• # Final Exam Review Packets

Posted by Allen Olsen on 6/4/2019

Here is a copy of the Final Exam Review Packet that I handed out in class today.

• # HW and classwork for 5/30 (A)

Posted by Allen Olsen on 5/30/2019

Today I will be out, but I will be in tomorrow and available to meet with you and answer questions after school, even though our class doesn't meet.

Remember that there is an in-class quiz on the Conics unit on Monday.  It will be taken from the set of previously-assigned homework questions below:

Sect. 8.1, pp. 570 - 71: 1 - 10; 11 - 29 (odd). (Parabolas)

Sect. 8.2, pp. 582 - 84: 1 - 6; 21 - 31 (odd); 37, 39. (Ellipses)

I anticipate four questions, one from each section of the HW.

Today, if you haven't done these problems, do them now, with help from others.  If you didn't finish all of them, do it now.

Today and over the weekend, here is what I would do to prepare for this quiz:

For parabolas, the diagram on page 565 and the tables at the bottom of pages 566 and 567 are useful; for ellipses, use the tables at the bottom of pages 575 and 577.

Know the equations 4px = y^2 for the parabola (vertex at (0, 0)) and x^2/a^2 + y^2/b^2 = 1 (vertex at (-a, 0)) for the ellipse.  You don't have to derive these; use them as a starting point.

1. p is the 'parameter', that is, the distance from the vertex to the focus on the axis.  In the ellipse, it is equal to (a - ae).

2. In both equations, the axis (or the major axis for the ellipse) is parallel to the x-axis.  The focus is to the right of the vertex if p>0, and to the left if p<0.

3. On the other side of the vertex from the focus is the directrix, with the equation x = -p/e (e=1 for the parabola and 0<e<1 for the ellipse).  The directrix is always perpendicular to the axis.

4. Finally, add (h, k) to the coordinates of all the points to get where they appear when the vertex is at (h, k) [parabola] or the center of the ellipse is at (h, k)

Be productive studying.  Thank you.

• # HW for 5/28

Posted by Allen Olsen on 5/28/2019

HW: We will have a Conics Test next Monday, June 3, in class.  Later this week we will review for it.  It will be based on correctly doing problems from the homework.  So if you have questions about the homework, you should ask during next class.  If you haven't done the homework, you should do it.

Final Exam Review:

Today we went through two problems from earlier in the year, to get a start at sorting through the topics that might be on the final exam.

Problem 1: f(x) = (3+x^2)/(4-x^2).  Find the domain and range of f(x).

To do this problem, you first need to know the Domain Rules:

1. √(x) has a domain of x ≥ 0.  2. 1/x has a domain of x ≠ 0.  3. log_b (x) has a domain of x > 0.

Second, you need to know how to find the inverse function.

Solution: The denominator factors as (2-x)(2+x).  The second domain rule applies, so the domain is x ≠ 2 and x ≠ -2, which can be written as D_f: (-∞, -2) ∪ (-2, 2) ∪ (2, ∞).

Next, find the inverse: y = (3+x^2) / (4 - x^2), y = f(x);  so x = (3+y^2) / (4 - y^2), y = f^(-1) (x).

Multiply this out to get x(4 - y^2) = 4x -x·y^2= 3+y^2.   Move the y^2 terms to the left, and the other terms to the right:

y^2 + x·y^2 = y^2(1+x) = 4x - 3.  So y^2 = (4x-3)/(1+x), and y = ± √[(4x-3)/(1+x)] = f^(-1) (x).  We want to find the domain of this inverse, to get the range of the original function f(x).

Use the first domain rule: (4x-3)/(1+x) ≥ 0.

There are only two ways for this to be true:

A. 4x-3 ≥ 0 AND 1+x > 0 (factors both positive); OR B. 4x-3 < 0 AND 1+x < 0 (factors both negative).

In class we showed that this led to either x ≥ 3/4, or x < -1.

Thus the range R_y: (-∞, -1) ∪ [3/4, ∞).  The function has a horizontal asymptote at y = -1, and two vertical asymptotes at x = -2 and x = 2.

I have asked you to write down in your notebook the elements of this problem that you need to work on.  This problem appeared on the first test of the year.

Key ideas for your Study Guide: Domain, Range, Rules for Domain, Finding the Inverse of a rational function, Solving Inequalities.  See pages 78 - 81, and particularly examples 3 and 4, for Domain/Range.  You should also look at Sect. 2.8, pp. 231 - 37, for how to solve inequalities.  Look at your Notes from September for the Rules for Domains.

Problem 2: Let f(x) = 2x^3 - 7x^2 - 10x + 24.  f(4) = 0.  Use this information to factor f(x) completely.

Solution: Use synthetic division with "4" in the box to get f(x) = (x-4)(2x^2 + x - 6) + 0.  Then use synthetic division or the quadratic formula to find that (2x^2 + x - 6) = (x+2)(2x - 3).

Answer: f(x) = (x-4)(x+2)(2x - 3).  This problem was on the Chapter 2 test last fall.

