Last Update: August-20-2010

First Update: March-20-2006

Inception: March-20-2001

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.

All Information is subject to change.

Go to The Start of Calculus I Lectures then scroll down to the current lecture video.

Textbook: This section of calculus will use:

Calculus Early Transcendentals, 5th edition

Author: James Stewart

Publisher: Thomson, Brooks/Cole

ISBN: 0534393217

(The original cd is NOT needed, and yes it is the fifth edition)

Partial Preview Copy of the 6th edition of Stewart

The first email, where to buy the text, style, prerequisites

Course information

Homework

Guide to Mathematical Grammar and Writing

A list of similar courses

Academic Calendar

Final Exam Schedule

Review the old exams for cumulative final:

Exam 1

Exam 2

Exam 3

2008-2010 Undergraduate Catalog

2010-2011 Undergraduate Online Catalog

Stand Up While You Read This Prolonged state of sitting in front of a screen is not good for your health.

If you want to take a break here are a few choices:

Calculus may help you understand the world !!

Optical Illusions

Brain Waves

More Illusions

Benefits of studying mathematics

Former graduates of this class

Neighbor's yard

Enough surfing, now go back, to your own.

The video content of this page works with Windows and has WMV (Windows Media Video) format. In particular you should not experience any problems with Windows Media Player and a variety of browsers (Internet Explorer, Firefox, Chrome) on Windows XP/ Vista/ Windows 7. Other operating systems (e.g Apple/Mac or Unix) are not compatible with the videos without additional software.

Suggested by Kevin Tabien Sept 2009.

1. Download MPlayer . Get the OS X 1.0 version.

2. Download DLL files . Get the file windows-all-20071007.zip ( or the most recent version).

3. Go to finder in the tab go, click the "go to folder option" and type "/usr/lib/" in here you will make a new folder called "win32". You may need to type in password for your computer and become administrator as the "usr" folder is hidden and writing or modifying files here can cause issues so only do what is needed. If you do not fee comfotable with this ask at the computer help window in Maes Building.

4. In this folder you will extract all the contents of the downloaded zip file.

5. Last is opening MPlayer and going to preferences and changing the video out put model to quartz/quicktime.

6. After all this just open the files using MPlayer and they should all work.

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.

(1) Computers, like everything else in life, eventually break down, by age, accidents, poor design, or malicious intent. The first line of defense is to have a back up copy of all personal files on several media as well as installation CDs and serial numbers of various software in case you need to rebuild your machine or file system.

(2) Multiple Internet security software products may be incompatible with each other. Investigate the safety of any software, including security software, before installation.

(3) Links to commercial sites here are not endorsements of their respective products. Do not give personal info or purchase any subscription.

(4) Assume your activity on computer is monitored. Assume posted personal information will become public.

(5) Prolonged time spent in front of a computer or a similar device or setting is unhealthy. A long list of serious medical problems are associated with such a habit (heart, eye, posture, weight, repetitive motion injuries, and addiction may result). If you have to spend substantial time at a computer make sure your workstation is designed properly and take frequent physically active breaks.

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.

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) Pause the video when you need to take a break. Use fast replay if the lecture is slow.

(7) 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.

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.

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.

The most common objection that students have to this course is "But I won't be able to ask my questions." This results from two misconceptions.

Misconception 1) This is an online course and in these courses communication with the instructor is a chore, or worse he is nowhere to be seen or heard of. But, this is not an online course. Classes are held as usual, and you have plenty of time to ask questions.

Misconception 2) In a traditional class I can always raise my hand to ask a question. But students typically don't feel comfortable to ask questions in the middle of a lecture. Realistically, a typical class has 40 students and lasts 50 minutes, rendering a question/answer format un-workable. With the video format you can pin-point your difficulty and discuss it with your instructor in the recitation or office hours.

That said there are downsides that you need to know about:

*) You cannot ask questions from the instructor in the middle of a video lecture.

*) This course does take more time initially compared to traditional lecture format. But you will have a stronger foundation for the long haul.

*) Most students are accustomed to being taught in person and in a social environment, sourrounded by classmates. A few students are accustomed to being instructed by reading a text. The video format is, psychologically, an experience somewhere between the two. You need to get used to it.

*) The video lecture is slower that traditional lecture. This is partly caused by the hardware. Use fast replay to adjust.

*) The field of view is small. A typical computer screen is only a small fraction of the blackboard. This causes a tiring flipping of pages or scrolling back and forth. Videos are immense cpu and memory intensive applications and this shortcomming has not been resolved by the current technology.

