These lectures cover the basic material for Calculus III at an introductory level. Students study the lectures at home and do the homework assignments. Class time will be used for problem solving and presentation by students. Videos are narrated and appear as handwritten in digital ink.

Last Update: August-10-2009

First Update: March-20-2006

Conception: March-20-2001

All information is subject to change. Attend all classes to be informed of the latest decisions and activities.

The Start of Calculus III

The Upcoming Test is on material between the green bands.

The Current Lecture is on material between the red bands.

Final Test MOnday Dec 14, 11-12:30, in class

You get 5 points added to your final for bringing printout of evaluation completion form for this course.

Go to Online Course Evaluation

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.

(1) Computers, like everything else in life, eventually break down, by age, accidents, poor design, or malicious intent. The first line of defence is to have a back up copy of all personal files as well as installation CDs and serial numbers of various software.

(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 make purchases based on links provided here.

(4) Consider that your activity is monitored. Assume posted personal information becomes public.

(5) Prolonged use of computers and similar devices are associated with a long list of medical problems (weight gain, heart condition, posture problems, repetative motion injuries, addiction, etc). If you spend a significant amount of time with computers consider investing in a properly designed work-station and take frequent physically active breaks.

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, 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. Also try cumulative tests.

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.

Chapter 12: Lines and Planes. This is part of Calculus II. Students need to review this chapter before the course starts.

Chapter 13: Vector Functions.

Chapter 14: Partial Derivatives.

Chapter 15: Multiple Integrals.

Chapter 16: Vector Calculus.

In class we will try to do many more problems than assigned in the printed syllabus.

Problems set for the 5th edition.

Problems set for the 6th edition. This is the one we will use.

After each video lecture

a) redo the examples done in lecture

b) do the applicable homework assigbment

c) redo the examples in the text

For Fall or Spring semesters:

4 hours to view the videos

4 hours to attend the classes

4 hours to do the homework

For Summer I or II:

6 hours for each category

Calculus I and II; prefereably in the last two long semesters with a real grade of B or better. You need to know the basic differentiaon and integration concepts and formulas as well as Chapter 12. Chapter 12 is the natural beginning of Calculus III but it is part of Calculus II.

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

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

2. Computer Science Department labs, MAES Building, 2nd floor, bring your own headset, not so crowded.

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

1: Calculus as a Hybrid or U-Try Style course. Description of the general format, policy, schedule, benefits, and requirements for the course. Please read carefully.

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 (and they do have a penalty!)

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) Giving multiple answers without discussion/notification 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.

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

Graded exams should be returned to the instructor after inspection.

3:

4:

1:

2: University Undergraduate General Catalog.

3: Fall 2009 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.

A series of videos for reviewing Pre-Calculus is included. This corresponds approximately to Chapter 1.

A series of videos for reviewing Calculus I is included. This corresponds approximately to Chapter 2-6.

You may want to come back to them for a refresher or a simple description of a topic.

a) Gilbert Strang Calculus. Text.

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.

To see the 3D (IMAX-Type) images, if one is provided, you need special glasses and also re-set the view by choosing images appropriate to your glasses. So, let's say you have red-cyan glasses then first choose a 3D shape, then choose View-> Anaglyph, then choose Red-Cyan Anaglyph.

a) 3D curve drawing applet

Simple online interactive program for drawing curves, principle unit vectors, r, r', r", and osculating circle in 3d.

Go to Show->Curve Input to give parametric formulas, go to Show->Vectors to ask for velocity, acceleration, unit vectors, osculating circle.

You can click and drag the curve to change the view point or orientation.

b) 3D Grapher

Simple online interactive program for drawing curves in 2d and surfaces in 3d.

c) Applets for Calculus 3

d) Surfaces and slices

This applet shows the surface as well as slices by planes parallel to coordinate planes. Click on "Cross Section Applet".

e) Comprehensive 3D-plots Most of what we want to do with a surface can be visualized here.

