Online Video Lectures for Calculus with Analytic Geometry I, MATH 2413 These lectures cover the basic material for Calculus I at an introductory level. Students study the lectures at home and do the homework assignments. Class time will be used for discussion, problem solving, and presentation by students. Videos are narrated and appear as handwritten in digital ink.
Last Update: January-10-2009
First Update: March-20-2006
Inception: March-20-2001

The Current Lecture is on material between the red bands.

The Upcoming Test is on material between the green bands.

Final is on Wednesday May 6 during class time. Good luck on the test. Remember that substituion details should be shown for problems of 5.5.
We did not get to finish the last two sections. Study those before related sections of Calculus 2 or 3.
Do not forget to complete the course evaluation form.
Compatibility:
The video content of this page works Windows and has WMV (Windows Media Video) format. Other operating systems (e.g Apple/Mac) may not be compatible with the videos. In particular you should not experience any problems with Windows Media Player and a variety of browsers (Internet Explorer, Firefox, Chrome) on Windows XP or Vista.

Basics:
Here you will find the daily lectures. Clicking on any video link on Windows opens up windows media player (WMP), or a designated compatible player, on your computer and starts the presentation. A typical video file is about 20-30 MB and takes about 3 minutes on a basic DSL/Cable line to download. Maximize the window size for WMP and adjust the volume. You can scroll back and forth in the file (up to the point it has been downloaded). You can also use the fast forward. To save the file for viewing at a later time (for example in case you have a dial-up) Right Click and use Save Target As.

Caution:
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Errors:
Errors and typos do occur in these lectures. I appreciate if you bring them, and any other suggestions you have, as well as a list of malfunctioning links, to my attention. After some videos you may see a Correction notice related to such errors.

Effectiveness:
To improve your experience with this course consider the following:
(1) Take notes while you view the videos, as in a traditional lecture.
(2) Pause the video, try to solve the presented problem by yourself, and then compare your solution with the presented solution.
(3) Become part of a study group.
(4) View the lectures on a large screen monitor, perhaps with your study group.
(5) Participate in class presentation.
(6) Make flash cards by noting solved problems you have studied (in videos, class, text), note the location of solution on the card, and time required for writing the solution anew from scratch. Store the flash cards in a box. Every week give yourself a timed test by randomly picking about 7-10 cards from the recent collection. Review the sections where you did not reproduce the solution correctly. Try cumulative tests once a month.

Does hybrid style work?
An experiment was conducted in 2006-7. The students who took Calculus 2413 in fall of 2006 were tracked into spring of 2007. Those who took the hybrid format section in Calculus I had an average* of 2.25 in Calculus 2. The others had an average of 1.65 in Calculus 2.
Some of the possible explanations for the significant improvement of the grades of the hybrid group are:
1- Higher time requirement.
2- Statistical fluctuation. We did not have data on the student backgrounds. It is possible that this particular class had substantially better preparation from the start. At a minimum we either needed the result of a placement test or data on the number of students who had taken calculus in high school.
3- Placebo effect. Also known as halo or pet-project effect.
4- Class participation. It is known that moderately high levels of adrenaline can make a lasting impression of events in your brain. That is why you remember traumatic or pleasant events for a long time. A class presentation typically causes some levels of anxiety, and hence moderately high levels of adrenaline, for the prepared presenter. Hence the student is more likely to retain the material and perform better in the long run.

* The average was obtained by setting, A=4, B=3, C=2, D=1, O=all other grades(F,Q,W,NG,I,...)=0.
For comparison and calibration the grade distribution of the hybrid group in fall of 2006 was set to be similar to the distribution of grades of all students in Calculus I fall 2005. None of calculus II classes were of hybrid type. There were 40 students in the hybrid class and 180 students in total.

Design:
To see how to make similar videos and associated issues check the following sites. These products were not used here but could be useful for an extended version of hybrid courses. As expected, the hardware and software change rapidly and searching for new approaches should be considered.
mathcasts.org discussion of hardware and software.
xthink.com handwriting recognition for math.
techsmith.com hints and user forum for screen recording.
www.imaginationcubed.com An early prototype of shared whiteboard page with built-in recorder, no voice, log-in is not required.
For a combination of voice and shared whiteboard you can use the following (log-in required, no recorder, options/software change frequently):
www.live.com (or msn.com or hotmail.com) provides "shared online activities" on the Instant Messenger, including shared whiteboard. Call, and then on Call Window go to Shared Fun Activities (fifth icon)/Whiteboard.
www.skype.com, go to Tools/Do More/SketchPad or similar software.

Accessibility:
The videos do not have subtitles as yet. At this time hearing impaired students need an interpreter to view videos.

Audience:
This course is primarily intended for students majoring in Mathematics, Physics, Engineering, and Computer Science. Some branches of Chemistry, and Geology, as well as some specialties in Biology, Psychology, and Business may also require this course.

Text:
We will follow Calculus, Early Transcendentals, by James Stewart 6-th edition. (If you want to use 5-th edition please be prepared to cross check with a classmate.)

Syllabus:
We will cover most of Chapters 2-6.
Chapter 1: Review of Precalculus.
Chapter 2: Limits and Derivatives.
Chapter 3: Differentiation Rules.
Chapter 4: Applications of Differentiation.
Chapter 5: Integrals.
Chapter 6: Applications of Integration.

Homework Assignment:
Typically you need to do three times as many problems as you see listed here to absorb the presented concepts. In class we will try to do more problems than assigned in the printed syllabus.
Homework assignment list for the 6th edition.

Weekly Time Requirement:
For Fall or Spring Semesters:
4 hours to view the videos.
4 hours to attend the classes.
4 hours to do the homework.
For compressed Summer I or II classes:
6 hours for each category.

Prerequisites:
You have several resources to help you decide the courses you can take:
General Catalog, advisors, instructors, textbook, first chapter, first few lectures and tests, classmates, upper-class students, Internet, etc.
It is essential that you allocate some effort to make sure that you are spending your time and energy in a beneficial way.

