Sets related to functions
1.17 Math problems
1.17 Math problems
We’ve now reached the first section of problems in this book. The purpose of these problems is to give you a way to comprehensively practice your math fundamentals. In the real world, you’ll rarely have to solve equations by hand, however, knowing how to solve math equations and manipulate math expressions will be very useful in later chapters of this book. At times, honing your math chops might seem like tough mental work, but at the end of each problem, you’ll gain a stronger foothold on all the subjects you’ve been learning about.
You’ll also experience a small achievement buzz after each problem you vanquish.
I have a special message to readers who are learning math just for fun: you can either try the problems in this section or skip them. Since you have no upcoming exam on this material, you could skip ahead to the next chapter without any immediate consequences. However (and it’s a big however), those readers who don’t take a crack at these problems will be missing a significant opportunity.
Sit down to do them later today, or another time when you’re properly caffeinated. If you take the initiative to make time for math, you’ll find yourself developing lasting comprehension and true math fluency. Without the practice of solving problems, however, you’re extremely likely to forget most of what you’ve learned within a month, simple as that. You’ll still remember the big ideas, but the details will be fuzzy and faded. Don’t break the pace now: with math, it’s very much use it or lose it!
By solving some of the problems in this section, you’ll remember a lot more stuff. Make sure you step away from the pixels while you’re solving problems. You don’t need fancy technology to do math; grab a pen and some paper from the printer and you’ll be fine. Do yourself a favour: put your phone in airplane-mode, close the lid of your laptop, and move away from desktop computers. Give yourself some time to think. Yes, I know you can look up the answer to any question in five seconds on the Internet, and you can use live.sympy.org to solve any math problem, but that is like outsourcing the thinking.
Descartes, Leibniz, and Riemann did most of their work with pen and paper and they did well. Spend some time with math the way the masters did.
P1.1 Solve for x in the equation x2− 9 = 7.
P1.2 Solve for x in the equation cos−1 Ax − φ = ωt.
P1.3 Solve for x in the equation 1x= 1a+1b.
P1.4 Use a calculator to find the values of the following expressions:
(1)√4
33 (2) 210 (3) 714 − 10 (4) 12ln(e22) P1.5 Compute the following expressions involving fractions:
(1) 1 2+1
4 (2) 4
7−23
5 (3) 134 + 13132 P1.6 Use the basic rules of algebra to simplify the following expressions:
(1) ab1
ab2cb−3 (2) abc
bca (3) 27a2
√9abba (4) a(b + c)− ca
b (5) a
c√3 b
b43 a2
(6)(x + a)(x + b)− x(a + b)
P1.7 Expand the brackets in the following expressions:
(1) (x + a)(x − b) (2) (2x + 3)(x − 5) (3) (5x − 2)(2x + 7) P1.8 Factor the following expressions as a product of linear terms:
(1) x2− 2x − 8 (2) 3x3− 27x (3) 6x2+ 11x− 21 P1.9 Complete the square in the following quadratic expressions to obtain expressions of the form A(x − h)2+ k.
(1) x2− 4x + 7 (2) 2x2+ 12x + 22 (3) 6x2+ 11x− 21 P1.10 A golf club and a golf ball cost $1.10 together. The golf club costs one dollar more than the ball. How much does the ball cost?
P1.11 An ancient artist drew scenes of hunting on the walls of a cave, including 43 figures of animals and people. There were 17 more figures of animals than people. How many figures of people did the artist draw and how many figures of animals?
P1.12 A father is 35 years old and his son is 5 years old. In how many years will the father’s age be four times the son’s age?
P1.13 A boy and a girl collected 120 nuts. The girl collected twice as many nuts as the boy. How many nuts did each collect?
P1.14 Alice is 5 years older than Bob. The sum of their ages is 25 years.
How old is Alice?
P1.15 A publisher needs to bind 4500 books. One print shop can bind these books in 30 days, another shop can do it in 45 days. How many days are necessary to bind all the books if both shops work in parallel?
Hint: Find the books-per-day rate of each shop.
P1.16 A plane leaves Vancouver travelling at 600 km/h toward Montreal.
One hour later, a second plane leaves Vancouver heading for Montreal at 900km/h. How long will it take for the second plane to overtake the first?
Hint: Distance travelled is equal to velocity multiplied by time: d = vt.
P1.17 There are 26 sheep and 10 goats on a ship. How old is the captain?
1.17 MATH PROBLEMS 81
P1.18 The golden ratio, denoted ϕ, is the positive solution to the quadratic equation x2− x − 1 = 0. Find the golden ratio.
P1.19 Solve for x in the equation 1 x+ 2
1− x= 4 x2. Hint: Multiply both sides of the equation by x2(1− x).