Up to 4 functions and their tangent planes, partial derivatives, and gradient vector can be dislayed.

f) Applets for calculus

Click on image of applet to download.

g) Interactive Text for Calculus 3

Mathematica-based interactive text for calculus 3.

h) Interactive exercises for calculus 3.

i) MIT tool case for calculus

j) A 4-dimensional cube

k) Gallery of Curves and Surfaces For IMAX effect choose a surface, choose View-> Anaglyph, use your glasses.

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.

3D Graphing Calculator software for drawing in 3d .

SketchUp 3D building/engineering drawing package. (trial license)

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 and . MIT MATH

b) Stanford .

c) Yale .

d) Academic Earth.

e) YouTube.

f) TED .

g) Apple .

h) GapMinder .

Read Section 13.1

Cal III Video 1a, Space Curves, Part 1.

vector valued functions, component functions, parametric equation, limit, continuity and derivative of vector functions, examples, helix, line segments.

Corrections:

1-At minute 18, -4 is written as 4 (or vice versa)

2-The tape abruptly terminates (due to a crash). This tape does not have fast forward or scroll. However you can pause it.

Cal III Video 1b, Space Curves, Part 2.

Shadow of helix on coordinate planes, curves as intersection of two surfaces, curves embedded on a surface, introduction to computer aided graphing via Mathematica 6.

3D curve drawing applet

Simple online interactive program for drawing curves, principle unit vectors, r, r', r", and osculating circle in 3d.

Go to Show->Curve Input to give parametric formulas, go to Show->Vectors to ask for velocity, acceleration, unit vectors, osculating circle. You can click and drag the curve to change the view point, orientation.

Read Section 13.2.

Cal III Video 2, Derivatives and Integrals of Vector Functions.

Derivative, tangent vector, unit tangent vector, tangent line, differentiation rules, scalar, dot, and corss product rules, definite integral.

Read Section 13.3 to the middle of page 864 and do problems 1-12

Cal III Video 3a, Arc Length, Part 1.

Pythagorean Theorem, finding the length of an arc, arc length element, arc length function, parametrization with respect to arc length, length vs displacement.

Read Section 13.3

Cal III video 3b, Curvature Part 2.

Curvature, the three principal unit vectors: principal unit tangent vector, principal unit normal vector, principal unit binormal vector, osculating circle and plane, brief formulas and description of applications in physics, dynamics, and geometry (no theoretical derivation)

Add the following problems to your list for this section, Problems 20, 24, 40.

Text Edition 6ET, Section 13.3, Parts 2-3, Pages 831-836, Problems from 17-60, Page 836

3D curve drawing applet

Simple online interactive program for drawing curves, principle unit vectors, r, r', r", and osculating circle in 3d.

Go to Show->Curve Input to give parametric formulas, go to Show->Vectors to ask for velocity, acceleration, unit vectors, osculating circle. You can click and drag the curve to change the view point, orientation.

Read 13.4.

Cal III Video 4a, Motion, Velocity and Acceleration, Part 1

Position, velocity, acceleration, speed, trajectory problems, Newton's Second Law of Motion, initial conditions, turning points, range, Problem 28.

Text Edition 6ET, Section 13.4, Part 1, Page 838-842, Problems from 1-32, Page 847

Cal III Video 4b, Tangential and Normal Components of Acceleration, Part 2

Tangential and normal components of acceleration via projection, tangential and component formulas via speed and curvature. Problem 33 text 5ET/ Problem 35 text 6ET.

Add the following problems to your homework list:

Text Edition 5ET, Section 13.4, Part 2, Page 874-876,

Text Edition 6ET, Section 13.4, Part 2, Page 842-846, Problems 16, 28, 29, 33, 36 Page 847

3D curve drawing applet

Simple online interactive program for drawing curves, principle unit vectors, position, velocity, acceleration and osculating circle in 3D.

Go to Show->Curve Input to give parametric formulas, go to Show->Vectors to ask for velocity, acceleration, unit vectors, osculating circle. You can click and drag the curve to change the view point, orientation.