Official Policy/General Catalog description: Prerequisite of 2413 is grade of C or better in Pre-Calculus MATH 2312, or its equivalent.
Prerequisite of 2312 is: Grade of 270 or better in Math THEA or C in DMATH 0372. If THEA exempt then prerequisite is 500 MATH SAT or 19 MATH ACT.

The following is an instructor's personal opinion on prerequisites for 2413:
Short Version:
Take College Algebra (1314), Trigonometry (1316), and Basic Calculus (2376/1325), then take Calculus (2413).
Long Version:
2413 is essentially an open admission course. Students come to this course with a wide range of preparations:
AP Calculus or CLEP Calculus, or
High school Calculus, or
Calculus for Science (2376) or Business (1325), or
College level Algebra (1314) and Trigonometry (1316), or
College level Pre-Calculus (2312), or
High school level Pre-Calculus, or
High school level Algebra and Trigonometry, or
High school level Algebra.

Your chance of success is higher at the top of the above list and decreases as you come down the list. There is a major gap between those who took some form of calculus before and those who did not.
You are encouraged to make sure that you have taken the equivalent of following courses before attempting Calculus I, 2413:
elementary arithmetic, geometry, algebra, trig, -and- college algebra 1314, trigonometry 1316, and basic calculus 2376 or 1325. Notice that this is a non-standard recommendation. These courses do not count toward majors where 2413 is required.
Red Flags: OVER-reliance on any of the following items in your educational path is a cause for concern:
Multiple-choice tests, work sheets (used in place of lectures), short work sheets (where all you do is to write a final answer and do not really show work), learning from simplified handouts and notes (as opposed to reading substantial texts), calculators, skipping math courses for a semester, absentee teachers, substitute teachers, teachers who were not certified, teachers who were not math majors, classes where everyone passed the course, classes where much of the syllabus was not covered, and social promotion.
In above cases, it is more prudent to start with College Algebra 1314, follow with Trigonometry 1316, and FINISH the entire syllabus on your own in each case. Then take 2376 or 1325. Then approach 2413.
Notice that most college algebra classes are not intended to prepare students for calculus. (Majority of students in these classes will not take any advanced math course, so the course does not expose students to level of algebra needed for calculus.) Even Precalculus 2312 does not properly fit the requirement since it is only an abbreviated combination of Algebra 1314 and Trig 1316. And again, many students in 2312 do not intend to go to calculus. That is why you need to spend the extra effort on finishing the syllabi of 1314 and 1316, and/or, take 1325 or 2376. Note that 1325 and 2376 do not have any trig components.
In fact, once you miss the advanced math track in high school it is a substantial struggle to catch up and get back on it in university. However, if you like the math-centered sciences then it surely pays to spend this extra effort.
There are several levels and styles of understanding mathematics. At the highest level, math is a result of direct intuition. Next, it can be the result of studying mathematics texts. More commonly, it can be the result of listening and viewing drill oriented presentations, similar to what you see in the videos here.
As such, this course falls short too, and presents you with at least one red flag since it does not emphasize the reading of any text. If you are interested in mathematics you must get used to reading mathematics texts. Then you should to do some of the challenging exercises. Then you should try discovering mathematics.

TALH:
Students from Texas Academy of Leadership in Humanities, or any other accelerated program, who have successfully passed a calculus course in high school, are encouraged to take Calculus 2413. All other students from TALH are required to consult with the instructor before taking this course. Please read the section on Prerequisites, above, carefully.

Searching:
You may search this page for a mathematical phrase to find an applicable video. To search a page on Internet Explorer 6 click EDIT on the menu bar and then choose FIND. Other browsers have a similar facility. The search is exact. Searching is more convenient on Firefox than IE.

Software Requirements:
The following free software will be needed.

1. WMV files (all video lectures) require Microsoft's Windows Media Player, or any one of the many compatible players. Version 10 has a better fast forward function than version 11.

2. PDF files (class information) require Adobe Acrobat Reader. Be aware: Adobe Yahoo! Toolbar and Adobe Photoshop are automatically included in the download as well. Uncheck the boxes for those options before downloading.

3. JAVA applets (extras) require Sun Microsystems Java Runtime Environment (JRE). (Once you are in an applet page click somewhere on the demo and look for a slider or a point that you can grab by mouse and move it around. Or there may be an input box where you type in a formula and the page responds by analyzing your input.)

4. FLASH files (extras) require Adobe/Macromedia Flash Player.

You may find the following viewers useful if you browse Internet for course related material. Note that commercial players may try to associate various file formats with their own software.
Mathematica Player
Livemath
Quicktime
RealPlayer Caution! Main site tries to get you to subscribe to various unrelated software. Be careful! Instead you may try a more reasonable download at BBC .
Geomview for Mac OS X, UNIX, CygWin

Computer Labs:
1. Gray Library Media Room, 7-th floor, can borrow a headset, very crowded.

2. Computer Science Department labs, MAES Building, rooms 211-214, bring your own headset, not so crowded. (Do not confuse that lab with the Writing Center Lab next door.)

3. Mathematics Department Lab, Lucas 209, bring your own headset, not crowded.

Policies and Announcements:
Announcement 1: Calculus as a hybrid course. Description of the general format, policy, schedule, benefits, and requirements for the course.

Announcement 2: Writing Requirements for calculus. Good handwriting, good exposition, and good mathematical grammar count heavily. Putting an equal sign between items that are obviously not equal is a big No-No. Not putting an equal sign between the two sides of an equation is also unacceptable. Missing parentheses, misaligned equations, small handwriting, missing equal signs, missing operation or relational symbols (e.g. integral, limit, derivitiave, d/dx, ' , ", =, <, etc), and use of blunt pencils, are some of the items to be avoided in your writings and presentations. Bring three already-sharpened pencils to tests. Show all work. All algebra and calculus related work must be exhibited. Utilize the space/white paper next to problem before parking solutions on scratch sheets. Scratch sheet is mainly for trial and error or a first draft. It is not intended as the final resting place for solutions. If you run out of space and use the scratch sheet for your answers then do put your name on it. Put the scratch sheets in the middle of your exam and return all pages to the instructor. Graded exams should be returned to the instructor.