P1.20 Use substitution to solve for x in the following equations:
(1) x6− 4x3+ 4 = 0 (2) 1
2− sin x= sin x
P1.21 Find the range of values of the parameter m for which the equation 2x2− mx + m = 0 has no real solutions.
Hint: Use the quadratic formula.
P1.22 Use the properties of exponents and logarithms to simplify (1) exe−xez (2) xy−2z−3
x2y3z−4
−3
(3) (8x6)−23 (4) log4(√
2) (5) log10(0.001) (6)ln(x2−1)−ln(x−1) P1.23 When representing numbers on a computer, the number of digits of precision n in base b and the approximation error are related by the equation n = − logb(). A float64 has 53 bits of precision (digits base 2). What is the approximation error for a float64? How many digits of precision does a float64 have in decimal (base 10)?
P1.24 Find the values of x that satisfy the following inequalities:
(1) 2x − 5 > 3 (2) 5 ≤ 3x − 4 ≤ 14 (3) 2x2+ x≥ 1 P1.25 Two algorithms, P and Q, can be used to solve a certain problem.
The running time of Algorithm P as a function of the size of the problem n is described by the function P (n) = 0.002n2. The running time of Algo-rithm Q is described by Q(n) = 0.5n. For small problems, AlgoAlgo-rithm P runs faster. Starting from what n will Algorithm Q be faster?
P1.26 Consider a right-angle triangle in which the shorter sides are 8 cm and 6 cm. What is the length of the triangle’s longest side?
P1.27 A television screen measures 26 inches on the diagonal. The screen height is 13 inches. How wide is the screen?
P1.28 A ladder of length 3.33 m leans against a wall and its foot is 1.44 m from the wall. What is the height h where the ladder touches the wall?
h
1.44m 3.33m
P1.29 Kepler’s triangleConsider a right-angle triangle in which the hypotenuse has length ϕ = √5+12 (the golden ratio) and the adjacent side has length √ϕ. What is the length of the opposite side?
P1.30 Find the lengths x, y, and z in the figure below.
x
30
◦
A
B
C D
E q3
2
q3 2
z
y
45◦
P1.31 Given the angle and distance measurements labelled in Figure 1.24, calculate the distance d and the height of the mountain peak h.
h
1000m 800m d
20◦ 25◦
Figure 1.24: Measuring the height of a mountain using angles.
Hint: Use the definition of tan θ to obtain two equations in two unknowns.
P1.32 You’re observing a house from a blimp flying at an altitude of 2000 metres. From your point of view, the house appears at an angle 24◦below the horizontal. What is the horizontal distance x between the blimp and the house?
1.17 MATH PROBLEMS 83
24◦
2000 θ
x
P1.33 Find x. Express your answer in terms of a, b, c and θ.
a c b
x θ
Hint: Use Pythagoras’ theorem twice and the tan function.
P1.34 An equilateral triangle is inscribed in a circle of radius 1. Find the side length a and the area of the inscribed triangle A4.
1
a
Hint: Split the triangle into three sub-triangles.
P1.35 Use the power-reduction trigonometric identities (page 64) to ex-press sin2θ cos2θin terms of cos 4θ.
P1.36 A circle of radius 1 is inscribed inside a regular octagon (a polygon with eight sides of length b). Calculate the octagon’s perimeter and its area.
1 b b b
b
b b
b b
Hint: Split the octagon into eight isosceles triangles.
P1.37 Find the length of side c in the triangle:
b a = 10
A = 41◦ c B
C = 75◦
Hint: Use the sine rule.
P1.38 Consider the obtuse triangle shown in Figure 1.25.
(a) Express h in terms of a and θ.
(b) What is the area of this triangle?
(c) Express c in terms of the variables a, b, and θ.
Hint: You can use the cosine rule for part (c).
c h
a
b
A B
C
θ
Figure 1.25: A triangle with base b and height h.
P1.39 Find the measure of the angle B and deduce the measure of the angle C. Find the length of side c.
b = 30 a = 18
A = 25◦ c B
C
Hint: The sum of the internal angle measures of a triangle is 180◦. P1.40 An observer on the ground measures an angle of inclination of 30◦to an approaching airplane, and 10 seconds later measures an angle of inclination of 55◦. If the airplane is flying at a constant speed at an altitude of 2000 m in a straight line directly over the observer, find the speed of the airplane in kilometres per hour.
1.17 MATH PROBLEMS 85
( (
2000m
30◦ 55◦ 10seconds pass
P1.41 Satoshi likes warm saké. He places 1 litre of water in a sauce pan with diameter 17 cm. How much will the height of the water level rise when Satoshi immerses a saké bottle with diameter 7.5 cm?
Hint: You’ll need the volume conversion ratio 1 litre = 1000 cm3. P1.42 Find the length x of the diagonal of the quadrilateral below.
4
8 7
x 12 α1
α2
11
Hint: Use the law of cosines once to find α1 and α2, and again to find x.