Read 14.1.

Cal III, Video 5a, Functions of Several Variables, Part 1 .

Functions of two variables, doamin, range, independent and dependent variables, verbal, numerical, algebric and visual representation of a function, graphs, wire frame, color maps, contour curves, level surfaces, isobars, isothermals, topographic maps, functions of three variables.

Correction: The tape ends abruptly. It is continued in Part 2.

Cal III, Video 5b, Functions of Several Variables, Part 2

Mathematica command for drawing raised contours on surfaces,iso bars and isothermals, contours of functions of three indepdent variables.

Multiple 3D-plots

Once you have tried it with pencil and paper enter your function here to compare.

Slicing a saddle contour plot of a saddle surface.

3D Grapher

Simple online interactive program for drawing curves in 2d and surfaces in 3d.

Read 14.2

Cal III, Video 6, Limits and Continuity. 75 min, 35mb

Review of limits, two pathological examples, definition of limit in 2D, how to show limit does not exist, how to show limit does exist, continuity, continuous operations, building continuous functions

Multiple 3D-plots

These hard to analyze functions can be visualized here. If you see a lot of wrinkles at the limit point, usually the origin, then the functions does not have a limit.

Read 14.3

Cal III, Video 7, Partial Derivatives. 45 min, 20mb

Partial derivatives, notations, geometrical interpretation, implicit differentiation, higher order derivatives, Clairaut's Theorem on equality of mixed derivatives, connection to partial differential equations or PDEs, Laplace's equation, harmonic functions, wave equation, heat equation.

Read 14.4

Cal III, Video 8, Tangent Planes and Differentials. 50 min, 25mb

Eqauation of tangent plane to a surface, linearization, differentiable functions, Theorem: if partial derivatives are continuous then function is differentiable, increments, differentials, total differential

Use this applet to see the tangent plane.

Comprehensive 3D-plots

Bottom left menu give choice of slicing or tangent plane.

Read 14.5

Cal III, Video 9, The Chain Rule. 70 min, 30mb

Review of chain rule from Calculus 1, chain rule for different tree diagrams, application of chain rule to implicit differentiation, Implicit Function Theorem.

Read 14.6

Cal III, Video 10, Directional Derivatives and the Gradient Vector. 75 min, 35mb

Directional derivatives, the gradient vector, steepest ascent/descent direction, tangent and normal lines to level curves, tangent planes to level surfaces, normal lines to level surfaces, orthogonal families of curves and paths of steepest ascent.

Coorection. At the end I say "orthonormal" families of curves, the correct phrase is "orthogonal" families of curves.

Comprehensive 3D-plots

Bottom left menu give choice of directional derivative and slicing with arbitrary vertical planes. You can right click on the bottom left page to change the direction of red arrow.

Read 14.7

Cal III, Video 11, Maximum and Minimum Values of Multivariable Functions. 75 min, 50mb

Relative/Local vs absolute/global maximum or minimum, critical points, saddle points, second derivative test, closed/bounded/compact sets, finding extremum on compact sets.

Extras:

To see the 3D (IMAX-Type) images you need special glasses, and on most sites you need to re-set the view by choosing images appropriate to your glasses. So if you have red-cyan glasses choose "red-cyan anaglyph".

Gallery of Curves and Surfaces Choose a surface, choose View-> Anaglyph.

For example go to : Space Curves -> Viviani, then go to View->Anaglyph, then slowly drag the mouse over the image to make it rotate.

Comprehensive 3D-plots Go to: View Settings -> Select 3D Views -> red-cyan anaglyph (there are two options: on white or on black background)

Read 14.8

Cal III, Video 12, Optimization subject to Constraints, Lagrange Multipliers.30 min, 12 mb

Lagrange multiplier method for solving maximization and minimization problems subject to constraints.

Extras:

You can use Comprehensive 3D-plots to visualize the optimization with constriants.