Announcements 3: Calculator Policy. Advanced calculators and cell phones are not allowed on tests. Only basic scientific calculators are allowed. These calculators cost about 10 dollars, and do NOT have the following capabilities: Graphing, solving equations symbolically, programming, integrating, differentiating, text storage, or wireless connection. You do need to practice using the calculator early on and become confident about its operations. Do keep its manual and find the website of the calculator for future reference. The calculators with multi-line display can show your steps and reduce data entry error. They cost about 20-30 dollars.

Announcement 4: Free Tutoring Lab in LUCAS 209 opens about 10 days after classes start. This is a good place to form study groups, consult with tutors, use computers, etc.

Announcement 5: University Undergraduate General Catalog.

Announcement 6: University Schedule. See the early pages (5-10) for important dates. This information is subject to correction and change.

Announcement 7: Small classes may be cancelled. Classes that have less than 10 students by 12th class day are usually cancelled.

Additional Resources:
1) Review of Precalculus:
A series of videos for reviewing Pre-Calculus is included. This corresponds approximately to parts of Chapter 1. You may want to come back to them for a refresher or a simple description of a topic.

2) On-line texts, notes, courses, and lectures (Please do NOT print the texts on public printers!)
a) G.S. Gill Calculus Bible. Text.

b) Gilbert Strang Calculus. Text.

c) H. J. Keisler Calculus. Text.

d) P. Dawkins Online Math Notes. Notes.

e) L. S. Husch Visual Calculus. Notes, step by step presentations, demos.

f) Calculus at University of British Columbia. Interactive text.

g) S. Hollis Video Calculus at University of Houston. Narrated slides of calculus I and II.

h) A. Banner Calculus at Princeton University. Video of Calculus I and II.

i) Stewart Calculus. Collection of resources by the publisher/author of the text we use for this course.

j) Hoffman and Bradley Calculus 7th ed. Calculus for Business, Economics, and the Social and Life Sciences.

k) Leah Keshet Calculus 102 at UBC. Calculus for Business, Economics, and the Social and Life Sciences.

l) Christof Schiller Motion Mountain. Physics Text.

m) Ben Crowell Light and Matter. Physics Text.

n) Home-School math resource list.

3) Extras, Interactive Demos, and Calculators
Various links to applets and interactive pages are placed after many of the video lectures. These pages help you understand calculus and perform many calculations. You may want to put your favorite in your browsers Favorites list. Some of the more useful ones are listed here:
Function analyzer. Evaluating, graphing, differentiating, integrating, graphing, solving, Taylor series.
Multiple Function Graph Explorer 1.
Multiple Function Graph Explorer 2.
Terminator.
Online Calculator.
Calc98 a downloadable calculator.
Power Calculator a downloadable calculator.

4) Advanced Software
For more advanced work, access to Maple, Matlab, and Mathematica is available in the computer lab. There are also a number of high quality free software available for download. Among them:
Scilab a numerical software package.
Octave a numerical software package.
Maxima a symbolic software package.
OpenOffice a presentation software.
TeX a text processing software.
5) Course Repositories
Several major universities have made some of their introductory courses available to the public. You may want to search for your favorite topic in the following links.
a) MIT .
b) Stanford .
c) Yale .
d) Academic Earth.
e) YouTube.
f) TED .
g) Apple .
h) GapMinder .
Precalculus Review
Plan on reading all the material in Chapter 1 in parallel with the course.
Use the Pre-calculus Review Lecture Videos if needed.

Read Section 1.5.
Book Edition 5: Page 55-63.
Book Edition 6: Page 52-59.
If you need a review then view the following video.
Review of Exponential Functions from Precalculus. Definition of exponential function. Extension to negative, zero, rational, and irrational exponents Properties of graphs of exponential functions. Base e exponential functions. Applications to simple population problems. Doubling time. Applications to radioactive problems. Half life.

Extras:
applet shows y=a^x becomes tangent to y=x+1 when a=e.
Optional: Review of Functions from Precalculus. Simple description of functions, domain, range, arithmetic operations, composition, decomposition, substitution.

Extras:
Here you can test your knowledge of basic functions (linear and quadratic) by this association game.

Here you can experiment with a JAVA applet showing arithmetic operations on functions (addition, subtraction, multiplication, division, as well as inverse functions and composition). Choose two functions from the drop down menu, or type them yourself. For example you can type 5*x^3+20*cos(pi*x)+e^(-2*x). Notice that we use ^ for power, * for multiplication, pi for pi=3.1415926535897932..., and e for e=2.718281828459045... . Set boundaries for the picture by modifying xmin, xmax, ymain, ymax, and choose which activities you wish to graph. Try to guess what the graph is going to look like first and then verify it.
Read Section 1.6. Part I, Functions and Inverse Functions
5th Edition Pages 63-67.
6th Edition Pages 59-63.
If you need a simpler description view the following video.
Optional: Review of Inverse Functions from Pre-Calculus Simple description of inverse functions. Domain, range, graph, slope, existence, horizontal line test.
Correction: there is a typo around minute 8:50 where I write
y goes to (y-2)/3
the correct form is
y goes to (y-3)/2
Read Section 1.6. Part II, Logarithmic Functions and Natural Logarithms.
5th Edition Pages 67-71.
6th Edition Pages 63-67.
If you need a simpler description of view the following video.
Review of Logarithmic Functions from Pre-Calculus Simple description logarithms, rules of logarithms, base 10 and base e, graphs.

For NOW we will skip Part III of 1.6, Inverse Trigonometric Functions.
5th Edition Pages 72-74.
6th Edition Pages 67-70.
We will come back to it later. However if you want to get a head start you may want to view the following videos.
Optional David E. Joyce Online Trig Course.

