Homework 12/17

p.262 #41, 43-46

Homework 12/3

p.226 #27, 24, 33, 38

Homework 11/27

p.168 #30-33

Homework 11/13

Finish problems 25 & 29.

Homework 11/8

Finish p.146 #6, 18, 24

Homework 11/7

p.146 #17

Homework 11/5

p.139 #18, 24

Due Monday

p.139 #5-6

Book section on product rule, quotient rule, and chain rule (incl. derivatives of the trig fucntions).

In case you missed it...

Georgia 42
Florida 30

Things due Monday

The derivatives of f(x) = arccsc x & f(x) = arccot x

Take home optimization test

Book section on optimization (make sure you give examples of a wide variety of problems)


Also, you have an in-class optimization test Monday and a memorization quiz Tuesday.

I'll be online most of the weekend.

Homework 10/25

p.344 #29-34, 37

Homework answers

15)
Primary: C = 2x^2 + 8xy
Limiting: 20 = yx^2; y = 20/x^2
Primary (one-variable): C(x) = 2x^2 + 160/x

17)
Primary: P = ab(a-b)
Limiting: a+b = 8; b = 8-a
Primary (one-var.): P(a) = a(8-a)(2a-8); P(a) = -2a^3 + 24a^2 - 64a

18)
Primary: A = (1/2) (l^2) sin (theta)
Limiting: None (l is a constant)

20)
Primary: P = 100 sqrt(x) + 150 sqrt(y)
Limiting: x + y = 40; y = 40 - x
Primary (one-var.): P(x) = 100 sqrt(x) + 150 sqrt (40-x)

22)
Primary: P = xy^2
Limiting: (x/2)^2 + (y/2)^2 = r^2; y^2 = 4r^2 - x^2 (limiting eq. is the pyth. thm.)
Primary (one-var.): P(x) = 4xr^2 - x^3

27)
Primary: A = xy
Limiting: y = (4-x)/(2+x)
Primary (one-var.): A(x) = x(4-x)/(2+x)

28)
Primary: V = hx^2
Limiting: 108 = 4x + h; h = 108 - 4x
Primary (one-var.): V(x) = (108 - 4x) x^2

33)
Primary: S(x) = (x - x1)^2 + (x - x2)^2 + ... + (x-xn)^2
Limiting: None

(The value of x that minimizes S(x) is (x1 + x2 + ... + xn)/n, i.e. the average of x1, x2, ..., xn.)

36)
Primary: C(x) = 30 sqrt(10000 + x^2) + 15 (200 - x)
Limiting: None

44)
Primary: T(x) = (1/3) sqrt (90000 + x^2) + (1/8) (200 - x)
Limiting: None

Upcoming Assignments

Tomorrow (Tues.): Optimization test

No later than Wednesday: book section on f(x), f '(x), & f ''(x)
(You must turn this in before Thursday so that I can give you feedback before the weekend. You may come and work on it after you are dismissed on Wednesday.)

Monday (10/29): book section on optimization.

The following Monday (11/5): book section on product rule, quotient rule, and chain rule (incl. derivatives of trig. functions)

Homework 10/12

Do problems #43 & 44 in the packet.

Homework 10/11

Do problems 15 & 29 in the packet.

Reminder

By popular demand, we will be having a memorization quiz on Wednesday.

I won't be online tonight until about 9:00 due to ultimate. If you need help before that (or just want to play some ultimate), I will be playing at 7:00 at the Dalton State College intramural fields (next to the tennis courts). Remember: these write-ups will count as a quiz grade.

Homework for the long weekend

Solve the problem you selected in class and write it up for display. While correctness will be the main component of the grade, a substantial portion will also come from how well your solution is presented. Make sure you present the problem well enough that someone unfamiliar with our class can understand the problem (even if they wouldn't understand the calculus you use to find the solution). They should also be able to understand how you set up the equation you are trying to minimize or maximize as that (probably) won't involve calculus. Also make sure you devote some words to explaining the steps you take to find the solution and that you explain the solution in the context of the problem (i.e. what the number you found means in terms of the original problem). Make sure you answer what the problem asks. if the problem asks for dimensions, give the dimensions; if it asks for area, give the area (and the dimensions that produce that area).

Since these will be displayed, make sure you only write on one side of the paper and make your solution visually appealing (diagrams and such). I'd also prefer that you use ink on unlined paper (without scribbles). Neatness will factor into the grade.

I will be checking the blog periodically Friday, Saturday, and Tuesday, but will not have access to the internet Sunday or Monday.