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

This course is for students in math, engineering, physics, computer science, biology+psychology, biology+chemistry, chemistry, biochemistry, forensic chemistry. Some students from other disciplines may be required to take this course as well. All students are graded on the same basis even if they decide to change their major.

The following lectures were based on Calculus, Early Transcendentals, 5th edition, by James Stewart.

We will cover most of Chapters 2-6.

Chapter 1: Precalculus, review as needed.

Chapter 2: Limits and Derivatives.

Chapter 3: Differentiation Rules.

Chapter 4: Applications of Differentiation.

Chapter 5: Integrals.

Chapter 6: Applications of Integration.

Homework.

Do what you can.

Do at least one problem from each basic problem category.

Do another in case you had a hard time with the first one.

Redo problems in lectures, classes, text.

Prepare self-test problems by writing them on flash cards and entering where the solution is to be found.

For Fall or Spring Semesters:

4 hours to view the videos.

4 hours to do the homework.

4 hours to attend the classes.

For compressed Summer classes:

6 hours for each category.

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 substantial effort to make sure that you are spending your time and energy in a beneficial way.

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.

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. This cuts both ways.

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.

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.

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, so you do not practice writing or presenting math), memorizing from simplified handouts and notes (as opposed to reading and understanding substantial texts), using 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.

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.

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.

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.

You can search this long page for a mathematical phrase to find an applicable video. Use CTRL F to open the search box for your browser. Each browser also has its own "Find in Page" option. Search is exact, similar to what a basic word processor provides. There is usually no spelling suggestion facility.

The following free software will be needed. They are usually already installed on a new computer.

testing your browser pluggins

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

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

1: Calculus as a hybrid / U-Try style course. Description of the general format, policy, schedule, benefits, and requirements for the course from Spring 2010. New policy will be announced.

2: Writing Requirements for calculus. Good handwriting, good exposition/presentation, and good mathematical grammar count heavily. Here is a sample of common writing mistakes that should be avoided (if you want to be taken seriously!).

a) Putting an equal sign between items that are obviously not equal.

b) Missing one side of an equation or not putting an equal sign between the two sides of an equation.

c) Misaligned equations.

d) Small handwriting.

e) Missing paranteses.

f) Missing operation symbols (e.g., limit, derivitiave, integral, d/dx, ' , ", etc).

g) Missing relational symbols (e.g., =, <, >, etc).

h) Use of blunt pencils. Bring three already-sharpened pencils to tests.

i) Multiple answers or not clarifying your final answer.

In addition:

Show all work. All algebra and calculus related work must be exhibited. Use one or two columns to present solutions.

If blank space is provided on exam paper then you are expected to write your solution next to the problem.

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 and refer to it on the exam paper. Put loose sheets, such as the used scratch sheets or formula sheet, in the middle of your exam and return all pages to the instructor.

If blank space is not provided next to questions then additional blank sheets will be given. Write your solutions on additional pages in order and label each part. If you want to skip a portion of a problem, and come back to it later, then start the next question on a new page to be able to keep the pages in order. Put exam paper on top and staple all papges including the formula sheet, if any. Make sure you do not write where you expect to staple.

Please put un-used scratch sheets in a designated file for re-use.

Graded exams should be returned to the instructor after inspection.

3:

Casio FX-115ES Manual This is easier to search than the printed one.

Casio FX-115ES Appendix Most of the words/buttons here are actually images and therefore not searchable.

Sharp EL-W516 Manual

Remember the text in PDF files is searchable. Look for the binocular icon or press CTRL f.

4:

1:

2: University Undergraduate General Catalog.

3: Fall 2010 important dates, final exam schedule, academic calendar, general info

4: Academic Calendar

5: Code of Conduct Handbook and Academic Honesty Policy. . Students should conduct themselves in a manner so that their work will not be questioned.

6: Classes that have less than 10 students by 12th class day are usually cancelled.

Are you ready for Calculus? A self test. Brian Hassard, SUNY, Buffalo.

