## Practice Makes Perfect

**Square a Binomial Using the Binomial Squares Pattern**

In the following exercises, square each binomial using the Binomial Squares Pattern.

Exercise 2

((q+12)^2)

**Answer**(q^2+24q+144)

Exercise 4

((x+frac{2}{3})^2)

**Answer**(x^2+frac{4}{3}x+frac{4}{9})

Exercise 6

((y−6)^2)

**Answer**(y^2−12y+36)

Exercise 8

((p−13)^2)

**Answer**(p^2−26p+169)

Exercise 10

((4a+10)^2)

**Answer**(16a^2+80a+100)

Exercise 11

((2q+13)^2)

Exercise 12

((3z+15)^2)

**Answer**(9z^2+65z+125)

Exercise 14

((2y−3z)^2)

**Answer**(4y^2−12yz+9z^2)

Exercise 15

((15x−17y)^2)

Exercise 16

((18x−19y)^2)

**Answer**(164x^2−136xy+181y^2)

Exercise 17

((3x2+2)^2)

Exercise 18

((5u^2+9)^2)

**Answer**(25u^4+90u^2+81)

Exercise 19

((4y^3−2)^2)

Exercise 20

((8p^3−3)^2)

**Answer**(64p^6−48p^3+9)

**Multiply Conjugates Using the Product of Conjugates Pattern**

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

Exercise 21

((m−7)(m+7))

Exercise 22

((c−5)(c+5))

**Answer**(c^2−25)

Exercise 23

((x+34)(x−34))

Exercise 24

((b+frac{6}{7})(b−frac{6}{7}))

**Answer**(b^2−frac{36}{49})

Exercise 25

((5k+6)(5k−6))

Exercise 26

((8j+4)(8j−4))

**Answer**(64j^2−16)

Exercise 27

((11k+4)(11k−4))

Exercise 28

((9c+5)(9c−5))

**Answer**(81c^2−25)

Exercise 29

((11−b)(11+b))

Exercise 30

((13−q)(13+q))

**Answer**(169−q^2)

Exercise 31

((5−3x)(5+3x))

Exercise 32

((4−6y)(4+6y))

**Answer**(16−36y^2)

Exercise 33

((9c−2d)(9c+2d))

Exercise 34

((7w+10x)(7w−10x))

**Answer**(49w^2−100x^2)

Exercise 35

((m+frac{2}{3}n)(m−frac{2}{3}n))

Exercise 36

((p+frac{4}{5}q)(p−frac{4}{5}q))

**Answer**(p^2−frac{16}{25}q^2)

Exercise 37

((ab−4)(ab+4))

Exercise 38

((xy−9)(xy+9))

**Answer**(x^{2}y^2−81)

Exercise 39

((uv−frac{3}{5})(uv+frac{3}{5}))

Exercise 40

((rs−frac{2}{7})(rs+frac{2}{7}))

**Answer**(r^{2}s^2−frac{4}{49})

Exercise 41

((2x^2−3y^4)(2x^2+3y^4))

Exercise 42

((6m^3−4n^5)(6m^3+4n^5))

**Answer**(36m^6−16n^{10})

Exercise 43

((12p^3−11q^2)(12p^3+11q^2))

Exercise 44

((15m^2−8n^4)(15m^2+8n^4))

**Answer**(225m^4−64n^8)

**Recognize and Use the Appropriate Special Product Pattern**

In the following exercises, find each product.

