## How to study for calculus?

Calculus is a branch of mathematics that deals with rates of change and continuity. It is a fundamental tool in many areas of science, engineering, and economics. To be successful in calculus, there are a few key things you can do to help you understand the material and excel in your studies.

Start with a solid understanding of the basics: Before diving into calculus, make sure you have a good grasp of basic algebra and pre-calculus concepts such as functions, trigonometry, and exponential and logarithmic functions. Without a solid foundation, it will be difficult to understand the more advanced concepts in calculus.

Practice, practice, practice: As with most subjects, practice is key to success in calculus. Make sure you are working through plenty of problems and exercises to help solidify your understanding of the material. Try to work on problems that are similar to the ones you will encounter on your exams.

Understand the concepts, not just the formulas: Calculus is more than just a set of formulas to memorize; it's a way of understanding how the world around us works. Try to understand the concepts behind the formulas rather than just memorizing them. This will help you apply the concepts to new situations and make connections to other areas of mathematics and science.

Use visualization and geometric intuition: Calculus is a visual subject and using visualization can help you to understand the concepts better. Try to visualize the concepts as much as possible and to understand how the graphs of functions change as you vary their parameters.

Stay organized and take good notes: keep your notes in an organized fashion and maintain a clear record of the main theorems, results, and formulas.

Stay positive and avoid procrastination: Stay positive, don't give up easily, and avoid procrastination. It's important to stay on top of the material and not to fall behind as it makes it much harder to catch up.

Remember the big picture: Remember the big picture and how the material you're learning fits into it. Knowing the big picture will give you a better understanding of the material, and it will also help you understand the importance of calculus in many areas of science and engineering.

Seek out additional help when needed: If you find yourself struggling with the material, don't hesitate to seek out additional help. Talk to your instructor or TA, attend office hours, or consider working with a tutor. If you need to work with a tutor we can help you. Please contact us.

These are some of the key strategies that can help you be successful in calculus. Remember that success in calculus requires dedication, hard work, and perseverance. Stay motivated, stay organized, and always strive to understand the concepts behind the formulas. With the right mindset and approach, you can excel in your calculus studies.

## MATH 135 Waterloo

MATH 135 (Algebra for Honours Mathematics) is one of the most difficult first-year courses at the University of Waterloo. This course covers the following topics:

• Logic, proofs, mathematical induction
• Divisibility, primes, GCD
• Extended Euclidean algorithm, linear Diophantine equations
• Linear congruences, Fermat’s little theorem, Chinese remainder theorem
• Public key cryptography, RSA, including fast exponentiation
• Complex numbers, the complex plane, polar representation
• De Moivre’s theorem, the Fundamental theorem of algebra
• Polynomials, factorization, roots, error-correcting codes
• Equations over finite fields, partial fractions

To be successful in MATH 135 you really need to know the theorems, definitions and methods very well. The proofs are the most important part of Math 135.

## Half angle formula

Trig identities are crucial part of trig functions and one of the most important identities in trigonometric functions is the half angle formula. In this post, we talked about different trig identities and formulas. First we recommend to visit the previous posts to review identities, including the double angle formula. Here we talk about half angle formulas. We introduce them, then we see how to use them with some examples and applications.

## What is the half angle formula?

There are 3 main half angle formulas for sin, cos and tan:

• $\sin^{2} u=\frac{1-\cos (2 u)}{2}$
• $\cos^{2} u=\frac{1+\cos (2 u)}{2}$
• $\tan^{2} u=\frac{1-\cos (2 u)}{1+\cos (2 u)}$

Next we see some good and useful examples of half angle formula. First, try to solve them by yourself and then check the solutions carefully.

## Examples of half angle formula:

Example 1: Find $\sin (\dfrac{\pi}{12})$.

Solution: To solve this problem, we use the half angle formula for the function $\sin$.
$$\sin^{2}(\dfrac{\pi}{12})=\frac{1-\cos (2 \dfrac{\pi}{12})}{2}=\frac{1-\cos (\dfrac{\pi}{6})}{2}$$
We know that $\cos (\dfrac{\pi}{6})=\dfrac{\sqrt{3}}{2}$. So $$\sin^{2}(\dfrac{\pi}{12})=\dfrac{1-\dfrac{\sqrt{3}}{2}}{2}=\dfrac{2-\sqrt{3}}{4}$$
Since $\pi/12$ is in the first quadrant, then $\sin (\dfrac{\pi}{12})$ is positive, so
$$\sin^{2}(\dfrac{\pi}{12})=\sqrt{ \dfrac{2-\sqrt{3}}{4} }$$

Example 2: Prove the identity $2\cos^2 x \sec(2x)=\sec(2x) +1$.