P1.43 Find the area of the shaded region.
2cm 2cm 2 cm
Hint: Find the area of the outer circle, subtract the area of missing centre disk, then divide by two.
P1.44 In preparation for the shooting of a music video, you’re asked to suspend a wrecking ball hanging from a circular pulley. The pulley has a radius of 50 cm. The other lengths are indicated in the figure. What is the total length of the rope required?
40◦
2m 4m
Hint: The total length of rope consists of two straight parts and the curved section that wraps around the pulley.
P1.45 The length of a rectangle is c + 2 and its height is 5. What is the area of the rectangle?
P1.46 A box of facial tissues has dimensions 10.5 cm by 7 cm by 22.3 cm.
What is the volume of the box in litres?
Hint: 1 L = 1000 cm3.
P1.47 What is the measure of the angle θ in the figure below?
60◦ 100◦
θ
Hint: At the intersection of two lines, vertically opposite angles are equal.
P1.48 A large circle of radius R is surrounded by 12 smaller circles of radius r. Find the ratio Rr rounded to four decimals.
1.17 MATH PROBLEMS 87
Hint: Draw an isosceles triangle with one vertex at the centre of the R-circle and the other vertices at the centres of two adjacent r-circles.
P1.49 The area of a rectangular figure is 35 cm2. If one side is 5 cm, how long is the other side?
P1.50 A swimming pool has length ` = 20 m, width w = 10 m, and depth d = 1.5m. Calculate the volume of water in the swimming pool in litres?
Hint: 1 m3 = 1000 L.
P1.51 How many litres of water remain in a tank that is 12 m long, 6 m wide, and 3 m high, if 30% of its capacity is spent?
P1.52 A building has two water tanks, each with capacity 4000 L. One of them is 14 full and the other contains three times more water. How many litres of water does the building have in total?
P1.53 The rectangular lid of a box has length 40 cm and width 30 cm. A rectangular hole with area 500 cm2must be cut in the lid so that the hole’s sides are equal distances from the sides of the lid. What will the distance be between the sides of the hole and the sides of the lid?
Hint: You’ll need to define three variables to solve this problem.
P1.54 A man sells firewood. To make standard portions, he uses a stan-dard length of rope ` to surround a pack of logs. One day, a customer asks him for a double portion of firewood. What length of rope should he use to measure this order? Assume the packs of logs are circular in shape.
P1.55 How much pure water should be added to 10 litres of a solution that is 60% acid to make a solution that is 20% acid?
P1.56 A tablet screen has a resolution of 768 pixels by 1024 pixels, and the physical dimensions of the screen are 6 inches by 8 inches. One might conclude that the best size of a PDF document for such a screen would be 6 inches by 8 inches. At first I thought so too, but I forgot to account for the status bar, which is 20 pixels tall. The actual usable screen area is only 768 pixels by 1004 pixels. Assuming the width of the PDF is chosen to be 6 inches, what height should the PDF be so that it fits perfectly in the content area of the tablet screen?
P1.57 Find the sum of the natural numbers 1 through 100.
Hint: Imagine pairing the biggest number with the smallest number in the sum, the second biggest number with the second smallest number, etc.
P1.58 Solve for x and y simultaneously in the following system of equa-tions: −x − 2y = −2 and 3x + 3y = 0.
P1.59 Solve the following system of equations for the three unknowns:
1x + 2y + 3z = 14, 2x + 5y + 6z = 30,
−1x + 2y + 3z = 12.
P1.60 A hotel offers a 15% discount on rooms. Determine the original price of a room if the discounted room price is $95.20.
P1.61 A set of kitchen tools normally retails for $450, but today it is priced at the special offer of $360. Calculate the percentage of the discount.
P1.62 You take out a $5000 loan at a nominal annual percentage rate (nAPR) of 12% with monthly compounding. How much money will you owe after 10 years?
P1.63 Plot the graphs of f(x) = 100e−x/2and g(x) = 100(1 − e−x/2)by evaluating the functions at different values of x from 0 to 11.
P1.64 Starting from an initial quantity Qo of Exponentium at t = 0 s, the quantity Q of Exponentium as a function of time varies according to the expression Q(t) = Qoe−λt, where λ = 5.0 and t is measured in seconds.
Find the half-life of Exponentium, that is, the time it takes for the quantity of Exponentium to reduce to half the initial quantity Qo.
P1.65 A hot body cools so that every 24 min its temperature decreases by a factor of two. Deduce the time-constant and determine the time it will take the body to reach 1% of its original temperature.
Hint: The temperature function is T (t)=Toe−t/τand τ is the time constant.