Suppose you want to optimize z=x^2+y^2 subject to (x-1)^2+(y-3)^2=10.

1- On the web site define function 1 as z=x^2+y^2.

2- Re-write your constraints as two or more functions e.g.

2a- solve your constraint for one of variables, say for y, and define function 2 as y=3+sqrt(10-(x-1)^2) go to ViewSettings, go to FunctionType select y=, go to Function 2 and enter the function y=3+sqrt(10-(x-1)^2) .

2b- similarly enter function 3 as y=3-sqrt(10-(x-1)^2).

3- Click on Graph to view all functions. The constriant will appear as a cylinder slicing the surface of the paraboloid. You are searching for max or min on the slice curve.

Read 15.1

Cal III, Video 13, Double integrals over rectangular domains. 30 min, 15 mb

Volume of a solid over a rctangualr domain, double integrals, double Riemann sum, midpoint rule, average value of a function.

Read 15.2

Cal III, Video 14, Iterated integrals over rectangular domains, Fubini's Theorem, separable functions. 20 min, 12 mb

Read 15.3

Cal III, Video 15, Iterated integrals over general domains, exchange of order of integration. 20 min, 12 mb

Read 15.4

Cal III, Video 16, Iterated integrals in polar Coordinates. Conversion formulas between polar and cartesian. 40 min, 24 mb

OPTIONAL : Read 15.5

To be uploaded. Video 17. Applications to calculation of mass, moments, second moments or moments of inertia, radius of gyration.

POSTPONE: Read 15.6 Edition 5

Edition 6 postpones this section to Chapter 16.6

POSTPONE Cal III, Video 18. Calculation of surface areas, two derivations for the formula.

Read 15.7 Edition 5

Read 15.6 Edition 6

Cal III, Video 19. 38 mb, 60 min.

Triple integrals in Cartesian coordinates, iterated integrals, Fubini's Theorem, six ways of doing a triple integral, applications: volume, mass, moments, centroid, second moments.

Book Edition 5ET. Read the first half of 15.8

Book Edition 6ET. Read Section 15.7

Cal III, Video 20. 25 mb, 40 min.

Triple integrals in Cylindrical Coordinate Systems. Transformation formulas from cylindrical to cartesian. Solution of two problems.

Book Edition 5ET. Problems 8, 10.

Book Edition 6ET. Problems 18, 20.

Book Edition 5ET. Read the second half of 15.8

Book Edition 6ET. Read Section 15.8

Cal III, Video 21. 19 mb, 30 min.

Triple integrals in Spherical Coordinate Systems. Transformation formulas from spherical to cartesian. Solution of several problems.

3D Grapher

Interactive program for drawing in 3d.

A review exchange of order of integration, cylindrical integration.

The Upcoming Test is on items between green bands below.

Read 16.1

Cal III, Video 22. 16 mb, 30 min.

Vector Fields.

Vector fields, examples: wind pattern, velocity field, force field, gravitational field, electrical field, magnetic field, current, flow lines, stream lines, direction field, gradient vector field, conservative vector field, potential function.

Read 16.2.

Cal III, Video 23. 30 mb, 50 min.

Line Integrals.

Line integral with respect to arc length along a curve, piecewise smooth curves, line integral with respect to x, y, or z, orientation of a curve, three dimensional cases, line integral of vector fields along a curve, 4 representations, applications: work, mass and center of mass of a wire.

Read 16.3

Cal III, Video 24a. 36 mb, 65 min.

Fundamental Theorem of Line Integrals, Part 1.

Line integral of conservative vector field (CVF), path independence, line integral of a CVF on a closed curve, finding potential for a CVF, Clairaut's Theorem, simply connected regions and sufficient conditions for a vector field to be a CVF, Law of Conservation of Energy.

Cal III, Video 24b. 10mb, 15 min.

Summary of Fundamental Theorem of Line Integrals.

< Read 16.4

Cal III, Video 25a. 10mb, 15 min.

Green's Theorem, Part 1.

