equal to one radius. This means that the arclength of the portion
of a circle (of radius r subtended by an angle of θ is
rθ. We also noted that if we replace each x in the
equation y=f(x) by x-h and each y by y-k, then the graph of \( y-k =f(x-h) \)
is the result of translating the graph of \(y=f(x) \) horizonally by \(h\)
units and vertically by \(k\) units. An extra homework problem (not to be
turned in) was to sketch (labeling important points) the graph of a function
similar to
- \( y= 6\sin{(4(x-2))}+5. \)
Friday September 6
- Topics: Properties of functions
- Text: 1.1; 1.2
- Tomorrow: Polynomials, rational functions, exponential
functions, logarithms, trigonometric functions
- Comments: Today we reviewed the definitions and notation
for intervals of real numbers, developed geometric intuition for the meaning
of \(|a-b| \) and used it for geometric intuition of the set \(\{ x \in
\mathbb{R} : |x-a| \leq r \} \). We also reviewed linear functions and linear
equations and recalled the general form of a quadratic equation \(ax^2 +bx+c
=0 \) as well as the quadratic formula: \[ x= \frac{-b \pm \sqrt{b^2
-4ac}}{2a}.\]
Thursday September 5
- Topics: Basic logic; basic set theory; real numbers
- Text: Basic Set
Theory
- Tomorrow: Properties of functions; trigonometric
functions
- Comments: If you are interested in learning more about
logic and thinking mathematically, consider taking the Coursera course
"Introduction to Mathematical Thinking" offered by Keith Devlin. He is the
"NPR Math Guy" if you listen to public radio.
- Today we looked at basic information about sets. The most important
aspect of sets is membership. We have two standard ways of
writing sets:
- listing the members --- for example {1,2,3,4}
- predicate notation --- for example \( \{x \in A : x^2 \leq 1\} \)
We also defined various ways of combining sets:
- Union: \(A \cup B = \{x : x \in A \text{ or } x \in B \} \)
- Intersection: \(A \cap B = \{x : x \in A \text{ and } x \in B \} \)
- Cartesian Product: \(A \times B = \{(a,b) : a \in A \text{ and } b \in B
\} \)
The handout also defined a function f with domain the set A
and codomain the set B to be a subset of \(A \times B = \{(a,b) : a \in A
\text{ and } b \in B \} \) satisfying the property that there cannot be two
ordered pairs in the function that have the same first coordinate but different
second coordinates. This means that graphs of functions are literally pictures
of the ordered pairs that make up the function,
Thursday September 5
- Topics: Basic logic; basic set theory; real numbers
- Text:
- Tomorrow:
- Comments: If you are interested in
learning more about logic and thinking mathematically, consider taking the
Coursera course "Introduction to Mathematical Thinking" offered by Keith
Devlin. He is the "NPR Math Guy" if you listen to public
radio.
Tuesday September 3
- Topics: Logistics; sets; logic; numbers
- Text: Basics of logic
handout
- Tomorrow: Basic logic; basic set theory; real
numbers
- Comments: I will occasionally be posting mathematics in
these comments. Please let me know if the two statements after the bullets
below do not show up in mathematics on your browser.
There is a homework assignment for logic. See the homework area of this
webpage.
- Today we reviewed the course information sheet and then started
discussing the foundation of all mathematics --- logic. Logical statements
are statements that are exactly one of "True" or "False" and can either be
statements without variables (e.g., \(5 \leq 3 \), or statements with
variables as long as the variables are quantified (e.g., for all real numbers
\(x, \quad x^2 \geq 0 \)) is a true logical statement. There are two
quantifiers: universal ("for all", \(\forall \)) and existential ("there
exists", \(\exists \) ). Although we did not talk about it in class today
(but we will tomorrow), the rules for negating quantified logical statements
are:
- \( \lnot (\forall x, \ p_x ) \Leftrightarrow \exists x, \ \lnot p_x\),
and
- \( \lnot (\exists x, \ p_x ) \Leftrightarrow \forall x, \ \lnot p_x\)
We also discussed the truth values of conjunctions "p and q",
disjunctions "p or q",andimplications "if p, then q"
as well as one of the most important of logical facts that justifies making a
deduction (called modus ponens): If we know that "p implies q" and "p"
are both true logical statements, then we deduce that "q" must also be true.
We will soon start using greek letters in our discussions. Here is a handout
to help you familiarize yourself with Greek in mathematics .