Homework 10/2

p.210 #18, 19 (a & b)

Homework 9/28

p.203 #13-15 (change the x in 15 to x^2)

Answers to p.176 #3-4

If the graphs represent f(x):
3. f(x) is increasing on (-inf, -2), (2, inf); f(x) is decreasing on (-2, 2); f(x) has critical points at -2 & 2.
4. f(x) is increasing on (-1, 0), (1, inf); f(x) is decreasing on (-inf, -1), (0,1); f(x) has critical points at -1, 0, & 1.

If the graphs represent f '(x):
3. f(x) is increasing on (-3.5, 0), (3.5, inf); f(x) is decreasing on (-inf, -3.5), (0, 3.5); f(x) has critical points at -3.5, 0, & 3.5.
4. f(x) is increasing on (-inf, -1.3), (1.3, inf); f(x) is decreasing on (-1.3, 0), (0, 1.3); f(x) has critical points at -1.3, 0, 1.3.

Homework 9/27

p.176 #43-45, 47
Solve first as if the graphs were of f(x) and then as if they were f '(x)

I won't be home until late tonight as I will be attending a lecture by Alfie Kahn. I will be online this afternoon until about 5:00.

Homework 9/26

p.160 #7-11

ARC and IRC

Average rate of change: the average rate of change of a function f(x) over the interval [a, b] is the change in f(x) over the interval divided by the change in x over the interval, in other words, the rise over the run or the average slope of the function on the interval. Since linear functions (y = mx+b) have constant slopes, the average rate of change of a linear function over any interval is m, its slope. To find the average rate of change of a non-linear function over [a, b], you must first find the points (a, f(a)) and (b, f(b)). (These are the points on f(x) when x=a and x=b, respectively.) Next you find the slope between these two points. Voila, you have found the average rate of change over the interval.

Instantaneous rate of change: the instantaneous rate of change of a function f(x) at a point x=c, is the slope of the line tangent to f(x) at x=c. To find this slope, you will need to first find the derivative of f(x) since the derivative tells you the slope of f(x) at any point. Once you have f '(x), plug in c for x to get f '(c). This value is the instantaneous rate of change of f(x) at c.

Put more simply, if you drove your car for 2 hours and covered 80 miles the average rate of change of your position (which in this example would be your average velocity) would be easy to find since you would need only divide 80 by 2 to obtain an average rate of change of 40 m/h. The instantaneous rate of change, on the other hand, would be different depending upon whether you wanted the instantaneous rate of change after 40 minutes, or after 50 minutes, or at any other time since your velocity at any given instant would be constantly changing. If you could model the distance you had traveled as a function of time, however, you could find your instantaneous speed at any given time by finding the derivative of the function.

Further reading if you are still having trouble:
http://www.mathematicshelpcentral.com/
http://www.ugrad.math.ubc.ca/

I will be on later tonight. If you have any questions, leave a comment.

The Taylor Tests are graded...

The scores varied from 50 to 100, with most of the class falling in the 70's and 80's. These are very good scores for a class's first Taylor Test, though there is certainly room for improvement. Make sure you study for tomorrow's test (which will not be multiple choice).

Practice Test Answers

1. 2 cos(2x) or 2 cos^2(x) - 2 sin^2(x)
2. 2x csc(5x) - 5x^2 csc (5x) cot (5x)
3. (2x+1)^1/2 + x(2x+1)^-1/2 or (3x+1)/(2x+1)^1/2
4. -4/(2x-1)^2
5. (1/2)x^-1/2 - (1/2)x^-3/2 or (x-1)/2x^3/2
6. -4x tan(3-x^2) sec^2(3-x^2)

1. 22
2. 19
3. 4x-3

I'll be on tonight if you have any questions & I'll be in my room in the morning around 7:30.

Homework 9/18

Finish p.150 #7-34

This (remaining) week's schedule

Wednesday (shortened period): Review/Practice Test

Thursday: Memorization Quiz, Taylor Test #1 (quiz grade)

Friday: Test

Homework 9/17

Finish p.130 #12-19, 23-25, 43-46, then do p.131 #29-30, 51-52

Extra Credit Post!

Anyone who leaves a comment on this post by 6:00 today (Saturday) receives free Extra Credit for being conscientious and thinking about calculus before Sunday night.

Homework 9/14

Complete problems 23-36 on the handout and do p.130 #11, 13, 15

I'll be online some Saturday and Sunday; leave a comment if you have a question & I'll respond whenever I see it.

Homework 9/12

p.121 #13-20, 27-34

Homework 9/10

p.87 #11-16, 23-26
p.235 #21-24
p.150 #1-2

I will be in my room in the morning until 7:45. After that, I will be in the auditorium.