Optional Review of Inverse Trig Functions from Pre-calculus: arcsin, arccos, arctan, graphs, and special values. Review from pre-calculus.

Optional: Review of Problems on Inverse Trig Functions from Pre-Calculus. Review of sample problems.


Calculus One
Calculus Video Lecture 1. Tangent lines and velocity. Introduction to calculus. Two branches of calculus: Differential and Integral calculus. Finding slope and equation of the tangent line to the graph of a function at a given point by a limit process. Synonyms for slope: speed, rate, gradient, slant, incline, steepness, grade, pitch. Angle of inclination or tilt angle. Finding velocity of a particle given its position as a function of time.

Read Section 2.1
Calculus Video Lecture 2a. The Limit of a Function. Limits, one-sided limits, infinite limit, vertical asymptotes, The Squeeze Theorem, basic limit laws.
Book Edition 5ET Sections 2.2 Page 92 (and parts of 2.3) do problems from Page 102.
Book Edition 6ET Sections 2.2 Page 88 (and parts of 2.3) do problems from Page 96.


Calculus Video Lecture 2b. An example. Finding limits using a calculator through a table of values, effective use of a calculator, finite precision issues.
Extras:
one-sided limits and piecewise defined functions with this applet.
Calculus Video Lecture 3. Using Algebra to Find the Limit of a Function. Finding limits using factorization and rationalization.
Text edition 5ET Section 2.3, Page 109-113.
Text Edition 6ET Section 2.3, Page 99-108.


Extras:
Several limit problems done using algebra (symbolically)

More on limit problems done using algebra (symbolically)
Calculus Video Lecture 4. Vertical asymptotes and infinite limits. Sections 2.2 and 2.3. Vertical asymptotes of rational, trigonometric and logarithmic functions.

Book Edition 5ET Do problems 23-32, 36, 39 Page 103.
Book Edition 6ET Do problems 25-34, 38, 41 Page 98.


Extras:
Applet for study of asymptotes of certain rational functions.
asymptote analysis applet for factored rational quadratics.
Graph analyzer with parametic input.
Calculus Video Lecture 5. The Precise definition of a Limit. Finding limits using the epsilon-delta definition. The reason for rigorous approach to mathematics. A counter-intuitive case: Sum of a series may depend on the order of summation. Infinite limits.
Book Edition 5ET Section 2.4, Page 114-124.
Book Edition 6ET Section 2.4, Page 109-118.


Extras:
This applet helps you to understand the precise definition of limits.
Calculus Video Lecture 6. Continuity. Continuity at a point, discontinuity, types: removable, infinite, oscillatory, one-sided continuity, continuity on an interval, continuous operations with continuous functions, continuity of basic functions, The Intermediate Value Theorem.
Book Edition 5ET, Section 2.5, Page 124-135.
Book Edition 6ET, Section 2.5, Page 119-130.


Extras:
Study one-sided limits and continuity of piecewise defined functions with this applet.
The following applet requires a bit of programming on your side to define a piecewise defined function. It is a good idea for you to get used to programming as early as possible.
This applet helps you investigate the continuity of a function. Click Options. Click "Define Function" button. If you want to use the function that is there already you need to change "pi" to Pi, or a number, e.g. 3.1415, for the applet to work (it has a typo). Notice how the functions are defined. The syntax almost means:
for x < 1 use y= x^4+0.5,
for 1< x < 2 use y=0.5*x+1,
for 2< x use y=cos(3*x)+1.
Calculus Video Lecture 7. Limits at Infinity, Asymptotes. The behavior of a function as x goes to positive or negative infinity, left tail and right tail, infinite limits, unbounded and bounded oscillations, horizontal asymptotes, examples from rational, root, exponential and trigonometric functions, precise definitions.
Corrections: At minute 32 of the video I draw y=-2/3 instead of y=2/3 as the horizontal asymptote.
Book Edition 5ET, Section 2.6, Page 135-149,
Book Edition 6ET, Section 2.6, Page 130-143,


Extras:
If you have a low degree rational function in a factored form use the following applet.
Bad Link: In this applet you can change parameters of already defined functions and see how asymptotes and the shape of the functions are influenced.
Note:
In Edition 5 Derivatives are introduced in two sections, 2.7 and 2.8.
In Edition 6 these are combined into one section namely 2.7.

Calculus Video Lecture 8. Slopes, Tangents and Velocities A review of what we have done so far using the limits notation. Finding slopes of tangent lines using limits. Four basic examples. Finding instantaneous velocity using limits. Projectile, turning point, maximum height, time to strike.
Book Edition 5ET, Section 2.7, Page 149-157.
Book Edition 6ET, Section 2.7, Page 143-153.


Extras:
Here you can see a JAVA applet showing the slope of sliding secant lines.
Calculus Video Lecture 9. Derivatives  basic notation, equation of tangent line, applications.
Book Edition 5 ET, Section 2.8, Page 158-165.
Book Edition 6 ET, Section 2.7, Page 143-153.


Calculus Video Lecture 10. Derivative as a function Derivative as a function, Newton and Leibniz notations, operators, differential operators, differentiability, differentiable at a point, differentiable on an interval, when derivative fails to exist, sharp corners and cusps, one-sided derivatives, Theorem: differentiability implies continuity.
Book Edition 5 ET, Section 2.9, Page 165-176.
Book Edition 6 ET, Section 2.8, Page 154-165.


Extras:
Here you can see JAVA applet showing a function and its derivatives. Choose a function from the drop down menu, or type one yourself, for example you can type 5*x^3+20*cos(pi*x)+e^(-2*x), (notice that we use ^ for power and * for multiplication, the software understands pi and e), set boundaries for the picture and drag the horizontal slider. Try to guess what the graph is going to do and then verify it.

