Mathematics 180E (Fall 2013)

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Contents

• Course Logistics
• Office Hours --- Finals Week Office Hours
• Homework Assignments
• Day-by-Day Notes
• Handouts
• Exam Objectives
• Useful Links
• Examination Archive
• Interesting Links
• Emergency Procedures

Course Logistics

• Course Information (Printable)
• Tentative Course schedule
• Grade Information
• Math Tutoring Hours (Thompson 390)

Day-by-Day Notes

Day and Date

Topics:

Text:

Tomorrow:

Comments:

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 has been canceled.

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:

1. Carefully read the problem all the way through.
2. Find, label and describe all quantities relevant to the problem --- especially those that have rates of change specified or implied in the problem statement.
3. If possible, draw a picture and label the appropriate parts of the picture.
4. Find an equation relating the above quantities.
5. Use the Chain Rule to differentiate both sides of the equation.
6. 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

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.

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.

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:

\(\lim_{x \rightarrow c}f(x) = L\) 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: First we worked a number of the limit problems from section 2.5 that were due today. Then 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: trigonometric 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.

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:

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 \(\text{for each } y \in R, \ f(f^{-1}(y)) = y \).

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, algebraic functions 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.

Thursday September 5

Topics: Basic logic; basic set theory; real numbers

Text: Basic Set Theory

Tomorrow: Properties of 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.

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", \(\exists \) ). 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", implications "if p, then q", andequivlances "p if and only if 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

• Course Information Sheet (Tuesday September 3)
• Basic Logic (Tuesday September 3)
• Basic Set Theory(Friday September 6)
• Greek in mathematics (Friday September 13)

Exam Objectives

• Exam 1 Objectives
• Exam 2 Objectives
• Exam 3 Objectives
• Exam 4 Objectives
• Exam 5 Objectives

Generally Useful Links

• University Calendars

Writing and Studying Tips

• How to Study An exceptionally good study guide from William Rapaport.
• Basic Tips for writing mathematics
• ``The Elements of Style'' by William Strunk, Jr. and many other classical reference works. Strunk's "Style" is the standard reference for expository writing and is now available in searchable electronic form. (Courtesy of Project Bartleby).

Homework Problems

Examination Archive