This is not a hard problem, but you need to practice problems like this so that they go well on the Final Exam.

Key Ideas for your Study Guide: Synthetic Division; Factor Theorem; Remainder Theorem; Factoring Nonmonic Trinomials.  This is all in Section 2.4 of your textbook, pp. 192 - 200, which you should read carefully and do the example problems.

• # HW for 5/21

Posted by Allen Olsen on 5/21/2019

Objective: Find equations for ellipses and circles.

For the following assignment, the tables at the bottom of pages 575 and 577 are useful.

HW: Sect. 8.2, pp. 582 - 84:

1 - 6; 21 - 31 (odd); 37, 39.

• # HW for 5/16 (A), 5/17 (D)

Posted by Allen Olsen on 5/17/2019

HW for the weekend:

Carefully read example 5 on page 569, and then do problems 49 through 52 on page 571.

Read problem 63 on pages 571 - 72 about the design of a suspension bridge, and play with the parabola that is described in the problem.  See if you can understand what the problem is asking.  Or, if you prefer, try Problem 64 about a bridge arch.  These problems are best done with several students together.

If you are having problems with 11 - 29, look at example 3 on page 568.

See you on Monday.

• # HW for 5/16

Posted by Allen Olsen on 5/16/2019

For A block: I am out today.   When your celebrating is done, now read the following:

Your classroom assignment today is to work constructively together to complete last night’s homework.

If some of you want to show your work on the board for others to see, that is fine.  But the goal should be to get everyone to the point that all have finished that homework.  I will ask to see your homework for a homework grade when I see you on Monday.  If you don't use this time productively, I will mark you down for it.  If you have difficulties, send me an email and I will do my best to help.  I expect to be back tomorrow, so you could ask me questions then.  But get this done.  I don't want to hear that "I didn't get how to do the problem."  You have a ton of resources -- each other, me, the textbook -- engage, persist, and you will be able to figure it out!

If you finish the homework from last night, you can start reading section 8.2 of the textbook, or work on the weekend homework below:

HW for the weekend:

Carefully read example 5 on page 569, and then do problems 49 through 52 on page 571.

Read problem 63 on pages 571 - 72 about the design of a suspension bridge, and play with the parabola that is described in the problem.  See if you can understand what the problem is asking.  Or, if you prefer, try Problem 64 about a bridge arch.  These problems are best done with several students together.

If you are having problems with 11 - 29, look at example 3 on page 568.

• # HW for 5/15

Posted by Allen Olsen on 5/15/2019

Objective: Find the vertex, focus, directrix and focal width of parabolas.

HW: Sect. 8.1, pp. 570 - 71:

1 - 10; 11 - 29 (odd).  Use the diagram on page 565, and especially look at the tables at the bottom of pages 566 and 567.

If you get stuck on these problems, look in the TEXTBOOK and READ or SKIM Section 8.1 for the definitions of terms or for examples that can show you how to do these problems.

• # HW for 5/14

Posted by Allen Olsen on 5/14/2019

Objective: Calculate inverse matrices using row operations.

HW: p. 558, Chapter 7 Review Exercises:

29, 30, and 32.

• # HW for 5/7

Posted by Allen Olsen on 5/7/2019

Some announcements.

1. Tomorrow Juniors must be at the Mock Crash.  If you are not a Junior, you need to come to class.

2. On Friday morning, D block will meet in Room 801, instead of our regular room.  This is because part of the US History AP Exam will be given in Room 803.

3. The Chapter 7 quiz will be on Thursday for A Block, in Room 803, and on Friday for D Block, in Room 801.  This is a non-calculator Quiz.  If you are unable to take this quiz, you must make arrangements in advance with me, or possibly lose credit.  in general, you have two days at most to make up a test that you miss.

4. Continue to study for the exam, doing the problems outlined in the previous posting.

• # HW for 5/6

Posted by Allen Olsen on 5/6/2019

Objective: Prepare for Chapter 7 Quiz.

I announced this for Wednesday, but because of the Mock Crash that all Juniors must attend during the first three blocks on Wednesday, I will have to move this exam to Thursday (A Block) and Friday (D Block).  Some of you also have AP exams this week.  I will need you to take this test by Friday at the latest.  If this is not possible, you need to contact me by email immediately and then talk to me face-to-face before the exam is given to set a retake date.

HW: Ch. 7 Review exercises: be able to do problems in the following groups:

Group 1: matrix multiplication: problems 3 - 10

Group 2: Evaluating determinants: problems 13 - 14

Group 3: Reducing augmented matrices to row echelon form: 16 - 18

Group 4: Solving systems of equations using Cramer's Method: 19 - 22

Group 5: Solving systems of equations using Gaussian Elimination: 23 - 28

Group 6: Solving systems of equations using reduced row echelon form: 33 - 36.

These are the only problems you need to work on in the chapter review, but you need to be able to do them.

I'm telling you this because I noticed that some of you were doing problems involving matrix addition, or inverse matrices.  You don't have to do these problems.