Visual differentiation applet. The trace will draw the derivative of the function.
Visual differentiation matching puzzle.
Calculus Video Lecture 11. Derivatives of Polynomials and Exponential Functions Derivative of monomials x^n, derivative of x^n for rational and negative n, linearity, derivative of polynomials, derivative of exponential functions, 2^x, 3^x, a definition of e, derivative of e^x, examples

Book Edition 5ET Section 3.1, Page 183-192, Assignment 12.
Book Edition 6ET Section 3.1, Page 172-183, Assignment 11.
Calculus Video Lecture 12. The Product and Quotient Rules The Product Rule, The quotient Rule, general power rule, power-quotient rule, examples, an explanation of product rule and quotient rule

Book Edition 5ET, Section 3.2, Page 192-198.
Book Edition 6ET, Section 3.2, Page 183-189.


Extras:
Applet illustrating the product rule.
Applet illustrates steps in differentiating calculus functions. Type your function. Be careful about missing multiplication sign and parenthesis. Choose the variable of differentiation. Press Apply button. Dx means derivative with respect to x, Dy is for y, etc. At each step applet shows which rule has been used.
Begin reviewing Trigonometry. Many of you have very little background in trig. But this is a topic that needs quite a bit of practice. On one hand you need to keep up with the course and on the other hand you need to spend time and make sure you understand what these trig functions represent. You need this information in Calculus II and then in Differential equations 3301 and then in circuit analysis etc. So plan in advance.
OPTIONAL: Pre-Calculus Optional Review Video Lecture. A Quick Review of Trigonometry 1 Angles, vertex, ray, opening, initial side, terminal side, positive or counter-clockwise direction, negative or clockwise direction, winding number, standard position, quadrants, circumference of a circle, measuring angles, degrees, radian, grad, circle, cycle, rotation, round, length of an arc, area of a sector, six trig functions in terms of adjacent, opposite, and hypotenuse, solving a right triangle, six trig functions for arbitrary angles, trig identities: Pythagorean, ratio, negative, sum of angles.

Extras:
the six trig functions in a unit circle setting.

the animated drawing of sine, cosine and tangent in a JAVA applet. click on the third big red box titled: "Applet: the graph of sin, cos, and tan".

the ordinary graph of sin, cos, tan, arcsin, arccos, arctan. Click on "Graphs of elementary trigonometric functions".

David E. Joyce Online Trig Course.
Calculus Video Lecture 13. The Derivatives of Trigonometric Functions Section 3.4, Page 211, Derivative formulas for sin, cos, tan, sec, csc, cot. Geometric proofs of several important trig inequalities for an angle x, measured in radians, in the first quadrant.

If 0 < x < pi/2 then
1) sin x < x < tan x,
2) cos x < (six x)/x < 1,
3) (x/2) sin x < 1 - cos x < x sin (x/2)
4) (sin x) /2 < (1- cos x) /x < sin(x/2)


Book Edition 5ET, Section 3.4, Page 211-214, Assignment 14.
Book Edition 6ET, Section 3.3, Page 189-197, Assignment 13.


Extras:
To experiment with just about any function and compare your results against an automatically generated solution you may use applet for differentiation, graphing, finding max and min. Experiment with the following buttons, “The Value or limit", "f'(x)", "The curve of". You will learn about other buttons later.
Calculus Video Lecture 14. The Chain Rule Composition of functions, the chain rule formula, examples, geometric explanation, and an indication of the simplified proof.


Correction: One item missing from this lecture is the derivative of a^x.
Here is the formula: (a^x)'= a^x ln a. (See Page 222 Edition 5/Page 201 Edition 6)
Here is the proof:
1) You can use chain rule to show [e^(bx)] ' = b e^(bx).
2) From properties of logarithm we know
a= e^(ln a) hence a^x=e^(x ln a).
3) Now differentiate both sides
(a^x)'= [e^(x ln a)]' = e^(x ln a) ln a = a^x ln a.

Book edition 5 ET, Section 3.5, Page 217-227.
Book edition 6 ET, Section 3.4, Page 197-207.
I recommend you do 1-46, 50. Those of you who have had calculus before try 58. The important issue is that you need to imptove your speed and accuracy. In most problems you need to see if f'(x) can be simplified and factored. In that case find roots of f'(x)=0.

Extras:
To experiment with a chain rule applet you can define two functions, such as y=f(x)=3-(x^2)/2 and y=g(x)=sin(x), and see the graphs of f, g and g(f(x)), as well as the tangent lines and their slopes, at a point (x_0, f(x_0)) (the red spot in the left picture), point (f(x_0), g(f(x_0))), on the second picture, and point (x_0,g(f(x_0))), on the third picture. Notice that you can drag the red spot. What does the color-coordination of various line segments mean? How does the third slope relate to the first two?

Here you can see a three dimensional Java applet of chain rule. Check both Normal Mode and Delta Mode buttons. You can modify angle of view by pressing up/down/left/right buttons, and move on curves by pressing x+ and x- buttons. Can you decipher the picture?
Calculus Video Lecture 15. Implicit differentiation, derivatives of inverse trig functions, orthogonal families of functions Finding dy/dx given F(x,y)=0, application to finding derivatives of inverse functions, arcsin or asin or sin^(-1), acos, atan, acsc, acos, acot. Examples of orthogonal trajectories.

Book Edition 5ET Section 3.6 Page 227-236.
Book Edition 6ET Section 3.5 Page 207-215.
In addition I recommend you do 41-54, 59-62, 66-68.
Optional: Review of Inverse Trig Functions from Pre-calculus. arcsin, arccos, arctan. Graphs, special values. Review from pre-calculus.
Optional: Review of Problems on Inverse Trig Functions from Pre-Calculus. Review of sample problems.
Extras:
Here you can see Java graphs of basic inverse trig functions and explanation of their derivatives. (Click on the bold dot and move it up/down. Pay attention to the color-coordinated line segments.)
OPTIONAL Calculus Video Lecture 16. Higher Order Derivatives. Higher order derivatives, position, velocity, acceleration, graphical interpretation, derivative notation, factorial, high order derivatives of x^a, sin(x), cos(x), implicit higher order derivatives.

