Optimization

Calculus Links for Key Topics



Limits

http://www.zweigmedia.com/RealWorld/tutorials/frames2_6a.html

This website explains how to estimate limits numerically. It also gives the basic definitions of a

limit. There are some practice questions, as well.



http://www.zweigmedia.com/RealWorld/tutorials/frames3_7.html

This website is about limits and continuity. It gives the basic definition of continuous functions.

There are some practice questions.



http://www.calculus-help.com/funstuff/tutorials/limits/limit05.html

This is a video about determining continuity. It talks about some different types of discontinuity,

too. There are some pretty good visual descriptions.



http://www.zweigmedia.com/RealWorld/tutorials/frames3_8.html

This website is about understanding limits based on algebra. It also has some information about

closed form functions. There are a few practice questions. It explains how to evaluate a limit

using simplification and there are some questions about that.



http://www.calculus-help.com/funstuff/tutorials/limits/limit01.html

This is a video about some basics of limits. There are also several practice problems.



http://www.calculus-help.com/funstuff/tutorials/limits/limit06.html

This is a video about the intermediate value theorem. It gives the basic definition of the theorem.

There is a graphical explanation regarding the theorem that is pretty helpful.



http://www.youtube.com/watch?v=HYSI-AHUqRM&feature=related

This is a video about the basic ideas of limits. It also has information about calculating a limit.

You can follow along with some example problems.



http://www.youtube.com/watch?v=u06Yrvt2XLc&feature=channel

This video explains how to use the definition of a limit to prove a limit statement. You can

follow along as a man explains a problem. There is information about epsilon/delta definitions.



http://www.youtube.com/watch?v=JwecZoYct3Y&feature=channel

This is the second part of the above video. This video explains what limit theorems are and how

to use these theorems to find limits.



http://www.youtube.com/watch?v=GGbMec7vd0U&feature=channel

This is the third video from the series of videos about limits. There is information about

substituting numbers in order to determine limits. There are some practice problems that are

explained.



http://www.youtube.com/watch?v=NE7vF78y7lU&feature=channel

This is the fourth video from the series of videos about limits. This video talks about limits with

rational functions. It gives the basic definition of limits of rational functions. A few problems

regarding this subject are also explained. This video is about ten minutes long.

http://www.youtube.com/watch?v=UMxScwTpa3c&feature=channel

This is the fifth video from the series of videos about limits. The video explains how to find the

limit of a rational function at an undefined point. One example problem is solved and explained.

This video is about two minutes long.



http://www.youtube.com/watch?v=QZtAwLHvUAI&feature=PlayList&p=A33E14E1235B1D99

&playnext=1&playnext_from=PL&index=1

This video talks about limits and their notations. An example problem is used to demonstrate

how to write limit notations.



http://www.youtube.com/watch?v=TgO5kAuIojY

This video is about limits and continuity. The video provides some basic definitions and also

explains how to use different ways to solve problems. There are some graphical explanations.

The video is about six minutes long.



http://www.youtube.com/watch?v=trAmOJ8pHNc

This video is about the limits of functions. It is about nine minutes long. A few example

problems are used to illustrate different information regarding limits. There is some information

regarding L‟Hopital.



http://www.coolmath.com/limit1.htm

This website provides basic information about limits. A geometric example is used to explain

what limits are. There are some numerical examples, as well. Explanations and answers are

given. There are also some graphical examples.



http://www.youtube.com/watch?v=W0VWO4asgmk

This is another video about the basics of limits. This video is about seven minutes long.

Graphical representations accompany sample problems.



http://www.youtube.com/watch?v=YRw8udexH4o

This video provides more examples regarding limits. The video shows how a function

approaches a limit at a certain point. There are two example problems that have graphical

representations. The video is about seven minutes long.





Implicit Differentiation

http://www.youtube.com/watch?v=23nBGx1avXI&feature=channel

This video teaches how to use implicit differentiation to find derivatives. A practice problem is

explained. There is also graphical analysis. Another practice problem is given and solved. Next,

there is a problem about finding the equation of a tangent line. The movie is about eight minutes

long.



http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/implicitdiffdirectory/ImplicitDiff.h

tml

This website gives problems regarding implicit differentiation. It shows how to solve a practice

problem explicitly, and re-explains how to solve it implicitly. Step-by-step processes are given.

Sixteen practice problems are given, along with detailed solutions. There is also a link near the

bottom of the website that gives access to a greater variety of practice problems.



http://www.hostsrv.com/webmaa/app1/MSP/webm1010/implicit

You can use this website to check your answers for implicit differentiation problems. You can

type in an equation relating “y” and “x” and then see how to get y‟.

Optimization

http://tutorial.math.lamar.edu/Classes/CalcI/Optimization.aspx

This website explains the basics about optimization. The information provided is pretty extensive.