Converting a line integral around a closed curve to a double integral over the region bounded by that curve. Several examples are done.

OPTIONAL Cal III, Video 25b. 10mb, 15 min.

Green's Theorem, Part 2.

Proof of Green's theorem.

!-- RED IS STARTING HERE -->

The Current Lecture and assignments are on the items between the red bands below.

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Read 16.5

Cal III, Video 26a. 20mb, 30 min.

Curl and Divergence, Part 1.

Basic definitions and properties of gradient, curl and divergence. Applications of Clairaut's Theorem: div curl F=0, curl grad f=0.

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Cal III, Video 26b. 20mb, 30 min.

Curl and Divergence, Part 2.

Vector format of Green's theorem, Stokes' Theorem, Derivation of divergence theorem in 2 dimension, brief explanation of Gauss's theorem in 3 dimension.

End of Calculus III.

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. If you need a review then view the following video.

Review of Exponetial Functions from Precalculus. Definition of exponential function. Extension to negative, zero, fractional,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.

Do problems of Assignments 1, Page 62.

Assignment 1 Page 62 Problems 1, 3, 5, 7, 9, 13

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 function 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. (See Software Requirement 3.)

Scan Section 1.6.

If you need a simpler description of pages 63-67 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.

If you need a simpler description of pages 67-71 view the following video.

Review of Logarithmic Functions from Pre-Calculus Simple description logarithms, rules of logarithms, base 10 and base e, graphs.

Do problems of Assignments 2, Page 74.

Assignment 2 Page 74 Problems 35, 36, 37, 38, 37, 48, 50

For NOW we will SKIP the section of 1.6 that is about inverse trig functions (Pages 72-74) and come back to it later. However if you want to get a head start you may want to view the following videos.

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.

Calculus Video Lecture 1. Tangent lines and velocity. Section 2.1, Page 87, 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.

Do problems of Assignment 3 Page 91.

Calculus Video Lecture 2. The Limit of a Function. Sections 2.2, Page 92 and Section 2.3 Page 104, Limits, one-sided limits, infinite limit, vertical asymptotes, The squeeze Theorem, basic limit laws.

Do problems of Assignment 4 Page 103.

Extras:

Calculating limits by constructing and inspecting tables of values .

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

Calculus Video Lecture 3. Using Algebra to Find The Limit of a Function. Section 2.3, Page 112, Finding limits using factorization and rationalization.

Do problems of Assignment 5 Page 111 as well as problems 11-30. Try Problems 58, 59 from page 113.

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.

Do problems 23-32 Page 103. Try Problem 39.

Extras:

Applet for study of asymptotes of certain rational functions.

Calculus Video Lecture 5. The Precise definition of a Limit. Section 2.4, Page 114, 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.

Do problems of Assignment 6 Page 122.

Extras:

This applet helps to understand the precise definition of limits.

Calculus Video Lecture 6. Continuity. Section 2.5, Page 124, 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.

Do problems of Assignment 7 Page 133.

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 a number, e.g. 3, for the applet to work (it has an error). Notice how the functions is defined. The syntax means: for x"<" 1 use x^4+0.5 , else if x "<" 2 use 0.5*x+1, else use cos(3*x)+1.

Calculus Video Lecture 7. Limits at Infinity. Section 2.6, Page 135, 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.

Do problems of Assignment 8 Page 146.

Extras:

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

In this applet you can change parameters of already defined functions and see how asymptotes and the shape of the functions are influenced.

Calculus Video Lecture 8. Slopes, Tangents and Velocities Section 2.7, Page 149, 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.

Do problems of Assignment 9 Page 155.

Extras:

Here you can see a JAVA applet showing the slope of sliding secant lines.

Calculus Video Lecture 9. Derivatives Section 2.8, Page 158, Derivatives, basic notation, equation of tangent line, applications.

Calculus Video Lecture 10. Derivative as a function Section 2.9, Page 165, Derivative as a function, Newton and Liebniz 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.

Do problems of Assignment 11 Page 173.