P1.66 A capacitor of capacitance C = 4.0 × 10−6farads, charged to an initial potential of Vo = 20volts, is discharging through a resistance of R = 10 000 Ω (read Ohms). Find the potential V after 0.01 s and after 0.1s, knowing the decreasing potential follows the rule V (t) = Voe−RCt . P1.67 Let B be the set of people who are bankers and C be the set of crooks. Rewrite the math statement ∃b ∈ B | b /∈ C in plain English.
P1.68 Let M denote the set of people who run Monsanto, and H denote the people who ought to burn in hell for all eternity. Write the math statement ∀p ∈ M, p ∈ H in plain English.
P1.69 When starting a business, one sometimes needs to find investors.
Define M to be the set of investors with money, and C to be the set of investors with connections. Describe the following sets in words: (a) M \C, (b) C \ M, and the most desirable set (c) M ∩ C.
Chapter 2
Vectors
In this chapter we’ll learn how to manipulate multi-dimensional ob-jects called vectors. Vectors are the precise way to describe directions in space. We need vectors in order to describe physical quantities like the velocity of an object, its acceleration, and the net force acting on the object.
Vectors are built from ordinary numbers, which form the components of the vector. You can think of a vec-tor as a list of numbers, and vecvec-tor al-gebra as operations performed on the numbers in the list. Vectors can also be manipulated as geometrical objects, represented by arrows in space. The arrow that corresponds to the vector
~v = (vx, vy) starts at the origin (0, 0) and ends at the point (vx, vy). The
word vector comes from the Latin vehere, which means to carry. In-deed, the vector ~v takes the point (0, 0) and carries it to the point (vx, vy).
This chapter will introduce you to vectors, vector algebra, and vector operations, which are useful for solving problems in many areas of science. What you’ll learn here applies more broadly to problems in computer graphics, probability theory, machine learning, and other fields of science and mathematics. It’s all about vectors these days, so you better get to know them.
89
Figure 2.1: This figure illustrates the new concepts related to vectors. As you can see, there is quite a bit of new vocabulary to learn, but don’t be phased—all these terms are just fancy ways of talking about arrows.
2.1 Vectors
Vectors are extremely useful in all areas of life. In physics, for exam-ple, we use a vector to describe the velocity of an object. It is not sufficient to say that the speed of a tennis ball is 20[m/s]: we must also specify the direction in which the ball is moving. Both of the two velocities
~v1= (20, 0) and ~v2= (0, 20)
describe motion at the speed of 20[m/s]; but since one velocity points along the x-axis, and the other points along the y-axis, they are com-pletely different velocities. The velocity vector contains information about the object’s speed and direction. The direction makes a big difference. If it turns out that the tennis ball is coming your way, you need to get out of the way!
This section’s main idea is that vectors are not the same as numbers. A vector is a special kind of mathematical object that is made up of numbers. Before we begin any calculations with vectors, we need to think about the basic mathematical operations that we can perform on vectors. We will define vector addition ~u + ~v, vector subtraction ~u − ~v, vector scaling α~v, and other operations. We will also discuss two different notions of vector product, which have useful geometrical properties.
2.1 VECTORS 91
Definitions
The two-dimensional vector ~v ∈ R2is equivalent to a pair of numbers
~v ≡ (vx, vy). We call vx the x-component of ~v, and vy is the y-component of ~v.
Vector representations
We’ll use three equivalent ways to denote vectors:
• ~v = (vx, vy): component notation, where the vector is repre-sented as a pair of coordinates with respect to the x-axis and the y-axis.
• ~v = vxˆı+vyˆ: unit vector notation, where the vector is expressed in terms of the unit vectors ˆı = (1, 0) and ˆ = (0, 1)
• ~v = k~vk∠θ: length-and-direction notation, where the vector is expressed in terms of its length k~vk and the angle θ that the vector makes with the x-axis.
These three notations describe different aspects of vectors, and we will use them throughout the rest of the book. We’ll learn how to convert between them—both algebraically (with pen, paper, and calculator) and intuitively (by drawing arrows).
Vector operations
Consider two vectors, ~u = (ux, uy)and ~v = (vx, vy), and assume that α∈ R is an arbitrary constant. The following operations are defined for these vectors:
• Addition: ~u + ~v = (ux+ vx, uy+ vy)
• Subtraction: ~u − ~v = (ux− vx, uy− vy)
• Scaling: α~u = (αux, αuy)
• Dot product: ~u · ~v = uxvx+ uyvy
• Length: k~uk =√
~u· ~u =q
u2x+ u2y. We will also sometimes simply use the letter u to denote the length of ~u.
• Cross product: ~u × ~v = (uyvz− uzvy, uzvx− uxvz, uxvy− uyvx). The cross product is only defined for three-dimensional vectors like ~u = (ux, uy, uz)and ~v = (vx, vy, vz).
Pay careful attention to the dot product and the cross product. Al-though they’re called products, these operations behave much differ-ently than taking the product of two numbers. Also note, there is no notion of vector division.