Some example problems are provided and they also have detailed, step-by-step solutions. These

problems are pretty representative of the types of optimization problems provided in the math

textbook.



http://www.nipissingu.ca/calculus/problems/application_problems.html

This website gives five related rates and optimization problems. The questions all have links to

the solutions. The solutions are very detailed and well-explained. On the bottom of the page,

there are other links. One of them is a Related Rates and Optimization Tutorial. Another one is a

link to a practice test.



http://www.mathvids.com/lesson/mathhelp/413-applied-optimization-problems

This is a video with practice problems about optimization. These questions have a step-by-step

walk through. This video is about twenty-minutes long. There are graphical explanations. This

site is pretty good and the problems are well-explained.





Trig Identities

http://www.sosmath.com/trig/Trig5/trig5/trig5.html

Different trig identities are given. These include: reciprocal identities, Pythagorean identities,

quotient identities, co-function identities, even-odd identities, sum-difference identities, double-

angle formulas, etc.







The Chain Rule

http://www.zweigmedia.com/RealWorld/tutorials/frames3_2.html

This link provides an explanation of how the chain rule works. There are also practice problems

that students can solve by themselves in a multiple choice format.



http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/chainruledirectory/ChainRule.html

This has a brief explanation of the concept. It is then followed by practice problems arranged in

three difficulty levels and followed with a detailed explanation. I recommend this site for anyone

who needs a step by step explanation of the chain rule.



http://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx

This website is useful for those who understand the concept, but just want extra problems to

make sure that they know it. It has practice problems and detailed explanations. It also has

information on how to prove various derivative formulas.



http://archives.math.utk.edu/visual.calculus/2/chain_rule.2/index.html

I highly recommend this website for people who want to take a brief quiz on all aspects of the

chain rule. It‟s a good thing to look at before an upcoming quiz. It provides a ten question quiz

on the concept. It has a variety of concepts incorporated into it including exponential, trig, and

logarithmic functions. This problems are followed with solutions and hints.



http://www.1728.com/chainrul.htm

This gives a detailed explanation for a fairly simple problem. It shows an informal explanation of

the chain rule.

http://www.calculus-help.com/funstuff/tutorials/derivatives/deriv05.html

I highly recommend this film for people who need a fun, but detailed explanation of the chain

rule. This is a video walk through of the concept and has helpful hints.



Antiderivatives

http://www.allaboutcircuits.com/vol_5/chpt_6/10.html

This provides extremely basic rules about integration including the constant rule, rule of sums,

and rule of differences.



http://archives.math.utk.edu/visual.calculus/4/antider.1/

Provides a basic breakdown of what antiderivatives are and few theorems followed by practice

drills to test your knowledge.



http://www.zweigmedia.com/RealWorld/tutorials4/frames6_1.html

This website is for people who already have a basic knowledge of antiderivatives. It provides an

explanation of antiderivative notation and a word by word explanation of how to read an

antiderivative formula. It has explanations of how to solve indefinite integrals and examples.

http://www.zweigmedia.com/RealWorld/tutorials4/frames6_3.html

This website shows how to solve an indefinite integral using a numerical and graphical approach.

It has examples and several graphical representations.



http://www.math.ucdavis.edu/~kouba/ProblemsList.html

Provides various types of integration problems with full solutions.



http://www.bloomington.k12.mn.us/indschool/TJ/staff/foley/APCalc/ppt/Antideriv.pdf

Quick introduction and review of antiderivatives. This power point has an explanation of the

meaning of “the value of c” and several practice problems to follow.



http://www.zweigmedia.com/RealWorld/tutorials4/frames6_2.html

Introduces how to solve indefinite integral by using „u‟ substitution. It has a step by step process

of the concept with extra practice problems.



Related Rates

http://www.zweigmedia.com/RealWorld/tutorials/frames4_4.html

This website gives a step by step breakdown of related rates problems. Given the rate of change

for certain quantities, you are required to find the rate of change of related quantities. This

website incorporates problems using a variety of shapes.



http://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx

This website introduces how implicit differentiation can be used when dealing with related rates.

It includes several real life story problems in relationship with related rates problems. This

website gives DETAILED explanations of a wide range of scenarios. I highly recommend this

website for people who



http://mathdemos.gcsu.edu/mathdemos/relatedrates/relatedrates.html

Provides a range of visual aids that illustrate fundamental concepts for understanding how to

develop equations that model related rate problems. It also provides a 6 step process that outline

the steps in a related rates problem. This website is good for people who like visual

representations of problems.

http://video.google.com/videoplay?docid=2145093424601290341

Gives a very verbose and detailed tutorial of related rates followed by intermediate leveled

sample problems with visual representations. This video is extremely helpful if you find it

beneficial to listen to lectures where everything is drawn out for you step by step.



http://www.youtube.com/watch?v=WlBpPpE-XRg

This video gives an explanation of a fairly easy related rates problem involving spheres. The

instructor also demonstrates the scenario. This is something good to look at as an introduction of

a basic related rates problem.



http://www.youtube.com/watch?v=8M11k0WPcMc

This website is similar to the one above, except it provides an explanation for a more difficult

problem involving triangles. This video as well as the one above show how problems as solved

in a step by step process, something that is necessary to do on tests as well as the AP.

http://www.youtube.com/watch?v=jv4gTxWqeBE

Shows, in an informal way, how to do related rates problems using implicit differentiation. It

also has drawings of the situation which may make it helpful to understand.



http://www.youtube.com/watch?v=a-WPa6bRdv4

Introduces a step by step process of how to solve related rates problems. It also has several

helpful hints that are good to know for future problems.



http://www.youtube.com/watch?v=wTYvMpVITg8

A video of how to solve a related rates problem when dealing with a cone. It gives a very

detailed explanation.