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 Section 3.1, Page 183, 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

Do problems of Assignment 12 Page 191.

Calculus Video Lecture 12. The Product and Quotient Rules Section 3.2, Page 192, The Product Rule, The quotient Rule, general power rule, power-quotient rule, examples, an explanation of product rule and quotient rule

Do problems of Assignment 13 Page 197.

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 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 hypothenuse, solving a right triangle, six trig functions for arbitrary angles, trig identities: pythagorean, ratio, negative, sum of angles.

Extras:

Here you can see the six trig functions in a unit circle setting .

Here you can see 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".

Here you can see the ordinary graph of sin, cos, tan, arcsin, arccos, arctan . click on "Graphs of elementary trigonometric functions".

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)

Do problems of Assignment 14 from Page 216. I recommend you do 1-30,35-44. For graphing use some of java applets. Those of you who have had calculus before try 46,47.

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 Section 3.5, Page 217, Composition of functions, the chain rule formula, examples, geometric explanation, an indication of the simplified proof.

Do problems of Assignment 15 from page 225. I recommend you do 1-46, 50. Those of you who have had calculus before try 58.

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.

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.

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. Section 3.6, Page 227.

Do problems of Assignment 16 from page 233. In addition I recommend you do 41-50. Those of you who have had calculus before try 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.)

Notice:

Test #3 covers 3.1 through 3.6 up to "Orthogonal Trajectories", middle of page 231. Problems similar to 1-27 can be on test 3. The remaining material will be on Test #4.

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

Do problems of Assignment 17 from page 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. Do 1,2,5...20,28,30,36,40,43,51.

Calculus Video Lecture 17. Derivatives of Logarithmic Functions . Section 3.8, Page 244. 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

Do problems of Assignment 18 from page 249.

Calculus Video Lecture 18. Hyperbolic Functions Part 1. Section 3.9 Page 250. Definitions of Sinh, Cosh, Tanh, Csch, Sech, Coth. Graphs of the six functions. Differentiation formulas. Basic identity.

Do problems of assignment 19, Page 254.

Calculus Video Lecture 19. Basic exercises on Hyperbolic Functions. Section 3.9 Page 254. Solution of Problems 11, 32, 39, 41.

to be uploaded: Optional Video Lecture 20: Inverse Hyperbolic Functions Sinh^-1 or arcsinh and other inverses, their graphs, domains, derivatives, formulas in terms of natural log.

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.

This section is very interesting and challenging. If you were asking why are we 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 math to be a major part of your career then you must do well in this section. We will have three sections in calculus that are like this: related rates, graphing, and optimization.

Calculus Video Lecture 22. Related rates. Section 3.10 Page 256. 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. Solution 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.

Do problems of assignment 20, Page 260.

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 reactangle in a computer window, baseball runer.

Calculus Video Lecture 23. Linearization. Section 3.11 Page 262. Linear Approximation, tangent line approximation, Linearization, small angle approximation, differentials, relative error, percentage error. Solution of several sample problems.

Do problems of assignment 21, Page 267.

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).

Calculus Video Lecture 24. Maximum and Minimum Points. Section 4.1 Page 285. 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.

Do Problems of Assignment 22, Page 285.

Extras:

Remember you have this all inclusive applet for differentiation, graphing, finding max and min.

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. Section 4.2 Page 295. Rolle's Theorem, The Mean Value Theorem, a function with zero derivative is a constant, functions with identical derivative differ by a constant. Solution of problems 12, 20.

Do Problems of Assignment 23, Page 295.

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. Section 4.3 Page 304. Increasing/Decreasing test, The first derivative test, Concavity, The second derivative test, inflection point, a sample problem.

Do problems of Assignment 24 Section 4.3 Page 304.

Calculus Video Lecture 27. L'Hospital's Rule and Indeterminate forms. Section 4.4 Page 313. 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 discusion of 0^0 case is missing from the video. The treatment is similar to inf^0 case. Please read about it in the text.

