# 2.E: Exercises for Chapter 2

## Calculational Exercises

1. Express the following complex numbers in the form (x + yi) for (x, y in mathbb{R}:)

(a) ((2 + 3i) + (4 + i))

(b) ((2 + 3i)^2 (4 + i))

(c) (frac{2+3i}{4+i})

(d) (frac{1}{i}+frac{3}{1+i})

(e) ((−i)^{−1})

(f) ((−1 + i sqrt{3})^3)

2. Compute the real and imaginary parts of the following expressions, where (z) is the
complex number (x + yi) and (x, y in mathbb{R}:)

(a) (frac{1}{z^2})

(b) (frac{1}{3z+2})

(c) (frac{z+1}{2z-5})

(d) (z^3)

3. Find (r > 0) and ( heta in [0, 2pi) ) such that ((1 − i)/ 2 = re^{i heta}.)

4. Solve the following equations for (z) a complex number:
(a) (z^5 − 2 = 0)
(b) (z^4 + i = 0)
(c) (z^6 + 8 = 0)
(d) (z^3 − 4i = 0)

5. Calculate the
(a) complex conjugate of the fraction ((3 + 8i)^4 /(1 + i)^10 .)
(b) complex conjugate of the fraction ((8 − 2i)^10 /(4 + 6i)^5 .)
(c) complex modulus of the fraction (i(2 + 3i)(5 − 2i)/(−2 − i).)
(d) complex modulus of the fraction ((2 − 3i)^2 /(8 + 6i)^2 .)

6. Compute the real and imaginary parts:
(a) (e^{2+i})
(b) (sin(1 + i))
(c) (e^{3−i})
(d) (cos(2 + 3i))

7. Compute the real and imaginary part of (e^{e^{z}}) for (z in mathbb{C}.)

## Proof-Writing Exercises

1. Let (a in mathbb{R}) and (z, w in mathbb{C}.) Prove that
(a) ( Re(az) = aRe(z)) and ( Im(az) = aIm(z).)
(b) ( Re(z + w) = Re(z) + Re(w)) and ( Im(z + w) = Im(z) + Im(w).)

2. Let (z in mathbb{C}.) Prove that ( Im(z) = 0) if and only if ( Re(z) = z.)

3. Let (z, w in mathbb{C}.) Prove the parallelogram law (|z − w|^2 + |z + w|^2 = 2(|z|^2 + |w|^2).)

4. Let (z, w in mathbb{C}) with (ar{z}w eq 1) such that either (|z| = 1) or (|w| = 1.) Prove that ( left| frac{z−w}{1 − ar{z}w} ight| =1. )

5. For an angle ( heta in [0, 2pi),) ﬁnd the linear map (f_ heta : mathbb{R}^2 ightarrow mathbb{R}^2), which describes the rotation by the angle ( heta) in the counterclockwise direction.

Hint: For a given angle ( heta), ﬁnd (a, b, c, d in mathbb{R}) such that (f_ heta (x_1 , x_2 ) = (ax_1 +bx_2 , cx_1 +dx_2 ).)

## 2.E: Chemistry of Life (Exercises)

• • Contributed by OpenStax
• Concepts of Biology at OpenStax CNX

### 2.1: The Building Blocks of Molecules

At its most fundamental level, life is made up of matter. Matter occupies space and has mass. All matter is composed of elements, substances that cannot be broken down or transformed chemically into other substances. Each element is made of atoms, each with a constant number of protons and unique properties. Each element is designated by its chemical symbol and possesses unique properties. These unique properties allow elements to combine and to bond with each other in specific ways.

## Exercises

1. Select two scientific articles that use qualitative research. Read each article and consider the following:

• Is the research question suitable for qualitative research? Why is it suitable?
• Do the authors identify the underlying paradigm and/or theories that shape the studies?
• Is there any evidence of reflexivity?
• How is the emic perspective demonstrated in the articles? Are the ‘voices’ of study participants included? How did the researchers try to achieve ‘Verstehen’?
• How is the research contextualized?
• How would you assess the quality of each article?

2. Reflect on your own scientific training: do you relate more to one paradigm than another? What may influence this and how does it affect the way you approach research?

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## The Computer

2.1 Computer systems are constantly developing, and manufacturers keep encouraging us to buy the latest, fastest systems, with the newest designs of keyboards, screens, more memory and faster processors, and so on. Give examples of circumstances in which having the latest technology is advantageous, and give examples when it can cause difficulties.

When raw computing power is at a premium, having the fastest computing power available is ideal - this is true for scientific applications, numerical modelling and analysis, and so on. Such increases in power can turn what were once batch jobs into interactive ones, allowing the user to explore and develop 'what if' scenarios that can lead to a better understanding of the underlying problems.

Developers of software usually benefit from having a faster design-code-compile-debug cycle, allowing them to produce their code more rapidly. However, this can cause problems too in that they become used to the capabilities of their machines and expect that all users will have similar resources. This can lead to large programs that perform ponderously on the more conventional machines that ordinary users have to work on.

2.2 Identify input and output devices that could benefit users with special needs.

## Exercises: Chapter 2, Section 5

1. Let be an open set and a continusously differentiable 1-1 function such that for all . Show that is an open set and is differentiable. Show also that is open for any open set .