Exercise 45

a. ((p−3)(p+3))

b. ((t−9)^2)

c. ((m+n)^2)

d. ((2x+y)(x−2y))

Exercise 46

a. ((2r+12)^2)

b. ((3p+8)(3p−8))

c. ((7a+b)(a−7b))

d. ((k−6)^2)

**Answer**a. (4r^2+48r+144)

b. (9p^2−64)

c. (7a^2−48ab−7b^2)

d. (k^2−12k+36)

Exercise 47

a. ((a^5−7b)^2)

b. ((x^2+8y)(8x−y^2))

c. ((r^6+s^6)(r^6−s^6))

d. ((y^4+2z)^2)

Exercise 48

a. ((x^5+y^5)(x^5−y^5))

b. ((m^3−8n)^2)

c. ((9p+8q)^2)

d. ((r^2−s^3)(r^3+s^2))

**Answer**a. (x^{10}−y^{10})

b. (m^6−16m^{3}n+64n^2)

c. (81p^2+144pq+64q^2)

d. (r^5+r^{2}s^2−r^{3}s^3−s^5)

## Everyday Math

Exercise 49

**Mental math** You can use the product of conjugates pattern to multiply numbers without a calculator. Say you need to multiply 47 times 53. Think of 47 as 50−3 and 53 as 50+3

- Multiply (50−3)(50+3) by using the product of conjugates pattern, ((a−b)(a+b)=a^2−b^2)
- Multiply 47·53 without using a calculator.
- Which way is easier for you? Why?

Exercise 50

**Mental math** You can use the binomial squares pattern to multiply numbers without a calculator. Say you need to square 65. Think of 65 as 60+5.

- Multiply ((60+5)^2) by using the binomial squares pattern, ((a+b)^2=a^2+2ab+b^2)
- Square 65 without using a calculator.
- Which way is easier for you? Why?

**Answer**- 4,225
- 4,225
- Answers will vary.

## Writing Exercises

Exercise 51

How do you decide which pattern to use?

Exercise 52

Why does ((a+b)^2) result in a trinomial, but (a−b)(a+b) result in a binomial?

**Answer**Answers will vary.

Exercise 53

Marta did the following work on her homework paper:

[egin{array}{c} {(3−y)^2} {3^2−y^2} {9−y^2} onumber end{array}]

Explain what is wrong with Marta’s work.

Exercise 54

Use the order of operations to show that ((3+5)^2) is 64, and then use that numerical example to explain why ((a+b)^2 e a^2+b^2)

**Answer**Answers will vary.

## Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

## What is 2.7 percent of 6.4E+12 - step by step solution

If it's not what You are looking for type in the calculator fields your own values, and You will get the solution.

To get the solution, we are looking for, we need to point out what we know.

1. We assume, that the number 6.4E+12 is 100% - because it's the output value of the task.

2. We assume, that x is the value we are looking for.

3. If 6.4E+12 is 100%, so we can write it down as 6.4E+12=100%.

4. We know, that x is 2.7% of the output value, so we can write it down as x=2.7%.

5. Now we have two simple equations:

1) 6.4E+12=100%

2) x=2.7%

where left sides of both of them have the same units, and both right sides have the same units, so we can do something like that:

6.4E+12/x=100%/2.7%

6. Now we just have to solve the simple equation, and we will get the solution we are looking for.

7. Solution for what is 2.7% of 6.4E+12

6.4E+12/x=100/2.7

(6.4E+12/x)*x=(100/2.7)*x - *we multiply both sides of the equation by x*

6.4E+12=37.037037037*x - *we divide both sides of the equation by (37.037037037) to get x*

6.4E+12/37.037037037=x

172800000000=x

x=172800000000

## Components

**Interactive Textbook**

Two-calendar-year subscription

The online version of the student text delivers a host of interactive features to enhance the teaching and learning experience of the student text. It is accessed online through Cambridge GO using a unique 16-character code emailed to you upon purchase.

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The downloadable offline version of the student text enables students to take notes, highlight key passages, and bookmark pages. It is available from Cambridge GO using the unique 16-character code that is included with the purchase of the Print Textbook and/or the Interactive Textbook.

Revised, reviewed and updated, Cambridge Studies of Religion Stage 6 Fourth Edition provides new, contemporary and up-to-date content to engage Study of Religion students.