I'll be online off and on for most of the evening. Leave comments if you have questions. If you're having trouble registering, try registering for a gmail account or ask a friend to post for you.

Homework 9/7

p.99 #5-8,13-16

Reminder

I forgot to mention it today, but the graphs of y = sin x and y = cos x will also be on the memorization quiz tomorrow. You will need to label 1 & -1 on the y-axis, and the four quadrantal angles (pi/2, pi, 3pi/2, 2pi) on the x-axis.

Homework 9/6

p.194 #35-42

Homework 9/5

Do problems #24-27,37-38,40-44 on the handout

Memorization quiz tomorrow (and another one friday)

Don't forget. It'll be over everything from last friday's memorization quiz as well as the special limits, continuity on an interval, and everywhere continuous.

Homework 9/4

p.84 #29-32

Homework Help Thread

I'll be back on in about 15 minutes (around 11:00). Go ahead and post any questions you have and I'll start answering them when I get on, but remember: since I don't have the worksheet with me, you need to give me specifics as to the problem you're having trouble with. Also, if someone posts a problem that you've already solved & I'm not on yet, point them in the right direction (preferably without just telling them the solution).

Glory, glory

Homework 8/31

Do problems 35-44 on the handout.

Homework 8/30

p.65 #57-60
You'll need to use the two special limits (which will be on the mem. quizzes next week) that are on page 62.

Homework 8/29

p.65 #41-44

Test Tomorrow (8/28)

The Trigonometry test is tomorrow. You should study your practice test and worksheets.

Also, remember that the 5-2/5-5, 7-2/7-3, 7-4/7-5, and p.459 worksheets are due tomorrow as well as your unit circle. I will be online sometime after open house if you have any questions.

Homework 8/24

p.459 worksheet #21-22, 26-28, 32-35

Homework Help Thread

I'll be on for a while if any of you need some help (and are still conscious).

Friday's test rescheduled

It will instead be on Tuesday the 28th.

Homework 8/23

Complete all earlier worksheets and do p.459 (today's worksheet) #7-14.

Homework Help Thread

I'll be around for the next hour or so. Post any questions to the comments (make sure to include the question that you're working on since I don't have the worksheet with me).

Bo Jackson in Tecmo Bowl

He was unstoppable since Tecmo Bowl didn't have hip injuries.

Test on Friday (8/24)

There will be a test over trigonometry Friday. The test will cover:

Converting degrees to radians (and vice versa)
Finding the values of trig. functions of angles that are multiples of 30 & 45
Simplifying trigonometric expressions
Solving trigonometric equations
Solving right triangles

Memorization Quiz Friday (8/24)

The quiz will cover everything from the first memorization quiz as well as:

Sum and difference identities
Double angle identities
Half angle identities
Difference of squares, diff. of cubes, & sum of cubes

I am going to move converting degrees to radians (and vice versa) to the test.

Homework 8/21

7-2 #1-7
7-3 #1-3, 7-11
Finish finding exact values for the six trigonometric values of the unit circle.

Homework 8/20

Find sine, cosine, tangent, cosecant, secant, and cotangent of the following angles:
0, 30, 45, 60, 90, 120, 135, 150, 180.

Also, study for the memorization quiz.

Memorization Quiz on Tuesday

Things to be memorized include:
Definitions of all 6 trig functions
Reciprocal identities
Quotient identities
Pythagorean identities
45-45-90 & 30-60-90 triangles

Homework 8/15

Problems 1-8 on worksheet 11-4

Homework 8/14

Finish the selected problems on the handouts.

Homework 8/13

Finish the family of graphs problems from class.

Commenting on Posts

The way this blog is currently set up, comments are only permitted by registered users. If you want to comment on a post you will need to sign in or, if you have not commented before, register a blogger id (if you have a google id, you can use it to sign in to blogger). The website should be able to walk you through this process. When registering you will be asked what name you would like to appear on your comments: set up your account so that the name appearing on your comments is your first and last name. Any anonymous or pseudonymous comments will be deleted. As detailed in the syllabus, no comments will be allowed that disparage other students or give complete solutions to homework. Since this blog is associated with Southeast, any violations of the commenting policy will be subject to school discipline. I would encourage everyone reading this post to leave a comment (something as simple as asdf or n/t will suffice) in order to make sure that you understand how to do so.