Book Edition 5ET Section 3.7 Page 236-240.
The problems in this section get to be more varied, more interesting, and more challenging. Try at least two from each category in addition to the syllabus assignments for Book Edition 5 ET. Do 1, 2, 5-20, 28, 30, 36, 40, 43, 51.

Book Edition 6ET: This section is eliminated in this edition. However this topic is related to Chapter 11. A short piece shows up in Section 3.3 Page 194.
Calculus Video Lecture 17. Derivatives of Logarithmic Functions. Derivative of logarithm in base b, derivative of natural log, logarithmic differentiation, derivatives of various power types b^x, x^b, u^v, another description of e.

BOOK EDITION 5ET Section 3.8, Page 244-249.
BOOK EDITION 6ET Section 3.6. Page 215-220.
OPTIONAL Calculus Video Lecture 18. Hyperbolic Functions Part 1. Definitions of Sinh, Cosh, Tanh, Csch, Sech, Coth. Graphs of the six functions. Differentiation formulas. Basic identity.



Book Edition 5 ET Section 3.9 Page 250-256.
Book Edition 6 ET Section 3.11 Page 254-261.
OPTIONAL Calculus Video Lecture 19. Basic exercises on Hyperbolic Functions. Book Edition 5 ET Section 3.9 Page 250-256.
Book Edition 6 ET Section 3.11 Page 254-261.
Solution of Problems 11, 32, 39, 41.
To be uploaded: Optional Video Lecture 20: Inverse Hyperbolic Functions Sinh^-1 or arcsinh or asinh and other inverses, their graphs, domains, derivatives, formulas in terms of natural log.
Book Edition 5 ET Section 3.9 Page 250-256.
Book Edition 6 ET Section 3.11 Page 254-261.
To be uploaded Optional Video Lecture 21: More on hyperbolic functions

What does Cosh, the hyperbolic cosine, have to do with hyperbola and cosine? Connections between hyperbolic and circular trigonometric functions, Euler formula.
Book Edition 5 ET Section 3.9 Page 250-256.
Book Edition 6 ET Section 3.11 Page 254-261.
This section is very interesting and challenging. If you were asking why we are learning all these math formulas you get a partial answer in this section. Do read the text and spend time with the problems done in the text. Also expect a rough road and prolonged delays before you get the hang of it. If you want mathematics to be a major part of your career then you must do well in this section. We will have three sections in calculus I that are like this: related rates, graphing, and optimization.

Calculus Video Lecture 22. Related rates. Study of problems from geometry, physics, and engineering where several quantities are related to each other and we use information about their current values and current rates of change to find a missing rate of change. Solutions of three problems are presented. Read the text for introductory problems and basic advice on how to get started.

After you view several problems check to see how the following steps were carried out and try to apply the same general approach.
1) Make a listing of variables whose rates of change are given or requested.
2) Draw a clear picture, if applicable.
3) Give variables names (x,y,v,..) and show them on the picture.
4) Write the rates of change of variables and their current values in a table.
5) Discover the relationship between the variables. This is the hard part; it may need geometry, visualization, or basic science etc. Review geometry formulas under the front cover.
6) Differentiate this relationship with respect to time carefully. (For example remember (x^3)' is not 3x^2, it is 3x^2 x’!).
7) Substitute the values you know to find the one you do not know. The current values of variables do not get to be used until this final stage.


Book Edition 5ET, Section 3.10, Page 256-262.
Book Edition 6ET, Section 3.9, Page 241-247.
Redo all problems that are done in the text and in the video.

Extras:
Here you can see Java demo of the increasing length of shadow of a man as he moves away from a lamp post. (Drag the scale, on some computers this demo does not show well).

Here you can see Java demo of two ships moving away from each other.

Here you can see ten related rates demos. Description of each problem. Table of links at the bottom of page: Overhead kite/airplane, sand pile, sliding ladder, shadow of a walking figure, oil spill, rolling snowball, elliptical trip, opening a rectangle in a computer window, baseball runner.
Calculus Video Lecture 23. Linearization and Differentials. Linear Approximation, tangent line approximation, Linearization, small angle approximation, differentials, relative error, percentage error. Solution of several sample problems.

Book Edition 5 ET, Section 3.11, Page 262-269.
Book Edition 6 ET, Section 3.10, Page 247-253.


Extras:
Here you can see a java applet for linear approximation. You can choose or type a function f(x), in place of x^2, choose the base of operation, a, and the displacement, h, as in f(a+h) =approximately equals= f(a)+h f '(a). The horizontal slider changes h, the vertical slider changes the slope of the line. The blue inset is the picture of the error of approximating the graph by the line you have chosen, so if your line is y=mx+b the error function is y=f(x)-(mx+b).
OPTIONAL Calculus Video Lecture 18. Hyperbolic Functions Part 1. Definitions of Sinh, Cosh, Tanh, Csch, Sech, Coth. Graphs of the six functions. Differentiation formulas. Basic identity.


Book Edition 5 ET Section 3.9 Page 250-256.
Book Edition 6 ET Section 3.11 Page 254-261.
OPTIONAL Calculus Video Lecture 19. Basic exercises on Hyperbolic Functions.
Book Edition 5 ET Section 3.9 Page 250-256.
Book Edition 6 ET Section 3.11 Page 254-261.
Solution of Problems 11, 32, 39, 41.

Calculus Video Lecture 24. Maximum and Minimum Points. Definition of absolute or global maximum or minimum, relative or local max or min, extremum points, critical points, The Extremum Value Theorem, Fermat's Theorem, The Closed Interval Method. Solution of problems 47, 53, 56.

Book Edition 5 ET, Section 4.1 Page 279-289.
Book Edition 6 ET, Section 4.1 Page 271-280.