A complete, flexible and comprehensive print and digital resource package for the NSW Studies of Religion syllabus, this series has been designed to guide students to a greater understanding of the origins, beliefs, texts, ethics, practices and people of the world’s most wide-reaching religions.

## Description of the auditory oddball EEG (& MEG) Dataset

For the EEG-MEG workshop at NatMEG we recorded a dataset of a single subject to allow you to work through all the different steps involved in EEG-MEG analysis: from event related averaging to frequency analysis, source modeling and statistics.

### The oddball paradigm

The experiment that the subject performed is a slight adaptation of the classical oddball experiment. Using the oddball paradigm one can study the well-known EEG component called the mismatch-negativity (MMN). The classical auditory oddball experiment involves the presentation of a continuous series of identical tones at a relatively slow rate, say between one every two seconds to two every second. Every so often, say one out of ten, the tone is slightly different in pitch, duration or loudness. In our version, an oddball occurs after every 3 to 7 standard tones. The interval between each tone is jittered between 700 to 900 ms.

### Mismatch negativity

The auditory MMN then occurs as a fronto-central negative EEG potential (relative to the response to the standard tone), with sources in the primary and non-primary auditory cortex and a typical latency of 150-250 ms after the onset of the deviant tone. Not only is the MMN an indicator or auditory functioning, it has also been shown to be under influence of cognitive factors and indicative of cognitive and psychiatric impairments. For a recent and comprehensive overview please see Näätänen et al (2007).

### Cued motor preparation

For the purpose of analyzing oscillatory dynamics we wanted the oddball paradigm to include motor responses, specifically left and right button-presses with the left and right index finger, respectively. The subject was therefor cued to respond with their left or right index finger whenever, and as soon as possible, after a deviant tone occurred. The experiment therefore consistent of series of standard tones + deviant, preceded by a cue (left or right), and followed by a button-press.

### Beta suppression and rebound

It is well known that power in the beta band (15–30 Hz) decreases prior to movement onset but showed a marked sudden increase beginning approximately 300 to 400 ms after termination of EMG activity and lasting for over 500 ms. This post-movement beta rebound (PMBR) is localized to bilateral regions of the pre-central gyrus, but with greater lateralization to the contralateral hemisphere. Contrasting left versus right responses should therefor give us a nice lateralized beta rebound in the pre-central gyrus.

For a recent overview of sensorimotor rhythms, including the beta rebound, please see Cheyne (2013).

### Training with feedback and blink trials

Before the recording, the subject performed the experiment in a short training session to get acquainted with the task. Whenever the subject was too late in responding (>2 seconds), or pressed the wrong button, feedback was provided. In the actual experiment the subject was always on time and responded with the correct hand each time. Finally, after each response, a blink trial is presented in which subjects are asked to blink so that they can remain fixated on the fixation cross - without blinking - throughout the period in time in which we are interested in the brain signal.

### Stimuli

The standard tones were 400ms 1000Hz sine-waves, with a short 50ms ramping up- and down to avoid a clicking sound. The oddball tones were identical except for being 1200Hz. The auditory stimuli are presented with an external device that stores uncompressed audio files (.wav) that can then be presented with sub-millisecond precision (in effect instantaneous). Sound was presented through flat-panel sound showers. We measured the duration between the arrival of the triggers (see below) in the data and the arrival of the sound through the air to the ears of the subject. This takes 7.0 ms.

