Homework due 4/27/09

Do problems 11, 17 & 18 on page 298 of your textbook.

Recommended Practice

Problems #1, 2, 4, 10, 12, 14, 42, 43, 49, 63 & 64 on pages 306 - 307 are some good problems to test your understanding of the larger concepts at play in Integration. What you should understand is that our work with derivatives is not forgotten and that it is all connected.

Note: Check the conference folder on to get solutions to the problems above in pdf format.

Test on Thursday, April 23rd

You will have a test on all of the work we have done with Integration, i.e. the areas bounded by curves, their meaning given a rate of change function, the idea of the definite integral, finding definite integrals by estimation and then by using anti-derivatives (The Fundamental Theorem of Calculus), and indefinite integrals and what they represent. Various techniques of finding anti-derivatives will naturally be assessed on this test. Make sure to go back to the very beginning of this unit and re-trace your footsteps. Focus on both, the concepts and the skills.

Homework due 04/20/09

Do problems 17, 18, 27, 53, 55, 58, 75 (a) and (c), 84 (a) and (b), 78 and 87 on pages 318-320. You may be curious about the strange order of the problems towards the end, namely, 84, 78, 87, and know that it is intentional.

Homework due 04/13/09

Do problems # 22, 25, 26, 27, 30, 31, 32, 35, 40, 54, 57, 59, 66, 71, 73, 77 & 87 on pages 292-293.

Homework due 04/09/09

The homework is on the worksheet given to you in class. If you were absent then you can download the pdf version from the class conference folder.

Quiz on Tuesday, April 7th

You will be quizzed on all our work with areas bounded by curves, left/right-hand approximations, the definite integral and The Fundamental Theorem of Calculus. Please bring your formula sheets with you.

Homework due 04/03/09

Do problems 6, 8 & 9 on page 270. Then do problems 4, 5, 7, 10 & 29 on pages 273 - 274.

Homework due 3/31/09

Do problem 2, 5, 26 (a) and (b), and then 27-29 on pages 253-254.

Homework due 03/30/09

Make sure to finish problem 2 from class and then do problems 6, 7, 8 and 9 on page 246 of your textbook.

Homework due 03/27/09

Read pages 240-244 of your text book and take careful notes on the reading in your homework journal.

Archimedes' Method of Exhaustion

It is important to note that Archimedes' method of finding the area of a circle is not very different from the third century Chinese method. One difference is that the formula is stated as:

The area of any circle is equal to the area of a right triangle in which one of the legs is equal to the radius and the other to the circumference (Fauvel and Gray eds, 1987, pp. 148-150).

The link below shows a step by step process of how he achieved his results.

How Did Archimedes Find the Area of a Circle?

Homework due 03/16/09 (after Spring Break)

Finish problem 26 on page 212 that we started in class. Transfer the entire solution to your homework journal as this will count as a homework assignment. Have a nice break!

Test on Thursday, March 5th

You will have a test on Related Rates this Thursday. Please make sure you have reworked problems and get enough practice so that you are more efficient when working problems in class.

Homework due 03/02/09

Do problems 25, 27, & 33 on pages 212-213 of your textbook. Also, keep in mind that you will have a test on related rates on Wednesday of next week.

Answers for Question 24 on Page 212

(a) z^2 = 0.25 + x^2. Therefore, z = square root of (0.25 +x^2)

(b) dz/dt = 0.693 km/min

(c) 0.4 radians/min

Homework due 02/26/09

Do problem 20 on page 212.

Quiz on Friday, February 27th

You will have a quiz on Related Rates and also on the technique of Implicit Differentiation.

Homework due 02/24/09

Do problems 18, 19 and 21 on page 212 of your textbook.

Things to Remember When Approaching Related Rates Problems:

Make sure to first determine all that you know about a situation, i.e. determine what quantities are changing and which ones are fixed. It is best to draw various diagrams in certain cases. Write down any rates given to you as derivatives in Leibnitz's notation. Then determine what the question is asking for. If it is a rate then is there a particular moment they want the rate for or is it just an expression? If you are not using implicit differentiation then try and get the "setup" as I described it by making the rate you want equal to a product of two rates, one given and one to be determined. If you are using implicit differentiation then make sure you are differentiating with respect to the desired variable (in most cases time).

Take-Home Test Discussion Post

You may address queries about questions on the Optimization test by commenting to this post. Please be careful not to share answers or entire solutions. Responses should be providing hints by guiding peers towards what to think about when attacking a problem. You may also want to refer to the post that provides tips for handling optimization problems.

Implicit Differentiation

Please make sure to review your class notes and understand the new technique we learned today so that you are not completely lost in Thursday's block class.

Homework due 02/05/09

Do problems 21 and 32 from pages 203-204.

Homework due 02/02/09

Do problems 12, 13, 16 & 19 on page 203.

Homework due 01/30/09

Do problems 11 and 17 on page 203.

Tips for Optimization Problems:

1. (a) Write down all that you know about a particular situation. This includes equations you can form and the constants and variables involved.

(b) Sketch any diagrams that help establish the way in which the quantities are related.

2. Figure out what quantity you are trying to maximize or minimize.

3. Find a function for this quantity in terms of the variables involved. In the case of more than one variable, try and express the functions in terms of only one variable through substitution.

4. Consider a reasonable domain for your function given the context of the problem. Check end behavior given the domain.

5. Find all Critical Points.

Homework due 01/29/09

After looking over the entire structure of the problem we answered in class which was to find the dimensions of the aluminum can, 12 oz in volume, that would minimize the surface area (amount of aluminum) used. Make sure you understand how we managed to verify that our values after using calculus were in fact the ones that would minimize the surface area. Then, consider the dimensions of the can we have found to be most economical and consider it containing soda. Next answer the following:

An actual can of Coca Cola is not a perfect cylindrical container and its dimensions do not correspond to those that minimize the material cost of manufacturing the can found above. The radius of an actual can is 3.413125 centimeters and height is 12.22375 centimeters. Why do you think that an actual Coca Cola can does not have the dimensions found by the optimization problem above?

History of Calculus - January 22nd

Dr. Adrian Rice has been a professor at Randolph-Macon College in Ashland, VA and is an emminant historian of mathematics, with 18th and 19th century math as his specialty. He received his Ph.D. from Middlesex University, London, in 1997. He also holds masters and bachelors degrees from King's College London, and University College London, respectively. He came to Randolph-Macon in September 1999 after having served as a visiting professor at the University of Virginia from 1998-1999, and a lecturer at Middlesex University from 1995 to 1998. For more information on Dr. Rice and his publications please visit http://faculty.rmc.edu/adrice/public_html/ .

He will be talking to us about the history of calculus; the circumstances surrounding its emergence, the key personalities involved and what other breakthroughs it has shaped. Please bring any questions you might have about the history of calculus and why it would be important to learn it. You can also generate some questions by commenting on this post if you would like for certain things to be discussed during his talk. We will meet in SAC on Thursday, January 22nd.

Take-Home Test Discussion Post

You may comment on this post to pitch questions about the test. The test is due Friday, January 23rd.

Homework due 01/15/09

Do problems on page 136-137; # 35, 37 & 44 - 47.

Homework due 01/13/09

Do problems 1, 2, 5, 6, 7 & 9 on page 136 of your textbook. You may want to keep your formula sheets or your notes from class on the rules of differentiation handy when attempting these.