Proof: We start from the left side and then we prove the right side. For the $\cos^2 x$ we use half angle formula, so $$2\cos^2 x \sec(2x)=(1+\cos (2 x))\sec(2x)=\sec(2x)+\sec(2x)\cos (2 x)=\sec(2x) +1$$

We add more examples of the half angle formula for you to practice. To learn about this topic it is really important to try solve these problems by yourself.

Example 3: If $\sin x=\dfrac{2}{3}$ and $x$ is in the first quadrant, find $\sin(\frac{x}{2})$.

Example 4: Find the exact value of $\tan(\dfrac{\pi}{8})$.

Example 5: Find a half angle formula for $\cot x$.

If you need help to understand the trig identities, double angle formulas and half angle formulas, contact us.

## Double Angle Formula

In trigonometry, it is really important to know and able to use identities. One of the main and crucial categories of identities is Double Angle Formula. In this post, we talked about different trig identities and formulas. Double Angle Formulas are trigonometric identities that simplify a trigonometric function of $2x$ as of trigonometric functions of $x$. First we recommend to visit the previous post to review identities. Here in this post, first we recall the Double Angle Formula and then we see some examples.

## Double Angle Formulas:

Here we have main Double Angle Formulas.

• $\sin (2 u) =2 \sin u \cos u$.
• $\cos (2 u) =\cos ^{2} u-\sin ^{2} u$.
• $\cos (2 u) =2 \cos ^{2} u-1$.
• $\cos (2 u) =1-2 \sin ^{2} u$.
• $\tan (2 u) =\dfrac{2 \tan u}{1-\tan ^{2} u}$.

## Examples of Double Angle Formulas

Before checking the solutions, try to solve it by yourself first.

Example 1: If $\sin x=\dfrac{3}{5}$ and $x$ is in the first quadrant, find $\tan (2 x)$.

Solution: Since $x$ is in the first quadrant, both $\sin x$ and $\cos x$ are positive. Now we use the Pythagorean identity $\sin^{2} x+\cos^{2} x=1$ to find $\cos x$. From this identity we have, $\dfrac{9}{25} +\cos^{2} x=1$, so $\cos x=\dfrac{4}{5}$. Then $\tan x=\dfrac{\sin x}{\cos x}=\dfrac{\dfrac{3}{5}}{\dfrac{4}{5}}=\dfrac{3}{4}$. Now by the last Double Angle Formula, we have $\tan (2 u)=\dfrac{2 \dfrac{3}{4}}{1-\dfrac{9}{16} }=\dfrac{\dfrac{3}{2}}{\dfrac{7}{16}}=\dfrac{48}{14}=\dfrac{24}{7}$.

Example 2: Prove the following identity $\dfrac{1-\cos 2x}{1+\cos 2x}=\tan^2 x$.

Solution: To solve this problem we use the double angle formula for $\cos 2x$. We simplify the left side to get the right side. By the double angle formula we have $\dfrac{1-\cos 2x}{1+\cos 2x}=\dfrac{1-(1-2 \sin ^{2} x)}{1+2 \cos ^{2} x-1}=\dfrac{\sin ^{2} x}{\cos ^{2} x}=\tan^2 x$.

Here we add more examples without solutions for you to practice.

Example 3: Prove the identity: $\dfrac{\cos 2x}{\cos x- \sin x}=\cos x+\sin x$.

The following example is really common in exams.

Example 4: Solve $\cos(2x)=\cos(x)$ when $0\leq x <2\pi$.

The following example is a bit challenging, but give it a try.

Example 5: Prove the identity: $\cos^4{x}=\dfrac{\cos(4x)}{8}+\dfrac{\cos(2x)}{2}+\dfrac{3}{8}$.

If you need help to understand the trig identities and Double Angle Formulas, contact us.

## Prime numbers

What is a prime number?

In mathematics, we say a natural number $p$ is a prime number if p has exactly 2 factors. For example, 11 is a prime number, because the only factors are 1 and 11. It is easy to see that the only prime numbers less than 10 are 2, 3, 5 and 7. The smallest prime number is 2. The only even prime number is also 2. Another equivalent definition of prime number is a natural number greater than 1 that is not a product of two natural numbers greater than 1. For example, 10 is not a prime number, because $10=2\times 5$.