### Triggers

The MEG system records event-triggers in a separate channels, called STI101 and STI102. These channels are recorded simultaneously with the data channels, and at the same sampling rate. The onset (or offset) can therefore be precisely timed with respect to the data. The following trigger codes can be used for the analysis we will be doing during the workshop:

- Onset of standard stimulus: 1
- Onset of oddball stimulus: 2
- Button-press onset of left hand: 256
- Button-press onset of right hand: 4096

- Data was sampled at 1000Hz.
- 306 channels MEG of which 102 are magnetometers, and 204 are planar gradiometers.
- 128 electrode EEG. The reference was placed on the right mastoid, the ground on the left mastoid. The locations of the electrodes are placed according to the 5% system, which is an extension of the standard 10-20 system for high-density EEG caps. You can find details in Oostenveld and Praamstra, (2001). In addition, the locations of the EEG electrodes was measured in 3D using the Polhemus system and recorded in the data.
- Horizontal EOG(1) electrodes were placed just outside the left and right eye. Vertical EOG(2) were placed above and below the left eye.
- Electrocardiogram (ECG) was recorded as a bipolar recording from the collarbones.
- Electromyography of the lower arm flexors of the left(1) and right(2) arm were recorded.

## Mini-Lesson Samples

View and download samples of the *TEKS Companion Guide* mini-lessons! To open these in a new browser window or tab, please right-click and select Open Link in New Tab (Mac: Press COMMAND and click).

TEKS | Sample Document |
---|---|

Grade 3 | |

3.3H: Comparing Fractions | View Sample |

Grade 4 | |

4.7ABC: Measuring Angles | View Sample |

Grade 5 | |

5.3H: Adding and Subtracting Fractions | View Sample |

Grade 6 | |

6.4E: Ratios and Percents | View Sample |

Grade 7 | |

7.4D: Solving Problems with Percents | View Sample |

Grade 8 | |

8.4A: Slope as Unit Rate | View Sample |

8.8C: Solving One-Variable Equations | View Sample |

Algebra 1 | |

A.3C: Key Features of Linear Functions | View Sample |

A.10B: Multiplying Polynomials | View Sample |

Algebra 1 Errata Sheet | View Errata Sheet |

## Announcements & Tutorials

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**CCSS:** 6.EE.B.7, 6.RP.A.1, 6.SP.B.5.C, MP1, MP4, MP56.RP.A.3, 7.RP.A.3

**TEKS:** 6.6C, 6.9A, 6.5A, 6.12C, 6.4B, 6.4E, 6.4G, 6.5A

Lesson: Getting in the Game

**Objective:** Students will use mixed skills to answer questions about the new movie *Free Guy*, which is based on a video game.

Tell students they are about to read an article about the new action movie *Free Guy*. Ask students: What is a stunt performer? *(a trained professional who performs daring acts, often as a career)* Would you be interested in this type of career? Why or why not? *(Answers will vary.)* Choose a few student volunteers to share their responses with the class.

Have students read the article individually. Then tell them that the first two paragraphs have a different main idea than the remainder of the article. Have students reread the article in pairs. Ask the pairs to summarize the first two paragraphs in 1 to 2 sentences and the remainder of the article in 3 to 4 sentences. Then choose volunteers to share their summaries with the class and discuss how the sections are different.

To solve the problems in this article, students will need to use the following skills: 1) writing and solving algebraic equations to represent a word problem, 2) writing, simplifying, and scaling up ratios, 3) finding mean, 4) rounding, 5) converting ratios to percents. If students need review on any of these topics, provide a few quick examples or practice problems to complete as a class.

Distribute calculators to students or allow them to use an online digital calculator. Split the class into pairs and have them solve the “Your Turn” question with their partners. After all pairs have finished their questions, have students partner with a new classmate. These new pairs can compare their answers and correct work based on the feedback and discussions. Remind students to use positive social discourse when challenging others’ answers or ideas. Then choose volunteers to share how they solved each question, ensuring that different solution strategies are presented.

Since each “Your Turn” question requires a different skill, some students may need support in determining what the questions are asking. To scaffold the word problems for students, help them analyze the text by giving them strategies for marking up each word problem. Encourage students to circle words they don’t recognize, underline important details or words, and highlight the final question. You can also display the chart and example from the “Annotating Word Problems” skills sheet for students to use as a guide when reading and solving each problem.