AP Calc Student Syllabus

AP Calculus covers beginning and intermediate level calculus with the goal of preparing students to take the Advanced Placement Calculus AB exam in the spring. As this is an AP course, it will focus not only on how to solve problems but also on the application of knowledge to real-world situations and on the theory that underlies the problem solving techniques. Students taking this course should be prepared to do a fair amount of memorization—in order to do well on the AP exam certain things (for example, trig identities and logarithm rules) must be memorized to avoid taking up to much time simply getting ready to solve the problems.

Students will be graded on daily work, homework, quizzes, tests, and/or projects. Students are responsible for doing the work assigned to them and for reading any supplementary texts assigned. Test grades will comprise 60% of the final grade, quizzes 30%, and homework/daily work 10%. Since this class is preparing you for a multiple-choice exam, tests will often be given in a similar format.

Students are expected to come prepared for class. This means you should have paper, pen or pencil, and your book. If you find that you do not have one of these items, you should make arrangements to acquire them before the bell rings. Students who are unprepared once class has started more than once in a week may be docked points from their daily grades. Students should be ready to work when the bell rings. This does not mean that you can run in when the bell rings, lay down your things and go to the restroom. If you plan on using the restroom, be sure to give yourself enough time to get to class before the bell. Remember: a tardy results in detention as per school policy.

This class has a class blog at apcalc08.blogspot.com. All homework for the class will be posted on the blog. In addition challenge problems will be posted on the blog periodically. Students are encouraged to use the comments on blog posts to discuss problems and help their fellow students. Rules for posting comments are as follows:

1. All students must post under the name they use in class (no anonymous comments).

2. No disparaging comments are to be made about other students or their work.

3. While you are encouraged to help your classmates with their work, you may not simply post the solutions to the problems. You need to help each other, not do each others work.

ETS Approved AP Calc syllabus

Below you will find the (slightly edited) AP Calculus syllabus that was approved by the ETS over the summer. This is not the student syllabus (the student syllabus will have details on grading and such and will be given a separate post), but it provides a general overview of how I envision the class proceeding. Just thought I'd post it for you to look over if you're interested.

--------------------------------

Instructional Design

The following document contains a brief outline of the sequence of topics I teach in my calculus course. I begin my course with an in-depth review of precalculus skills for two major reasons, the first, and most obvious, of which is that gaps in students’ understanding of these concepts, if not diagnosed, can cause undue difficulty in the assimilation of the concepts encountered in calculus. Moreover, these gaps, if not found early, can be masked as problems with calculus rather than what they actually are which makes them harder to fix later in the semester. The second reason is that this review allows me to treat subjects with which students are already familiar (my treatment of functions, for instance, is extremely general, requiring students not only to recognize relationships between non-mathematical sets as functions but also to properly describe such functions as being injective or surjective) at a greater degree of abstraction, thus allowing them to develop the abstract reasoning ability required for parts of calculus. Over the course of this review most students begin to realize that the ability to defend an answer and understand the underlying mathematics conceptually is valued more in my class than simply arriving at a correct answer through procedural knowledge. In addition, during this review I am able to stress the connections between functions and their graphs which I can then build upon in the calculus section of the course.

As regards my particular treatment of functions in this review, I first introduce functions as a specific type of a relation between sets using concepts from elementary set theory. After students are comfortable with this definition of a function (and the concepts of injective and surjective functions), I introduce the graph of a function as the locus of all points that are solutions to the given equation. This approach allows me to specifically emphasize the interplay between the graphical and numerical representations of the function and leads naturally into more advanced concepts such as the average rate of change of a function over an interval (which I approach both graphically and numerically) and elementary curve sketching techniques (where I can again emphasize the relationship between the numeric and graphical representations of a function through end behavior, an analytical concept). I find that this approach leaves students better able to deal with more advanced (often analytical) concepts relating to functions that arise later in the course such as the interpretation of the derivative and definite integral as functions (such as is found in the second fundamental theorem of calculus, for example). Throughout my review of functions (and later as related concepts arise) students are expected to be able to make verbal arguments of their solutions and approaches to problems and to communicate clearly (both verbally and in written assignments) both their positions on problems dealing with these concepts and, more importantly, the reasoning that led them to that position.

I try to continue this emphasis on conceptual understanding into the portion of the course in which we cover calculus. This takes several forms. Often, to introduce a new topic I first pose the topic as a subject for debate before the class. This helps both the speakers, who must come up with a convincing argument for their side, and those that don’t speak, who must find flaws in the speakers’ arguments, develop their reasoning skills. I also try to incorporate a more abstract understanding of the processes of calculus through extensions to topics. For instance, while my primary (conceptual) concern when we study limits is that students understand a limit as being able to get arbitrarily close to a value, I reinforce this understanding by covering the epsilon-delta definition of a limit. Though the use of that definition never shows up on one of my tests, I think it helps the students crystallize their more informal understanding of a limit.