What is a composite number?

If a number greater than 1 is not prime, we call it a composite number. The smallest composite number is 4.

Is 1 a prime number?

Note than 1 is not a prime number and it is not a composite number.

List of prime numbers 1 to 100:

In total we have 25 prime numbers less than 100. Here is the list:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

There are a total of 168 prime numbers less than 1000.

How many prime numbers are there?

There are infinitely many prime numbers. There are many proofs for this statement. Let’s prove it here with the most famous method:

On the contrary, let’s assume we have finitely many prime numbers, let’s call them $p_1,…,p_k$. Let $n=p_1\times p_2\times … p_k +1$, it is not difficult to see than $n$ is greater than all $p_{1},…,p_{k}$, so $n$ is not one of them, and then, $n$ is a composite number. So $n$ has a prime factor, let’s say $p_i$ is a factor of $n$. It means than $\dfrac{n}{p_i}$ is an integer, but $\dfrac{n}{p_i}=\dfrac{ p_1\times p_2\times … p_k }{p_i} +\dfrac{1}{p_i}=p_1\times p_2\times p_{i-1}\times p_{i+1}\times … p_k +\dfrac{1}{p_i}$, which is not an integers, that contradicts our assumption that we have finitely many prime numbers. So, it proves than There are infinitely many prime numbers.

How to check if a number is prime?

To see if a number $n>2$ is a prime number first we check if $n$ is divisible by 2, if it is divisible by 2, then is it not a prime number, then we check divisibility by 3, and then by 5,… . We don’t need to continue to $n$. It is enough to check until $\sqrt{n}$.

For example, to check the number 53, we check this test until $\sqrt{59}$. So we only need to check this test for number, 2, 3, 5 and 7.

What is prime factorization?

Prime Factorization” is factoring our number $n$ into prime factors (prime numbers multiply together to make $n$. For example, the prime factorization of 100 is $100=2^{2}\times 5^{2}$. Prime factors of a number $n$ are prime numbers that are multiplied together to get $n$. As an example, 2 and 5 are prime factors are 10. Note that Prime Factorization is unique.

## Trig Identities

Trigonometry is one of the main topics in math, and many students have issues with trigonometric topics. One of the main topics is trig identities. These trigonometric identities help us with many other topics, including trigonometric equations, derivative with trigonometric functions, integral of trigonometric functions and proving trig identities.

Here we have the list of main trig identities:

Reciprocal identities:

• $\sin u=\frac{1}{\csc u}$
• $\cos u=\frac{1}{\sec u}$
• $\tan u=\frac{1}{\cot u}$
• $\cot u=\frac{1}{\tan u}$
• $\csc u=\frac{1}{\sin u}$
• $\sec u=\frac{1}{\cos u}$

Pythagorean Identities:

• $\sin^{2} u+\cos^{2} u=1$
• $1+\tan^{2} u=\sec^{2} u$
• $1+\cot^{2} u=\csc^{2} u$

Quotient Identities:

• $\tan u=\frac{\sin u}{\cos u}$
• $\cot u=\frac{\cos u}{\sin u}$

Co-Function Identities:

• $\sin \left(\frac{\pi}{2}-u\right)=\cos u$
• $\cos \left(\frac{\pi}{2}-u\right)=\sin u$
• $\tan \left(\frac{\pi}{2}-u\right)=\cot u$
• $\cot \left(\frac{\pi}{2}-u\right)=\tan u$
• $\csc \left(\frac{\pi}{2}-u\right)=\sec u$
• $\sec \left(\frac{\pi}{2}-u\right)=\csc u$

Parity Identities: We know that $\sin x$ is an odd function and $\cos x$ is an even function.