Chris O’Hara also worked with the movie’s special effects teams to create thrilling action sequences with the actors. Read about visual effects designer Dan DeLeeuw in Scholastic *MATH*’s article “Super Special Effects.” Learn how DeLeeuw and his team used computer science to make the impossible possible for the Avengers: Endgame movie.

Have students choose their favorite movie and research facts about it like running time, budget, revenue earned, and more. Then have students create 5 mixed-skills questions about the movie and solve each question.

## Abstract

Although recent evidence has demonstrated the potent effect of physical exercise to increase the efficacy of cognitive training, the neural mechanisms underlying this causal relationship remain unclear. Here, we used multiscale entropy (MSE) of electroencephalography (EEG)—a measure of brain signal complexity—to address this issue. Young males were randomly assigned to either a 20-day dual *n*-back training following aerobic exercise or the same training regimen following a reading. A feature binding working memory task with concurrent EEG recording was used to test for transfer effects. Although results revealed weak-to-moderate evidence for exercise-induced facilitation on cognitive training, the combination of cognitive training with exercise resulted in greater transfer gains on conditions involving greater attentional demanding, together with greater increases in cognitive modulation on MSE, compared with the reading condition. Overall, our findings suggest that the addition of antecedent physical exercise to brain training regimen could enable wider, more robust improvements.

## Scoins.net

This page practises straightforward differentiation, chain rule and product rule.

I do not claim this is easy, and will provide good revision for C3 & C4 candidates

17 d ( sin 3 x) -1 is also cosec 3x

dx

Differentiation in context: finding the stationary values of a function

29 A box is built from a square sheet of side a. Square corners of side x are cut out and the result is folded and sealed.

a) Show that the box has volume V=(a-2x).(a-2x).x

b) By finding the stationary values of V, find the maximum volume of the box

c) Show that the maximum surface area for the box occurs when it is a cube.

30 A ball flies through the air according to the function

2U² y = xT – 5 x² (1+T²) where U and T are constants for each flight.

a) Find where dy/dx = 0, which is at the top of the flight, and check that this is half the range (the larger x-value that makes y=0).

b) Treat 2U² y = xT – 5 x² (1+T²) as a quadratic in x and write the discriminant (the b² – 4ac part). For equal roots, the discriminant must be zero. Find the value of T when U=12 and y = 3.

c) Treat 2U² y = xT – 5 x² (1+T²) as a quadratic in T and write the discriminant. For U=12 and y=3, find the x-value that gives equal roots.

d) If y=0 and you consider x as a (product) function in T, show that T = ±1 gives the maximum.

## A minimum of blind calculation

FP2 further complex numbers

**WORK COVERED IN LAST TERM OF 2015-6**

**Lesson** 1: working out w=f(z) transformations by changing the subject of the equation

Starter: draw |z|=1, |z|=4, arg(z)=π/2, arg(z)=π/4, arg(z)=0

Write the Cartesian equations

Write Cartesian equation for |z-2|=3

Classwork/ homework: Ex.3H Q.1, 3, 4, 6

**Lesson** 2: how to picture complex functions of a complex variable, ℂ↦ℂ

Picturing complex functions of a complex variable works well only for simple functions. Even w=z 2 turns out complicated, and the w-locus overlaps itself for many z-locuses.

w=z* looks quite simple, but isn’t so much so. You can’t define dw/dz, because δw/δz=1 if δz is a small increase in z along the real axis, but something quite different, δw/δz=−1 if δz is a small increase in z along the imaginary axis.

So a lot of our work with z-planes and w-planes (pretty much all we do in FP2) is with simpler equations, i.e. *Möbius transformations*.

**Lesson** 3: the rules for Möbius transformations (when pole not on z-locus)

**WORK TO BE COVERED IN FIRST TERM OF 2016-7**

**Lesson 4: To get back up to speed on Möbius transformations**

**Do now**: On the whiteboards, write down:

1. What is the name for transformations like w = (az+b) ⁄_{(cz+d)}?

2. That transformation can also be written as z ↦ (az+b) ⁄_{(cz+d)}. What does the symbol ↦ mean, in words?

3. Those transformations distort some shapes. But what simple shape C do they always transform into (another example of) the same shape?