Handouts
Generally Useful Links
Writing and Studying Tips
Chapter.Section |
Type 2 Exercises |
Due |
Type 1 Exercises |
Due |
Notes |
Logic Handout |
2,3,5,7, 8 |
none |
1 |
Friday Sep 6 |
|
Set Theory Handout |
1,2,3 |
none |
none |
none |
|
1.1 |
5,15,20,55 |
Tuesday Sep 10 |
58 |
Tuesday Sep 10 |
|
1.2 |
10,17,21,22,56 |
Thursday Sep 12 |
none |
none |
|
1.3 |
4,6,9,13,14,25,33,35 |
Thursday Sep 12 |
34 |
Thursday Sep 12 |
|
1.4 |
2,9,25,28,29,31,33,34 |
Thursday Sep 12 |
30 |
Thursday Sep 12 |
|
1.5 |
10,16,25,31,45 (Extra Credit) |
Friday Sep 13 |
48 (Extra Credit) |
Friday Sep 13 |
|
1.6 |
1,4,8,16,20,28,32 |
Monday Sep 16 |
34 |
Monday Sep 16 |
|
2.1 |
1-4,7,11,15,17 |
Tuesday Sep 17 |
9 |
none |
|
2.2 |
3,4,20,28,50,61,65 |
Friday Sep 20 |
14 |
Friday Sep 20 |
|
2.3 |
7,10,15,16,25,26,37,38 |
Tuesday Sep 24 |
33 |
Tuesday Sep 24 |
|
2.7 |
5,8,9,13,18,28,29 |
Tuesday Sep 24 |
one of 28,38 |
Tuesday Sep 24 |
|
2.4 |
3,6,9,22,29,47,55,57 |
Thursday Sep 26 |
58,66 |
Friday Sep 27 |
|
2.5 |
4,8,17,21,30,31,36,37,40,41 |
Friday Sep 27 |
34,54 |
Friday Sep 27 |
|
2.6 |
5-10,15,18,30,35,36,41,47 |
Tuesday Oct 1 |
46 |
Tuesday Oct 1 |
|
2.8 |
5,14,17 |
none |
21-24 |
none |
|
2.9 |
5,6,13,17 |
none |
22 |
none |
|
3.1 |
1,4,9,15,23,27-30,33,39,40 |
Monday Oct 7 |
32 |
Monday Oct 7 |
|
3.2 |
1,4,7,...,43(by 3's), 50,51,54 |
Friday Oct 11 |
46 |
Friday Oct 11 |
|
3.3 |
7,10,13,15,17,23,24,28,49 |
Monday Oct 14 |
42 |
Monday Oct 14 |
|
3.4 |
1-29 odd, 31,36,41,48,49 |
Tuesday Oct 15 |
35 |
Tuesday Oct 14 |
|
3.5 |
2,5,11,16,19,24-51(by 3's) |
Thursday Oct 17 |
36 |
Thursday Oct 17 |
|
3.6 |
1-13 odd,21,24,28,39,44 |
none |
28 |
none |
|
3.7 |
13-40(by 3's),51,52,57,61,67-88(by 3's) |
Thursday Oct 24 |
68 |
Thursday Oct 24 |
|
3.8 |
21-40 odd |
Friday Oct 25 |
36 |
Friday Oct 25 |
|
3.9 |
4-64(by 4s) |
none |
66 |
Monday Oct 28 |
|
3.10 |
5,11,12,17,31,40,43,45,48,54 |
Monday Oct 28 |
54 |
Monday Oct 28 |
See note in Day-by-Day |
3.11 |
2,8,11b,13-35 odd,45 |
none |
40 |
none |
Be sure to do #40 |
4.1 |
5,7,9,18,19,29,30,31,39,47,57 |
Tuesday Nov 5 |
42 |
Tuesday Nov 5 |
|
4.2 |
1,5,23,24,29,48,53,55,57 |
Thursday Nov 7 |
74 |
Thursday Nov 7 |
Work must be neat |
4.3 |
1,3,11,12,13,15,19,21,29,43,47,55 |
Friday Nov 8 |
50 |
none |
|
4.4 |
1,3,13,15,19, 25,29,33,35,43 |
Tuesday Nov 12 |
56 |
none |
Read #61 |
4.5 |
7,13,15,23,29,45,59,62,74 |
Tuesday Nov 12 |
60(a) |
Tuesday Nov 12 |
|
4.6 |
4,5,13,19,25,29,41,46,59,69 |
Monday Nov 18 |
66 |
none |
|
4.7 |
3,15,19,21,23,29,33,39,43,51,59 |
Tuesday Nov 19 |
47 |
Tuesday Nov 19 |
|
4.8 |
none |
|
none |
|
|
4.9 |
9-69 (by 3's) |
Monday Nov 25 |
40,46 |
Monday Nov 25 |
|
5.1 |
3-51 (by 3's), 55, 75 |
Tuesday Nov 26 |
62 |
Tuesday Nov 26 |
|
5.2 |
9,11,17,21,27,39,44,58,61,67 |
Monday Dec 2 |
74 |
Monday Dec 2 |
|
5.3 |
9,10,15,16,17,18,39,42,45,47,59 |
Thursday Dec 5 |
60 |
Thursday Dec 5 |
|
5.4 |
3-33(by 3's),41,42,52,53 |
Friday Dec 6 |
53 |
Friday Dec 6 |
|
5.5 |
3-27 (by 3's) |
TBA |
25 |
TBA |
|