Extras:
Remember you have this all inclusive applet for differentiation, graphing, finding max and min.
Note; variable is x. If a rational exponent is used, as in x^(1/3), applet restricts domain to positive numbers.
Here is an interactive demo for investigating relationship between f and f ', max/min and tangent line .
Calculus Video Lecture 25. The Mean Value Theorem. Rolle's Theorem, The Mean Value Theorem, a function with a zero derivative is a constant; functions with identical derivative differ by a constant. Solution of Problems 12, 20.


Book Edition 5ET Section 4.2 Page 290-296.
Book Edition 6ET Section 4.2 Page 280-286.


Extras:
Here is a silly but good applet for the Mean Value Theorem or Rolle's Theorem . Go to the middle of page; play with both "Drag curve and Drag tangent" options.
Calculus Video Lecture 26. Influence of Derivatives on the Shape of a function. Increasing/Decreasing test, the first derivative test, Concavity, The second derivative test, inflection point, a sample problem.


Book Edition 5ET, Section 4.3 Page 296-307.
Book Edition 6ET, Section 4.3 Page 287-297.

Calculus Video Lecture 27. L'Hospital's Rule and Indeterminate forms. 0/0, inf/inf, 0*inf, inf-inf, 1^inf, inf^0, 0^0. Solution of 12, 23, 45, limit as x goes to infinity of (1+1/x)^x, and other sample problems.
Correction:
The discussion of 0^0 case is missing from the video. The treatment is similar to inf^0 or 1^inf case. Please read about it in the text.

Book Edition 5ET, Section 4.4 Page 307-315.
Book Edition 6ET, Section 4.4 Page 298-307.

Understanding section 4.5 is a very good indicator of how well you have understood the entire math you have learned so far. It is a culminating point for the first half of calculus.
Section 4.5 will be time consuming, challenging, and important, so plan in advance.
Read the text, redo the examples in the text, View the next three videos, and,
Do Problems 5, 13, 19, 21, 24, 27, 31, 41, 45, 59 from
Book Edition 5ET Section 4.5 Page 323
Book Edition 6ET Section 4.5 Page 314


Extras:
Remember you have this all inclusive applet for differentiation, graphing, finding max and min, etc. First do a problem yourself then make use of the applet to check your steps.
OPTIONAL Calculus Video Lecture 28. Curve Sketching 1, General Description. Point Plotting, Domain, Range, Intercepts, Symmetry, Asymptotes, First derivative analysis, Second derivative analysis.

Book Edition 5ET, Section 4.5, Page 316-324.
Book Edition 6ET, Section 4.5, Page 307-315.

OPTIONAL Calculus Video Lecture 29. Curve Sketching 2, Solution of two problems. Book Edition 5ET Section 4.5, Page 323, Problems 4 and 18.
Book Edition 6ET Section 4.5, Page 314, Problems 4 and 18.

OPTIONAL Calculus Video Lecture 30. Curve Sketching 3. Solution of two problems. Book Edition 5ET Section 4.5 Page 323 Problems 47 and 42.
Book Edition 6ET Section 4.5 Page 314 Problems 47 and 42.

OPTIONAL Calculus Video Lecture 31. Problem Review.
A couple of minutes of the video, in the middle of the third question (page 305/problem 38), is missing.
Rolle's Theorem, Section 4.2, Page 295, Problem 3,
Shape of a graph, Section 4.3, Page 304, Problems 15, 38, 40,
L'Hospital's Rule, Section 4.4, Page 313, Problem 48.

Extras:
Worked out exercises related to curve sketching, a. Detailed solution of 11 problems.

Worked out exercises related to curve sketching, b. Detailed solution of 4 problems.

Can you tell f from f ' from f ". A test of your understanding of shape of derivatives.

This is a very interesting section. You will see real applications here. Spend time on translating from English to Math. If you allocate time you will get the benefit in the long run.
Calculus Video Lecture 32. Optimization. Section 4.7 Page 337. Constraints, objective functions, Snell's Law, Fermat's Principle, Solution of a sample problem and 16, 25, 51.



Extras:
Rectangle of maximum area with a given perimeter. Similar to Problem 5.

Rectangle with the largest area inscribed in a semi-circle. Similar to Problem 19.

A gutter with maximum area.

Rectangle with largest area inscribed in a right triangle. Similar to Problem 24.

Moving a pipe around a corner between two hallways. A bit more challenging than others. Similar to Problem 54.
Read Section 4.8, age 334, Edition 6.
Calculus Video Lecture 33. Newton's Method.
Newton-Raphson Method for finding roots of equations, approximate methods, graphical explanation, algorithm, example, pathological cases.
Corrections: A statement made at the beginning of this lecture says there are no formulas for roots of polynomials of degree four and above. The correct description is: There are no formulas (similar to the quadratic formula) for equations of degree five and above.

An important issue here is the efficient use of a calculator. Make sure you know how to use your own calculator for storing a number and recalling it and grouping expressions in parentheses. There are keys for these activities but each calculator has a different style. On some calculators these are called STO, RCL, and (, ).
For example let x=0.123456789. Find (x+x^2)*sin(x+x^3), but enter x ONLY ONCE.

Extras:
Applet shows the successive steps of Newton's Method. Click somewhere on the X-axis.



Extra Extra: For your eyes only!.
Basin of Newton's Method. A polynomial has as many roots as its highest power. Depending on where you start Newton's method you will generally converge to one of the roots. This applet colors starting points according to which root they lead to. The whole operation is done on the complex plane (z = x + i y), and creates amazing pictures. Highly recommended for a rainy day. Some non-interactive low-resolution sample pictures are here. But it is better if you create your own and zoom on it, as that is done here. To zoom simply left-click and drag the mouse over the diagonal of the rectangular region you want to enlarge.

You can find more about fractals here.
Here is the resulting fractal image for f(z)=z^3-1.