4. That is true only when you interpret some things L which usually you’d interpret as different shapes as extreme examples of that simple shape. What things?

5. How do you tell when a transformation z ↦ (az+b) ⁄_{(cz+d)} will produce an extreme shape L?

6. What are the steps for finding the image shape by the geometrical methods?

7. What are the steps for finding the image shape by the algebraic method?

8. All of these transformations are combinations of three simpler sorts of transformations. What are those three?

Practice: Activity 10, page 10 of the booklet

**Lesson 5: what Möbius transformations do to regions how to interpret “diameter ends” of the z-locus when it is a line.**

If w = (z−i)/(z+i), then the z-locus |z|=2 (left plane) produces the w-locus shown in the right plane, above.

Choose a point (easy to calculate) inside, or a point outside, the circle |z|=2, and see what their images are in the w-plane. What region in the w-plane is the image of the *region* inside the circle |z|=2?

Activity 11, page 11 of workbook.

When the z-locus is a line, the “diameter ends” are:

1. the point on the z-line nearest to the pole

**Lesson 6: complex-number equations for perpendicular bisector in ℂ, half-lines in ℂ, arcs of circles in ℂ revisit algebraic method**

**Do now**: Fill in this flowchart for the geometric method of calculating Möbius transformations

(If the z-locus is a line, its “diameter-ends” are ∞ and the point on the z-line nearest to the pole, which is the foot of the perpendicular line from the pole to the z-line).

Why does it make a difference to the method whether the pole is on the z-locus?

If w=(z−i)/(z+i), calculate w for these z-values:

So in this sort of working, there is only one infinity. Lines go to Hell in many different directions, but they all end up at the same Hell. ∞ and −∞i are the same *for these purposes*.

ACTIVITY 12 from workbook: In your book, draw a diagram of the points z for which |z−1|=|z−3|.

Write down the general rule for the path of z if |z−a|=|z−b|.

ACTIVITY 13 from workbook: |z|=r is the circle with centre at the origin and radius r.

What is the path of z if |z−a−ib|=r?

If z=x+iy, use the fact that |x+iy−a−ib|=|(x−a)+i(y−b)| to write an equation in x and y for the path of z.

Work through Activity 13 in workbook

Do Activities 14 and 15 in workbook

**REVISIT ALGEBRAIC METHOD FOR MÖBIUS TRANSFORMATIONS** **EULER’S FORMULA**

w = (z−1)/(z+1). The locus of z is the circle |z|=1. Find the locus of w.

This is quick and easy by the algebraic method, so try the algebraic method. Then check by the geometric method.

Check the differences between geometric method and algebraic method. Review problems from the homework.

.

What are ln and exp? Brainstorm what you know.

log_{2}x is the power p to which you have to raise 2 to get x: 2 p =x

Graph of log_{2}x and graph of 2 x

The base for logs e (approx. 2.71828) is defined by: the rate of increase of e x is equal to e x itself, for all x

exp(x) is another way of writing e x . ln(x) is another way of writing log_{e}x.

log_{10}x used to be very important for practical calculations.

We’ve glossed over something: what does e.g. e √2 even mean? Is there any power of 2 that equals 3, for example? There’s another way of defining ln and exp which avoids this problem.

**Do now**: draw table of integrals of t n for n=−4 to n=4

Define ln x = integral from 1 to y of t −1 dt. Sketch graph of ln x

Deduce, graphically, ln(xy) = ln x + ln y (use example ln 6 = ln 2 + ln 3)

2×2=4, so ln(2×2)=ln(4). Divide ln(4) into two areas, the integral from 1 to 2, and the integral from 2 to 4.