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.

a) Gilbert Strang Calculus. Text. PDF file made from scanned image. Not searchable.

c2) A brief Introduction to infinitesimal Calculus By K. D. Stroyan

regular videos of Calculus 1 by John Griggs at NCSU Based on Stewart, but not related to the publisher:

i3) Calculus I

i4) Calculus II.

i5) Calculus III.

k) Calculus Refresher Brief Notes for Calculus I and part of II, By Paul Garrett, UMN.

l) Difference Equations to Differential Equations, An introduction to Calculus By Dan Aloughter, Furman University.

m) Uppsala Lectures on Calculus By Evgeny Shchepin. in PostScript, advanced calculus coverage.

n) Calculus By Benjamin Crowell, text in PDF.

o) Understanding Calculus web site by Faraz

p) Introduction to Methods of Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers by Sean Mauch, Caltech, PDF, 2500 pages.

q) Rigorous Calculus by. william V Smith, BYU.

s) Regular videos for a Trigonometry Course, By Renae Shrum at University of Idaho

w) Elements of Calculus I Applets arranged by topics.

z) Hoffman and Bradley Calculus 7th ed. Calculus for Business, Economics, and the Social and Life Sciences. PDF. Link may be down.

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. (Fractional powers of negative numbers should be handled carefully here.)

Multiple Function Graph Explorer 1.

Multiple Function Graph Explorer 2.

Integrator.

Online Calculator.

Calc98 a downloadable calculator.

Power Calculator a downloadable calculator.

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.

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 .

Plan on reading all the material in Chapter 1 in parallel with the course.

Use the Pre-calculus Review Lecture Videos if needed.

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.

applet shows y=a^x becomes tangent to y=x+1 when a=e.

Review of Functions from Precalculus. Simple description of functions, domain, range, arithmetic operations, composition, decomposition, substitution.

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

5th Edition Pages 63-67.

6th Edition Pages 59-63.

If you need a simpler description view the following video.

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.

David E. Joyce Online Trig Course.

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

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

The Upcoming Test is on items between green bands below.

Read Section 2.1

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 2.2 and 2.3

Do Problems of 2.2

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 2a. The Limit of a Function. Limits, one-sided limits, infinite limit, vertical asymptotes, The Squeeze Theorem, basic limit laws.

Extras:

Read 2.3

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.

Several limit problems done using algebra (symbolically)

postpone for now:Calculus Video Lecture 4. Vertical asymptotes and infinite limits. Vertical asymptotes of rational, trigonometric and logarithmic functions.

Book Edition 6ET Page 98.

Applet for study of asymptotes of certain rational functions (standard quadratics as well as up to factored cubics).

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.

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.

Study one-sided limits and continuity of piecewise defined functions with this applet.

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,

If you have a low degree rational function in a factored or standard form use the following applet.

In this applet you can change the parameters of certain functions and see how asymptotes and the shape of the functions are influenced. Use the last option for the function.

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.

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.

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 matching puzzle.

Read 3.1

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 6ET Section 3.1, Page 172-183,

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 6ET, Section 3.2, Page 183-189.

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.

OPTIONAL: Pre-Calculus 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.

the six trig functions in a unit circle setting.

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.

1) sin x < x < tan x,

Book Edition 6ET, Section 3.3, Page 189-197.

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.

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 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.

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?

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 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.

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.)

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.

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. In addition 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 6ET Section 3.6. Page 215-220.

Extras:

Drills on Logarithmic Differentiation

.

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.

In Edition 6 this topic is placed as section 3.11 after Linearization. Use the text for formulas for the inverse hypberbolic functions.

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 6 ET Section 3.11 Page 254-261.

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

Book Edition 5 ET Section 3.9 Page 250-256.

Book Edition 6 ET Section 3.11 Page 254-261.

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 6ET, Section 3.9, Page 241-247.

Redo all problems that are done in the text and in the video.

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

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 6 ET, Section 3.10, Page 247-253.

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).

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

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.

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 6ET Section 4.2 Page 280-286.

Here is an interesting 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 6ET, Section 4.3 Page 287-297.

OPTIONAL 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 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.

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

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 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.

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

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. Again, if you want to have math play a serious role in your future education you need to do well in this section.

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.

Read Section 4.8, age 334, Edition 6.

OPTIONAL 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.

For example let x=0.123456789. Find (x+x^2)*sin(x+x^3), but enter x ONLY ONCE.

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

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.

Here is the resulting fractal image for f(z)=z^3-1.

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.

Problematic link 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.

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.

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.

13 Exercises on substitution method.

10 Exercises on substitution method.

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.

Area between curves calculated by Riemann sums (1).

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.

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.

Washer Method

Washer Method

Visualization of slices of a wedge of a cylinder. Similar to Example 9 Page 452. Intersecting cylinders.

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.

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.