Here is a basic version of fractal image for f(z)=z^3-1 each color represents set of points which will lead to the same root.
Calculus Video Lecture 34. Antiderivatives. Basic anti-differentiation formulas, initial conditions, rectilinear motion, position, velocity, acceleration, initial position, initial velocity.
Book Edition 5ET, Section 4.10, Page 353-360.
Book Edition 6ET, Section 4.9, Page 340-347.


Extra:
Simple exercises on antiderivatives.
Calculus Video Lecture 35. Areas and Distances Part 1. Section 5.1, Page 369. Numerical approximation of area under a curve. Left point rule, right point rule, midpoint rule, an example, identities for sums of power, sum of squares, sum of cubes, area as a limit, exact area under a parabola.


Extras:
Applet helps you see the rectangles for the Riemann sums.

Calculus Video Lecture 36. Areas and Distances Part 2. Sigma notation, partition of an interval, area formulas using sigma notation, finding area under f(x)=2x^2+x over [1,3] by using partition, sum of powers formulas, and limits, calculating distance from velocity function, Theorem: Change in position in a time interval is equal to the area under the velocity function over the time interval.
Edition 5 ET Section 5.1, Page 369.
Edition 6 ET Section 5.1, Page 354.
Corrections:
1) At the beginning of the lecture I use the phrase "summation convention" that is incorrect and should be dropped. The correct phrase is just "sigma notation".
2) In the last few seconds of the video 4+1/3 is written as 14/3. It is of course 13/3.


Calculus Video Lecture 37. The Definite Integral. Riemann sum, integral sign, integrand, upper limit, lower limit, properties of the definite integral: reversal, linearity, concatenation, comparison.
Edition 5ET Section 5.2, Page 380.
Edition 6ET section 5.2, Page 366.

Calculus Video Lecture 38. Evaluating a Definite Integral. Problem 22 Section 5.2. An important example problem putting all basic concepts of area or definite integral calculation under one roof.

The Upcoming Test is on items between green bands below.
Final is on Wednesday May 6 during class time. Good luck on the test. Remember that substituion details should be shown for problems of 5.5.
We did not get to finish the last two sections. Study those before related sections of Calculus 2 or 3.
Do not forget to complete the course evaluation form.
Calculus Video Lecture 39. Fundamental Theorem of Calculus (FTC). Integration and differentiation are inverse processes. How to use FTC to calculate areas and definite integrals in a short time.
Book edition 5ET, Section 5.3, Page 394-404.
Book edition 6ET, Section 5.3, Page 379-390.
Extras:
Applet allows you to graph a function and the area under it.

Applet lets you manipulate a graph by hand and draw its derivative, integral, and tangent line.
Applet shows signed area under a specific function.
Calculus Video Lecture 40. Indefinite Integral and the Net Change Theorem. Indefinite integral as a notation for antiderivative. Table of basic indefinite integral formulas, Problems 9, 13, 14, 15, 21. The Net Change Theorem, displacement is the definite integral of velocity.
Book edition 5ET, Section 5.4, Page 405-414.
Book edition 6ET, Section 5.4, Page 391-400.

Calculus Video Lecture 41. Substitution Method. Substitution method for indefinite and definite integration. Two styles for calculating definite integrals. Several examples.
Book edition 5ET, Section 5.5, Page 414-422.
Book edition 6ET, Section 5.5, Page 400-408.


Extras:
13 Exercises on substitution method.
10 Exercises on substitution method.

The Current Lecture and assignments are on the items between the red bands below.
Final is on Wednesday May 6 during class time. Good luck on the test.
Give me all of your lecture notes on Wed. Come to L206 to collect it by May 10 otherwise it will be discarded.
Remember that substituion details should be shown for problems of 5.5.
Do not forget to complete the course evaluation form.
We did not get to finish the last two sections. Study those before sections of Calculus 2 or 3 related to volume calculation.
Calculus Video Lecture 42. Area Between Curves. Area between two curves, finding top, bottom, left, and right parts, criss-crossing curves, practice with x as a function of y, general regions.
Book edition 5ET, Section 6.1, Page 437-443.
Book edition 6ET, Section 6.1, Page 414-421.


Extras:
Area between curves calculated by Riemann sums (1).
Area between curves calculated by Riemann sums (2).
Calculus Video Lecture 43. Volumes and Method of Washers. Volume of a cylinder, volume of a solid, cross section, solids of revolution, volume by method of washers or annular rings, examples of non-rotational solids.
Book edition 5ET, Section 6.2, Page 444-455.
Book edition 6ET, Section 6.2, Page 422-433.


Extras
FLASH applet takes you through steps of an example for method of washers.
This applet creates solids of revolution from a curve.
This applet is similar to the last problem in the video but not identical.
A gallery of objects defined through their cross section. Good exercise for visualization. Read the description of the object. Try to visualize it and check the animation to see if you were correct.
Visualization of slices of a wedge of a cylinder. Similar to Example 9 Page 452.
Calculus Video Lecture 44. Volumes by Cylindrical Shells. Volume of a cylindrical shell, calculating volume by cutting a solid of revolution into cylindrical shells, examples, reason for having two different methods for calculating volume.
Book edition 5ET, Section 6.3, Page 455-459.
Book edition 6ET, Section 6.3, Page 433-437.


Extras:
Applet takes you through steps of calculating a volume by shells.

Calculus Video Lecture 45. Review of Chapter 5. Five multi-part Problems similar to problems
Book Edition 5ET; 9-12 page 391, 21-25 page 391, 19-40 page 403, 1-40 page 411, 1-70 page 421.
(The solution of the second problem is lengthy; you may want to do that at the end.)


Calculus Video Lecture 46. Review of Chapter 6.1-3. Three multi-part Problems similar to
problems 5-26 page 442, problems on page 452, problems 3-20 page 458.
Calculus Video Lecture 47. Work. Definition, physical units, several examples, problems 1, 14, 15.
Book edition 5ET, Section 6.4, Page 455-459.
Book edition 6ET, Section 6.4, Page 438-442.
Extras:
Several examples of calculation of work.

End of Calculus I