But the integral from 2 to 4 is the same as the integral from 1 to 2, only squashed down into half-height and stretched out into double-length.

If ln(e)=1, then ln(x)=k defined by e k =x

log_{10}x=m defined by 10 m =x

Define exp(x) as inverse function of ln. Sketch graph.

Deduce exp(x) = [exp(1)] x . Define e=exp(1). Then exp(x)=e x .

Deduce (d/dx) exp(x) = exp(x) and (d/dx) exp(Ax) = A exp(x)

**LN, EXP, AND EULER’S FORMULA**

**Do now**: Write down neatly in your book, on a new page, all the rules you know about ln x and e x

[Reminder: exp(x) is another way of writing e x ]

ln x = integral from 1 to x of (1/t) dt

— **Euler’s formula**

If z=r cis θ what are z 2 and z 3 ? and z ½ ?

Deduce cis x = [cis 1] x for all x

Define [exp(t)] i = exp (it) by (d/dt) exp (it) = i exp(it) and exp(0)=1

Then the path of exp(it) is at right angles to exp(it) for all t, so if we sketch exp(it), it goes round the unit circle at a rate of 1 radian per unit of time

Therefore exp(i) = cis 1, exp(it) = cis t [Euler’s formula], e iπ +1=0

exp(kiθ)=(cos θ + i sin θ) k =cos kθ + i sin kθ

[*This is called De Moivre’s theorem*]

(d/dt)cos t = − sin t (d/dt) sin t = cos t

cos t = ½(e it +e −it ) sin t = ½(e it −e −it )

Classwork and homework: From your “Further complex numbers: exercises and answers” booklet, do:

1. “Exercise on exp and ln functions”, ninth page in the booklet (with ” on bottom right corner of photocopy)

2. “Exercise 3A”, eleventh page in the booklet (with ” on bottom right corner of photocopy)

3. More Möbius questions if I’ve asked for them in my comments on your last homework

**Using Euler’s identity and De Moivre**

We know: **exp (iθ) = cis θ = cos θ + i sin θ** [Euler’s identity>

exp(kiθ) = **(cos θ + i sin θ) k = cos kθ + i sin kθ** [De Moivre’s theorem]

**(d/dθ)cos θ = − sin θ (d/dθ) sin θ = cos θ**

**Do now:** 1. Write the “student response” on your homework

2. What is e 2iπ ? What is e 4iπ ? What is e 6iπ ?

3. For homework you calculated sin θ = (e iθ &minus e iθ )/2i

Calculate a similar formula for cos θ

4. You know exp(i[θ+&phi]) = exp iθ . exp i&phi

so: cos (θ+&phi) + i sin (θ+&phi) = (cos θ + i sin θ).(cos &phi + i sin &phi)

Multiply out the right hand side to get formulas for cos (θ+&phi) and sin (θ+&phi) in terms of cos θ, cos &phi, sin θ, and sin &phi

Proof that multiplying complex numbers adds arguments

Ex. 3D: getting expressions for cos nθ and cos n θ using De Moivre

cos 30 + i sin 3θ = (cos θ + i sin θ) 3 *(by De Moivre)*

&emsp&emsp= cos 3 θ + 3i cos 2 θ . sin θ &minus 3 cos θ . sin 2 θ &minus i sin 3 θ *(by binomial expansion using Pascal’s triangle)*

Equate imaginary parts: sin 3θ = 3 cos 2 θ . sin θ &minus sin 3 θ

&emsp&emsp= 3(1&minussin 2 θ)sin θ &minus sin 3 θ = 3 sin θ &minus 4 sin 3 θ.

cos 4 θ = [½(e iθ +e &minusiθ ] 4 *(from Euler’s identity)*

&emsp&emsp= (1/16) (e 4iθ +4e 2iθ +6+4e &minus2iθ +e &minus4iθ ) *(by binomial expansion using Pascal’s triangle)*

&emsp&emsp= (1/16) (2 cos 4θ + 8 cos 2θ + 6) = (1/8) (cos 4θ + 4 cos 2θ + 3)

Classwork and homework: Q.2, 3, 5, 6, 7 of Ex.3D (p.36) Q.1, Q.2, Q.3 a, b, c, Q.4, of Ex.3I (p.61)

**Finding roots of complex numbers**

**Do now:** find roots of z 3 =8 and draw them

How many cube roots does every number have in ℂ? How many fourth roots, etc.? What do they look like?