Limit Comparison & Direct Comparison Tests

http://www.youtube.com/watch?v=xesQnFWw8f8&feature=related

Provides an informal, but very detailed explanation of how to use the direct comparison test to

find out when a series converges(interval of convergence).





Derivatives

http://www.zweigmedia.com/RealWorld/tutorials/frames2_1.html

(set up: explanation of average rate of change, example of application, practice problems;

analyzes average rate of change with graphs and algebraically)



http://www.zweigmedia.com/RealWorld/tutorials/frames2_2.html

(set up: explanation of instantaneous rate of change by comparing the topic to average rate of

change, provides algebraic and graphical practice, explains the definition of a derivative in

relation to limits, provides several examples throughout)



http://www.zweigmedia.com/RealWorld/tutorials/frames2_4.html

(set up: reiterates what a derivative function is in relation to a graph and breaks down the four

step process (algebraically), practice problems given to solve the derivative and are also in story

problem format)

http://www.zweigmedia.com/RealWorld/tutorials/frames2_4B.html

(set up: explains the short-cut to finding derivatives with the power rule with practice problems,

explains how to use the power rule when given negative exponents, explanation of what

differential notation is and how to approach more complex derivatives)



http://www.zweigmedia.com/RealWorld/tutorials/frames3_1.html

(set up: explains the idea that (1) the derivative of a quotient is not the quotient of the derivative

and (2) the derivative of a product is not the product of the derivatives; explains how to use the

product rule and apply it to practice problems; explains how to use the quotient rule and apply it

to practice problems; the “CTE” (Calculus Thought Experiment) shows you what to do when

you are given a function that seems to require both the product rule and the quotient rule)



http://www.zweigmedia.com/RealWorld/tutorials/frames3_3.html

(When dealing with the derivative of logarithms and exponents, there are different rules that

apply; Note: this link also has some explanation of the use of chain rule in relationship to

logarithmic and exponential functions; if you do not understand the chain rule, you can either

look at the basic links from the Chain Rule section or simply skip those sections of the webpage;

set up: shows how to take the derivative of a function that uses „log‟ or „ln‟ and provides

examples with practice problems; shows how to take the derivative of an absolute value with

examples; shows how to take the derivative of exponential functions)





Fundamental Theorem of Calculus

http://www.zweigmedia.com/RealWorld/tutorials4/frames6_4.html

(set up: introduces the fundamental theorem of calculus by applying knowledge about derivatives

and integration from previous units and linking it to the topic by using a sample problem; defines

the FTC in both algebraic and written notation; provides practice problems in both algebraic and

graphic form; explains the FTC with substitution and provides sample and practice problems)



http://www.youtube.com/watch?v=PGmVvIglZx8&feature=related

(video that is approximately 10 minutes long; defines the FTC and provides several practice

problems with step by step walkthroughs of each)





Finding Max and Min

http://www.zweigmedia.com/RealWorld/tutorials4/framesc5_1.html

(set up: defines relative max, min, and extrema in relation to a graph with practice problems;

defines absolute max and min in relation to a graph with practice problems; defines „candidates‟

for extrema with examples and practice problems)



http://www.zweigmedia.com/RealWorld/tutorials4/frames5_3.html

(Helpful for basic understanding of how certain points apply to a graph; set up: outlines the

features of a graph with important points and neatly organizes the important aspects of critical

points, inflection points, maxes and mins, behavior at infinity; provides several practice

problems)



http://www.zweigmedia.com/RealWorld/tutorials4/framesc5_2.html

(Requires understanding of max & min on a graphic standpoint; set up: shows how to apply max

and min on an algebraic level, which leads into solving optimization problems; it also provides a

general outline of how to solve them; there is step by step practice to solve the question)

http://www.themathpage.com/acalc/max.htm

(set up: explanation of extrema (max & min), critical point, concavity; shows how to solve for

them and required conditions; with examples and practice problems)



http://www.clas.ucsb.edu/staff/lee/Max%20and%20Min's.htm

(set up: a less interactive method of understanding the concept and application of max and min,

but still provides the same depth of explanation; explains how to find critical points with an

example and various methods; critical points are also analyzed by using the first derivative test

and second derivative test)







Inverse Trig

http://www.youtube.com/watch?v=e1L1_Lf0aGU

(video that is approximately 7 minutes long; shows how to take the derivative of inverse trig

functions and shows the basic derivative of the six main ones; provides several examples with

step by step walkthroughs of each)



http://www.youtube.com/watch?v=f5pvJBRWlb8

(video that is approximately 5 minutes long; similar to the video before, but with more

challenging problems)