Do problems of assignment 25 Section 4.4 Page 313.

For summer 2007 we skip section 4.5. You may skip video lectures 28,29,30. Your next test may have one question from section 4.3 video #26.

Understanding section 4.5 is a very good indicator of how well you have understood all the 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 Section 4.5 Page 323.

Extras:

Remember you have this all inclusive applet for differentiation, graphing, finding max and min, etc.

Calculus Video Lecture 28. Curve Sketching 1, General Description. Section 4.5 Page 316. Point Plotting, Domain, Range, Intercepts, Symmetry, Asymptotes, First derivative analysis, Second derivative analysis.

Calculus Video Lecture 29. Curve Sketching 2, Solution of two problems. Section 4.5 Page 323, Problems 4 and 18.

Calculus Video Lecture 30. Curve Sketching 3. Solution of two problems. Section 4.5 Page 323, Problems 47 and 42.

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.

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.

Do problems of Assignment 26 Page 336.

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.

Calculus Video Lecture 33. Newton's Method.

Section 4.9 Page 347. 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.

Do Problems of Assignment 27 Page 351.

An important issue here is the efficient use of a calculator. Make sure you know how to use your calculator for storing a number and recalling it and grouping expressions in parentheses. There are keys for these activities typically called STO, RCL, and ( ). For example suppose x=0.123456789. Find out (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.

Applet shows the successive steps of Newton's Method. You control all steps in this applet.

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 converge to a different root. 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. The software can zoom on an area and show more details of the fractals. 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 click and drag over a small diagonal line segment.

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. Section 4.10 Page 353. Basic anti-differentiation formulas, initial conditions, rectilinear motion, position, velocity, acceleration, initial position, initial velocity. Solution of 16, and sample problems.

Do problems of Assignment 28 Page 358.

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.

Do problems of Assignment 29 Page 378.

Extras:

Applet helps you see the rectangles for the Riemann sums.

For summer 2007 video lectures 36,37,38 are optional. You are encouraged to view them.

Calculus Video Lecture 36. Areas and Distances Part 2. Section 5.1, Page 369. 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.

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 notaion".

2) In the last few seconds of the video 4+1/3 is written as 14/3. It is of course 13/3.

Do problems of Assignment 29 Page 378.

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

Calculus Video Lecture 38. Evaluating a Definite Integral. Section 5.2, Page 391, Problem 22. An important problem putting all basic concepts of area or definite integral calculation under one roof.

Calculus Video Lecture 39. Fundamental Theorem of Calculus (FTC). Section 5.3, Page 394. Integration and differentiation are inverse processes. How to use FTC to calculate areas and definite integrals in a short time.

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. Section 5.4, Page 405. 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.

Do problems of Assignment 32 Page 411.

Calculus Video Lecture 41. Substitution Method. Section 5.5, Page 414. Substitution method for indefinite and definite integration. Two styles for calculating definite integrals. Several examples.

Do problems of Assignment 33 Page 420.

Extras:

13 Exercises on substitution method.

10 Exercises on substitution method.

Calculus Video Lecture 42. Area Between Curves. Section 6.1, Page 437. Area between two curves, finding top, bottom, left, and right parts, cris-crossing curves, practice with x as a function of y, general regions.

Do problems of Assignment 34 Page 442.

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. Section 6.2, Page 444. 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.

Do problems of Assignment 35 Page 452.

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. Section 6.3, Page 455. 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.

Do problems of Assignment 36 Page 459.

Extras:

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

Calculus Video Lecture 45. Review of Chapter 5. 5 Problems (some with three parts) similar to problems 9-12 page 391, 21-25 page 391, 19-40 page 403, 1-40 page 411, 1-70 page 421. (Second problem is lengthy, you may want to to that at the end.)

Calculus Video Lecture 46. Review of Chapter 6.1-3. continuation of previous video. 3 Problems (some with three parts) 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 1, 14, 15.

Extras:

Several examples of calculation of work. ---------------------