Finding cube (etc.) roots of any complex number, textbook p.38-39

**Edexcel proof of De Moivre’s theorem**

De Moivre’s theorem is: (cos θ + i sin θ) n = (cos nθ + i sin nθ)

Working backwards: Edexcel proof of De Moivre’s theorem, textbook p.28-29

## Need help please

Can someone kindly help me with the following question in bold. Am i on the right track with the formula 30=6e^kt? Please help me figure the right formula. Thanks so much.

**A new disease spreads to an area. At first, 4 people contract it. Seven days later, 2 more people contract it for a total of 6. Assume the spread is exponential. What is the exponential growth rate?**

** **

**1.) How many people will have contracted this disease in 30 days?**

#### Dr.Peterson

##### Elite Member

Start at the beginning, when there are 4 people, not 6. And don't try to answer question 1 yet before you can do that, you need to find what k is. For that, you'll be using the 6.

So, start with the equation N = 2e^(kt), which fits the initial amount. Then write an equation describing the 7-days-later case, and use that to solve for k.

Then you'll be ready to answer questions.

##### Elite Member

To OP, Dr Peterson meant to write that N = **4**e^(kt)

The reason it works is since at the start t=0 days and N, the number of people with the disease, is 4.

Note that N(0) = 4e^(k*0) = 4e^(0) = 4*1=4 which is correct, ie after 0 days there are 4 people with the disease. Now you need to find k. Use the fact that when t = 7, N=6. So solve 6 = 4e^(k*7) for k. Do not approximate k. Replace k into the formula and the find N(30).

#### Tara Marie

##### New member

##### Elite Member

Hi Tara Marie. That equation is false.

Were you trying to say k = 0.879009? That value is incorrect.

Please show your work solving this equation for k:

##### Elite Member

##### Elite Member

#### Tara Marie

##### New member

##### Elite Member

Hi. You just need more practice using logarithms and working with exponential growth/decay. (Don't feel bad, if you can't understand what Jomo did in post #6. Some of his results do not make sense.)

This is the equation we need to solve for k:

Note that k is currently in the exponent position. We need to get k out of the exponent position (our goal is write k=expression, not e^k=expression). We're going to use a basic property of logarithms, to get k out of the exponent position.

We first isolate the exponential term e^(7k), on the left-hand side. We do that by dividing each side of the equation above by 4:

Now that we have the exponential term by itself, we take the natural logarithm of each side:

Now we're ready to apply a basic property of logarithms. (You're expected to memorize this property and understand how to use it. One of the reasons you're struggling is because you haven't practiced enough using logarithms and properties to recall when and how to use them.)

Here is the basic property of logarithms that we need right now:

On the left-hand side, we see the natural logarithm of a power (that power is b^n). This property tells us -- when we take the logarithm of a power -- that we may move the exponent n out front as a factor, while changing the exponent on b to 1. In other words, it allows us to get n out of the exponent position.

In the equation we're trying to solve, we took the natural logarithm of a power:

We apply the property and write:

The equation we're trying to solve for k now looks like this:

You're expected to have also memorized that the expression ln(e) always represents 1 (and understand why). Therefore, we replace the expression ln(e) above with the number 1.

Finish solving for k. Then replace symbol k with your result below.

You can now find N, when t is 30, by setting t=30 above and evaluating the right-hand side.