While I believe that a true conceptual understanding of mathematics is necessary for students to be successful in my course and to retain their knowledge past my course, I am not a Luddite in that regard. Most of my students come to me with a basic working knowledge of how to use graphing calculators (which the school provides) and I incorporate those calculators whenever applicable. In problem sets on definite integrals, I usually throw in one or two functions not easily integrated so that students realize the value of the ability of the TI-83 to approximate such intervals. While I do have some assignments on which students cannot use calculators (to prevent students from becoming dependent upon them), many of my assignments not only teach students how to solve problems but also how to use the calculator intelligently (some students waste more time trying to do everything in a calculator than it would take to do the same task by hand) in doing so.

The book that acts as the primary resource for the class is the sixth edition calculus book by Larson, Hostetler, and Edwards. The school provides all students with TI-83 graphing calculators for use at school and at home.

Teaching Strategies

In my class I try not to simply teach but to facilitate learning. To this end, I frequently have students present problems to the class. After a homework assignment, I usually pick several particularly interesting problems from the assignment for students to present. If the assignment contains a particularly difficult problem, I sometimes have students work the problem as a class, with each student only allowed to contribute one step until everyone has contributed. I also assign research problems periodically throughout the course in which a student (or, in some instances, a group of students) is given a particular application of the topic we are studying which they must then research and then present to the class. These presentations (which include a written component as well as the actual presentation) are evaluated not by me, but by the other members of the class, who complete forms in which they not only rate the presentation but must then also comment on its particular strengths and weaknesses. As these presentations are usually on applications of calculus, the verifications that students provide of their solutions frequently make use of their graphing calculators; this is especially true of projects involving limits as students use everything from tables to graphs to programs they have written to illustrate (and provide verification of) their solution. I think that the use of calculators in this manner is particularly convincing to students and I encourage its use in these projects by having a ranking on the evaluation form for how convinced the observing student was of the accuracy of the solution.

Course Syllabus

I. Functions, Graphs, & Limits (8 weeks)

A. Functions

1) Functions as relations between sets (domain & range)

2) Definition of functions

3) Injective, surjective, bijective

4) Composition of functions

5) Inverses

B. Linear Functions

1) Slope as a rate of change

2) Parallel and perpendicular lines

3) Equations of lines

a) slope-intercept

b) point-slope

C. Non-linear Functions

1) Families of functions

(y=xⁿ, y=|x|, y=1/x, y=x^1/n)

2) Piece wise functions

3) Exponential functions

4) Logarithmic functions

a) Logarithmic functions

b) Properties of Logarithms

5) Trigonometric functions

6) Odd and Even functions

D. Limits

1) Intuitive definition of limits

2) Limit notation (including right and left hand limits)

3) Existence of a limit

4) Evaluating limits

5) Formal definition of limits

6) Limits involving infinity

7) Definition of Continuity

8) Discontinuity

a) Point discontinuity

b) Jump discontinuity

c) Infinite discontinuity

II. Derivatives (12 weeks)

A. Tangent lines

1) Local linearity

2) Secant lines

3) Tangent line at a given point

4) Instantaneous rate of change as limit of average rate of change

B. The Derivative

1) Definition of the derivative

2) Differentiability and continuity

3) The Power Rule

4) Properties of Derivatives

5) Derivatives of Trig. Functions

6) The Product of Quotient Rules

7) The Chain Rule

8) Derivative of exponential functions

9) Derivative of inverse functions

10) Derivative of logarithmic functions

11) Implicit differentiation

12) Related Rates

C. Applications of the derivative

1) Critical points and relative maxima

2) Graphical interpretation of the first derivative

3) Graphical interpretation of the second derivative

4) Applications of the derivative

5) Rolle’s Theorem

6) Mean value Theorem

7) Optimization

8) Linearization

III. Integrals (8 weeks)

A. Antiderivative

B. Approximating Area

1) Using a regular partition

a) Right and left endpoint

b) Midpoints

c) Using any point (proof of equivalence)

2) Riemann Sums

3) Definition of the definite interval

4) The Fundamental Theorem of Calc.

5) Definite Integrals and the Antiderivative

a. Average Value Theorem

b. Mean Value Theorem For integrals

6) The 2^nd Fund. Theorem of Calculus

C. The Integral

1) Properties of integrals

2) Integration by parts

3) Approximating def. integrals

a. Trapezoidal rule

4) Area between Z curves

5) Volume

a. solids of rotation

b. solids with known cross-sections

6) Differential Equations