Lesson+Summaries+-+Curve+Sketching

Mobius Strip Day! Ryan Smith Thursday, March 10, 2011

Today in class, we learned about the mobius strip, a one-sided, three-dimensional figure. We created own own mobius strips and tested the idea that it only had a single side. We then proceeded to cut the mobius strips that we made in various ways to see what shapes would form. We then put all the mobius strips onto the ceiling, and hopefully they're still there.

This is a link that shows how to make a mobius strip and some experiments to do with it. You can always make your own experiment to. []

There was no homework today.

**Here is the note we took on March 18th**
Here is a link to a good video if you need help on learning about oblique asymptotes: []


 * March 21, 2011**
 * Virginia Lee**
 * Topic: Increasing and Decreasing Functions**

Today, we started off the class with an activity on GSP, (Geometer's Sketchpad - which is conveniently available to you on all the OSCSS computers) and some review on increasing, decreasing and constant functions. We then did a worksheet where we sketched functions from their derivatives.

We can use the first derivative to determine the interval(s) over which the function is increasing or decreasing.

Ex 1: Find the intervals of increasing and decreasing for:

f (x) = 2x3 + 3x2 - 36x + 5 f ’(x) = 6x2+6x-36 0 = 6(x+3)(x-2) x = -2, 3 <- Called critical points


 * || -3 > x || -3 < x < 2 || x > 2 ||
 * f ’ (x) || + || - || + ||

∴ f (x) is increasing when x > -2 and when x>3 (-∞, 3) f (x) is decreasing when -2 < x < 3 (2,∞)

Ex 2: Sketch a continuous function if f ' (x) > 0 when x < 0, and f ' (x) 0 and when f (0) = 4





The derivative of a cubic function is a quadratic function! Taken from: []

Remember that the derivative of a cubic function is a quadratic function, whereas the derivative of a quadratic function is a linear function, and the derivative of a linear function is a constant function. Knowing this is helpful when sketching functions and their derivatives.

- At inflection points, note that the slope of the tangent is equal to zero.

2. Sketching Functions from Derivatives
 * HOMEWORK:** See [] and complete the questions. We also have a curve sketching quest on April 1st.
 * HANDOUTS:** 1. Increasing / Decreasing / Constant Functions (front), GSP: When are Functions Increasing or Decreasing (back)


 * Additional Links:** See [] for a concise explanation about increasing / decreasing / constant functions and the first derivative test (which we need to know in order to do the homework. [] also has a great explaination, with lots of examples!

March 22, 2011 by Rosetta D'souza
 * Maxima and Minima**

***Function must be defined over an interval.** f’(x) = 0
 * For functions defined as an interval [a, b], we can find the local and absolute max and min points.
 * Max and min points happen at either end points* or critical points**.**
 * Therefore, to find the absolute max and min points, we need to test the value of the function at the end points and the critical points.

Ex) Find the extreme values (absolute max and min points) of f(x) = -2x3 + 9x2 + 4, x ϵ [-1, 5] end points - x = -1, 5 f’(x) = 0 = -6x2 + 18x ∴ -6x (x - 3) = 0 ∴ x = 0, 3 - critical points f(-1) = 15 f(5) = -21 – absolute min f(0) = 4 f(3) = 31 – absolute max

Sketching a Function Graph From a Derivative Graph Handout

Homework: N/A

Extra link: [] explains the above in more detail.

[] []
 * Rabeea Fatima**
 * March 23, 2011**
 * Minima and maxima (contd**.**)**
 * Helpful Links:**
 * Homework sheet handed out.** **﻿**
 * Notes: ﻿ **
 * Notes: ﻿ **

Teamwork Tuesday
Today our first group task was to solve the speed challenge. The entire class was divided into groups of three or four members each and the challenge presented to us was:  - Place the numbers from 1 to 6 in the circles so that no two consecutive numbers are in circles joined by a line for the following diagram

Next, the topic we covered was optimization. For this activity, we worked in our groups again and using the following materials we made an open box: a pink sheet, scissors, ruler and tape.

Then, as a class we solved the following optimization problem: a piece of sheet metal 60cm by 30cm is to be used to make a rectangular box with an open top. Determine the dimensions that will make a box with the largest volume.  Volume = X (60-2x) (30-2x) = 4x^3-180x^2+1800

**· A fxn is at its max or min at its critical points (when f’(x)=0) or at its end points** V(x) = 4x^3-180x^2+1800 V’(x) = 12x^3-360x+1800 = 0
 * · Our interval 0X= 23.6, 6.3 [23.6 is invalid because it would make width negative which is impossible, our original width is 30 cm so 30-2(23.6) of width would not let us make a box at all]

V (6.3) = 4(6.3)^3 -180 (6.3)^2 +1800 <span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">= 5196 cm3

<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">So, the dimensions would be

<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">Height = x = 6.3 cm <span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">Length = (60-2x) = (60-2(6.3)) = 47.4 cm <span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">Width = (30-2x) = (30-2(6.3)) = 17.4 cm

<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">Next, each group figured out the steps to solve such optimization prlems. Then the class as a whole came up with the following concise set of steps to solve these problems <span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">1//.// **//Draw a diagram for the problem and define a variable.//** ==== <span style="color: windowtext; font-family: "Times New Roman","serif"; font-size: 12pt; font-weight: normal; line-height: 115%;">After that we re-did the box making activity, this time using a yellow sheet and found the dimensions <span style="color: windowtext; font-family: "Times New Roman","serif"; font-size: 12pt; font-style: normal; font-weight: normal; line-height: 115%;">needed to get a box with the largest volume. (answer: height=4cm, width=13.6 cm, length=20cm) ==== <span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">Then we spent the rest of the class working on the “Practice with optimization” sheet to understand how to set up functions for such problems. <span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">(Answers: //<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">1) A(x) = x(50-x) // <span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">//2) P(x) = 2x + 128/x// //<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">3) SA(x) = 2x^2 + 4000/x // //<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">4) D(t) = √ (√12.5-4t^2) + (√12.5-6t^2) // //<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">5) Cost(r) = 2πr^2a +2vol/r // //<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">6) P(x) = (50+x)(30-0.25x) // **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Useful links: ** <span style="color: blue; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">[] <span style="color: blue; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">[] <span style="color: blue; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: normal; margin: 0cm 0cm 10pt;">[]  **Homework:**   - Curve sketching assignment is due tomorrow March 30, 2011   - Optimization homework sheet is due Thursday March 31, 2011
 * //<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">2. Determine the function and the interval. // **
 * //<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">3. Find the first derivative (f’(x)) and set it equal to 0 to find critical points. Also find the end points. // **
 * //<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">4. Test the critical points from the interval and end points (if closed interval) by inputting them into the function. // **
 * //<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">5. Find the highest and lowest # from step 4, it is the optimal point. // **
 * //<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">6. Refer back to the initial question and answer it. // **

March 30th, 2011
Today was a work period, so we had a chance to work on the Curve Sketching assignment, the Optimization homework, and study for the test on Friday. Here are some useful links to help you with curve sketching and optimization. [] [] [] []

Remember that the Optimization homework is due by 9:00 am on Thursday, March 31st and the test is on Friday, April 1st!
STUDY HARD!!