There is a moment in Spring that counting cannot find nor an hourglass measure.
A very old man, whose face was as fleshless as the foot of a bird, sat in meditation upon the rocky shore of the flat and hazel-covered isle. A boy sat by his side. Behind the two, half hidden by the trees, was a little monastery. It had burned down but had been roofed anew with rushes by the boy, that the old man might find shelter in his last days. In the garden the lilies and the roses of the monks had spread out until their confused luxuriance met and mingled with the narrowing circle of the ferns. Beyond the ferns rose many hazels and small oak trees. In the distance was the sound of a flute.— W.B. Yeats
Exeter 1-5 and AMC 1-5 Feb. 3 Exeter probs limiting values
exponent probs
Last night reworked and written up. Go here for some help on Exeter 5. Feb 4 Exeter reference 5 more Exeter. Do more of the AMC questions.
Feb 5 Thurs class AMC questions Tues. Feb 8 Mon class AMC questions Wed. Feb9 Tues class AMC questions Thurs Feb 10 Wed class AMC questions Fro. Feb11 AMC questions Tues. Feb 15 Mon class AMC Test in Library at 8:30 Wed. Feb 15 Tues. class Wed. class parametric equations x = cos t; y = sin t
Plot this equations by hand.
Fri., Feb. 24 Thurs. class handout from book — Read and do 1-8 of example problems Mon. Feb 27 Worked examples of parametric to rectangular conversion Redo prob 7 by putting t values on each graph.
Continue problem begun in class, (see web pages).
B-level problems on assignments pages.
Tues. Feb 28 Mon. Class finish B-level problems on assignments pages.
Graph these two polar graphs (where q stands for theta, and sin2 means "sine squared): r = cos2 q - sin2 q
and
r = 3cos2 q sin q - sin3 q.
There is method to my madness here.
Plot this one dimensional parametric equation. Give the path and a sample of times along the path. x = 3 sin( pi t/5 )
Plot these parametric equations (with annotations)
x = sec t; y = tan t.
x = sin(3t); y = sin(t)
Have fun.
Wed., March 1 Mon.& Tues. Class If you want to prepare for the test. Review (or figure out) how to make lines and circles with both polar and parametric equations. With parametric equations figure out how to make ellipses, hyperbolas, and parabolas. Parametric and Polar Take home Test: Parametric project using lines, circles, and conics. Polar project using lines and circles. Friday, March 11 take home parametric Newton's method; functions; inverse functions Probs from handout, first assignment. Do as many as you need to do. Tues, March 15 Mon. class Write a computer program for Newton's method (or understand your old square root program). Assignment 2 from handout. Wed, March 16 Tues. class For x = -2, -1, 0, 1, 2 estimate the slope at the corresponding point on y = x^2. Do the same for sin x for x = nπ for integer n.
Sorry about the late posting.
Wed., March 22
functions, slopes of curves functions handout from class Thurs, March 23 1. finish function handout. 2. If f(x) = x^2, graph 1/f(x) and f(1/x). 3. What is the derivative graph (slope graph of x^3 - 3 x
4. Bring in a sketch of cos x and sec x on the same axes.
Thurs, March 23 absolute value, functions Graph |x| + |y| = 1;
Graph |x| - |y| = 1 ;
Graph |x| + |y| = 0;
Graph |x| - |y| = 0;
Graph ||x| + |y|| = 1Graph f(x) =|x+1|+|x-1| -1;
Dont forget the graph of cos x and sec x done on the same axes. Add tan x and cot x done one a different set of axes.
Fri., March 24 Thurs class Graph sine and cosine on 4 pairs of axes. One one pair, graph secant as the reciprocal of cosine; on a second graph csc x; on a third graph tan x; on the last graph cot x. Tues., March 28 Mon class Graph f(x) = 2^x and f(x) = 1/x^2 on the graph sheets I handed out. For each find 1/f(x), inverse f, and f'(x).
Graph 2 sin 2x on a large sheet of graph paper. On the same axes graph its reciprocal, its inverse, its derivative.
Tues., March 28 Tues class sin(1/x) Wed class come up with an interesting graph. Thurs class maxima and minima probs 1. Problem on web page which is find the area of the rectangle under y = 3 e^(-x2/5)
2. Find the area of the largest rectangle under y = cos x for x between -pi/2 and +pi/2.
Tues. April 12 Mon. class problem on web page Tues. class on web, find as many area and volume figures as you can. Fri., April 15 (tax day) Wed. class areas and volumes Fri. class Do Russian problems 1040 and 1456 Tips on 1040, Do 2-6 first, and use trig identities for the others. cos 2x = ... Tues. class log, trig review Do Russian problems 540 and 1525 Wed. class Do 1526 Friday Thurs. class Do 1091, 1092 Mon Fri. class Analytic geometry Find coordinates of orthocenter, circumcenter, centroid of A(0,0), B(6,0), C(4,2). Find incenter if you have time.
Finding centroid, orthocenter, circumcenter, and incenter (harder) given the vertices of the triangle 1.Find the equation of the GH line and the points it crosses the edges (same triangle you have been using).
2. There are 4 angles at the intersection of y =2x and y = 3x. Find the equations of the bisectors of all 4 angles.
Wed. class Foot, perpendiculars, radius of incircle. Triangle on web pages Thurs. class Begin with the triangle for which we found the special points such as the orthocenter (it is on Wednesday's web pages).
1. Find the three altitudes of this triangle.
2. Find the feet of the altitudes, ie., the points where the altitudes meet the edges.
3. Find the equations of at least two sides of the orthic triangle (which connects the feet).
4. Find the perpendi culars to the nearest side of the orthic triangle to a vertex.
5. Find the point of interesection of these perpendiculars which is a famous point of the triangle.
Fri. class Mon, Tues class Final Exam: You will received the coordinates for three points from me at the beginning of the period. All students will get different points and thus have a different triangle. You will be told to create a picture of your triangle with added features (such as the medians). After you complete your picture, you will get the test which will include questions asking you to find slopes, angles, distances, midpoints, the equations of lines and their intersections.
Alternate option. If you have time, you can do a project-test before the period. The skills required will be the same.
First option: Compute the point discovered by Josh Klehr for a triangle I give you.
Second option: This involves trisecting the angles of a triangle I give you and finding the intersection of the trisecting lines.
oops! The Dutch part of the problem does not work. You are done when you showthe it does not work.