For every , there is an with . By Theorem 2-11, there is an open set and an open subset such that and . Since clearly , this shows that is open. Furthermore is differentiable. It follows that is differentiable at . Since was arbitrary, it follows that is differentiable.

Let be a continuously differentiable function. Show that is not 1-1.

By replacing with a vector of variables, the proof of part (a) generalizes to the case where is a function defined on an open subset of where . to show that continuity of the derivative cannot be eliminated from the hypothesis of Theorem 2-11.

Clearly, is differentiable for . At , one has So satisfies the conditions of Theorem 2-11 at except that it is not continuously differentiable at 0 since for .

## The Computer

2.1 Computer systems are constantly developing, and manufacturers keep encouraging us to buy the latest, fastest systems, with the newest designs of keyboards, screens, more memory and faster processors, and so on. Give examples of circumstances in which having the latest technology is advantageous, and give examples when it can cause difficulties.

When raw computing power is at a premium, having the fastest computing power available is ideal - this is true for scientific applications, numerical modelling and analysis, and so on. Such increases in power can turn what were once batch jobs into interactive ones, allowing the user to explore and develop 'what if' scenarios that can lead to a better understanding of the underlying problems.

Developers of software usually benefit from having a faster design-code-compile-debug cycle, allowing them to produce their code more rapidly. However, this can cause problems too in that they become used to the capabilities of their machines and expect that all users will have similar resources. This can lead to large programs that perform ponderously on the more conventional machines that ordinary users have to work on.

2.2 Identify input and output devices that could benefit users with special needs.

## 3-6: More Guests

You just found a bigger dinner table, so now more space is available. Think of three more guests to invite to dinner.

• Start with your program from Exercise 3-4 or Exercise 3-5. Add a print statement to the end of your program informing people that you found a bigger dinner table.
• Use insert() to add one new guest to the beginning of your list.
• Use insert() to add one new guest to the middle of your list.
• Use append() to add one new guest to the end of your list. Print a new set of invitation messages, one for each person in your list.

## Fill in the Blank

A microscope that uses multiple lenses is called a _________ microscope.

### 2.3: Instruments of Microscopy

Which would be the best choice for viewing internal structures of a living protist such as a Paramecium?

1. a brightfield microscope with a stain
2. a brightfield microscope without a stain
3. a darkfield microscope
4. a transmission electron microscope

Which type of microscope is especially useful for viewing thick structures such as biofilms?

1. a transmission electron microscope
2. a scanning electron microscopes
3. a phase-contrast microscope
4. a confocal scanning laser microscope
5. an atomic force microscope

Which type of microscope would be the best choice for viewing very small surface structures of a cell?

1. a transmission electron microscope
2. a scanning electron microscope
3. a brightfield microscope
4. a darkfield microscope
5. a phase-contrast microscope

What type of microscope uses an annular stop?

1. a transmission electron microscope
2. a scanning electron microscope
3. a brightfield microscope
4. a darkfield microscope
5. a phase-contrast microscope

What type of microscope uses a cone of light so that light only hits the specimen indirectly, producing a darker image on a brighter background?

1. a transmission electron microscope
2. a scanning electron microscope
3. a brightfield microscope
4. a darkfield microscope
5. a phase-contrast microscope

## Lab Exercise with Quizzing

Read the materials, click on the lab exercise links and take the quizzes. Please note these will open in a new window.

Lab Exercise 1: Probe the Brain

PBS offers excursions into the working of the brain. In a virtual sense you will have the opportunity to “relive” the probing of the motor cortex that was done in the 1940s by Canadian brain surgeon Wilder Penfield.

Follow-Up Quiz

1. Which area of the brain appears to be probed in this demonstration?

2. When the probes are applied in this demonstration, the labels show that there is symmetrical lateralization of body part motor control.

3. Based on what you know about brain activity organization form reading the text, if the area is probed for the arm in the right hemisphere, this will control the ______.

Lab Exercise 2: Brain Atlas Exercise

The Scalable Brain Atlas (SBA) is a fully web-based display engine for brain atlases, imaging data and ontologies. Be sure to explore several of the brains available. As you explored the human brain, what did you find surprising? Do you know what each lobe is primarily dedicated to? Can you identify other structures?

Follow-Up Quiz

1. Which of the following brain atlases did this exercise contain?

2. Which mouse brain atlas would you use to look at the area of the brain important for memory formation and retrieval?

1. Alan mouse brain volumetric atlas
2. Waxholm space for the mouse
3. in vivo MRI template
4. DTI Atlas

3. The atlases show that human brains have completely different names for neural structures than other animals.

This Brain Atlas offers views of the normal brain, but also the brain affected by various disorders. Explore the human brain under different brain conditions by visiting the following labeled links:

Cerebrovascular Disease (select and click on any of the subcategories under this heading)

Neoplastic Disease (select and click on any of the subcategories under this heading)

Degenerative Disease (select and click on any of the subcategories under this heading)

Inflammatory or Infectious Disease (select and click on any of the subcategories under this heading)

As you explore these, can you identify any of the disorders? What is the typical brain deterioration in individual disorders?

Follow-Up Quiz

1. Which of the following types of neuroimaging techniques are NOT used in the images on this site?

2. When you click on “Can you name these brain structures?” under the “Normal Brain” section, what area is the bottommost arrow pointing to?

3. The website has a time-lapse clip that clearly illustrates a brain tumor.