• $\sin (-u)=-\sin u$
• $\cos (-u)=\cos u$
• $\tan (-u)=-\tan u$
• $\cot (-u)=-\cot u$
• $\csc (-u)=-\csc u$
• $\sec (-u)=\sec u$

Sum and Difference Formulas:

• $\sin (u \pm v)=\sin u \cos v \pm \cos u \sin v$
• $\cos (u \pm v)=\cos u \cos v \mp \sin u \sin v$
• $\tan (u \pm v)=\frac{\tan u \pm \tan v}{1 \mp \tan u \tan v}$

Double Angle Formulas:

• $\sin (2 u) =2 \sin u \cos u$
• $\cos (2 u) =\cos ^{2} u-\sin ^{2} u$
• $\cos (2 u) =2 \cos ^{2} u-1$
• $\cos (2 u) =1-2 \sin ^{2} u$
• $\tan (2 u) =\frac{2 \tan u}{1-\tan ^{2} u}$

Half Angle Formulas:

• $\sin^{2} u=\frac{1-\cos (2 u)}{2}$
• $\cos^{2} u=\frac{1+\cos (2 u)}{2}$
• $\tan^{2} u=\frac{1-\cos (2 u)}{1+\cos (2 u)}$

Sum to product formulas:

• $\sin u+\sin v=2 \sin \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right)$
• $\sin u-\sin v=2 \cos \left(\frac{u+v}{2}\right) \sin \left(\frac{u-v}{2}\right)$
• $\cos u+\cos v=2 \cos \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right)$
• $\cos u-\cos v=-2 \sin \left(\frac{u+v}{2}\right) \sin \left(\frac{u-v}{2}\right)$

Product to Sum Formulas:

• $\sin u \sin v=\frac{1}{2}[\cos (u-v)-\cos (u+v)]$
• $\cos u \cos v=\frac{1}{2}[\cos (u-v)+\cos (u+v)]$
• $\sin u \cos v=\frac{1}{2}[\sin (u+v)+\sin (u-v)]$
• $\cos u \sin v=\frac{1}{2}[\sin (u+v)-\sin (u-v)]$

The quadratic formula is a formula that helps us to find the solutions of an equation of degree 2. First of all, we need to turn our equation of degree 2 to standard form $ax^2+bx+c=0$.

The quadratic formula is: $\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$.

How to use the quadratic formula?

Let’s see some examples of the quadratic formula.

Example 1: Solve the quadratic equation $x^2+5x+6=0$.
Solution: In this quadratic equation a=1, b=5 and c=6. We put these numbers into the quadratic formula and we get:
$\dfrac{-5\pm \sqrt{25-24}}{2}=\dfrac{-5 \pm 1}{2}=-2,-3$.

Example 2: Solve the quadratic equation $x^2+5x+7=0$.
Solution: In this quadratic equation a=1, b=5 and c=7. We put these numbers into the quadratic formula and we get:
$\dfrac{-5\pm \sqrt{25-28}}{2}=$. As you see, under the square root we have a negative number, so there we have no solution for this quadratic equation.

When do we use the quadratic formula?

The quadratic formula is the most general formula to find solutions to quadratic equations. You can use it when you have a quadratic equation.

## How to study for math waterloo competitions?

There are many math competitions in waterloo (Pascal, Cayley, Euclid, Fryer, Galois, Hypatia, Gauss, and Fermat). In each of them, you have math questions that you need to answer. Most of the problems are not difficult but still, you need a lot of preparation to be successful. With the math you learn in school, you are not able to solve many problems and get a good result out of these exams. Here are core topics for all of these competitions:

1- Number theory: Gcd, Lcm, Euclid division algorithm, Modulo arithmetic, divisibility, Fermat’s little theorem, Euler theorem, and diophantine equations
2- Combinatorics: Permutation and combination (with repetition problem section), circle computation, binomial theorem, sets, graph theory, recursive counting.
3- Algebra: Polynomials, Quadratic equations, root coefficient relationship, maxima and minima, Solving equations, factoring.
4- Geometry: Solutions of the triangle, Ptolemy theorem, Ceva’s theorem, Menelaus theorem, and other theorems.
5- Problem-solving techniques: Induction, proof by contradiction.

For all of these competitions (Pascal, Cayley, Euclid, Fryer, Galois, Hypatia, Gauss, and Fermat), you need these topics.

## Topics in introductory Calculus

Calculus is one of the most important topics in mathematics and usually many students are not good at it. Here there are topics that are so important in calculus that most of the students need to be great at it:

• Functions (Definition, domain, range, one-to-one, onto and etc)
• Limits (Definition, one-sided limit, properties, squeeze theorem, continuity and etc)
• Derivatives (Definitions, slope, product and quotient rule, chain rule and etc)
• Applications of derivatives (Maxima and minima, first derivative test and etc)
• Analyzing functions
• Integrals

We offer tutoring for calculus.

Mathematics has an important role in our technological and scientific age. Taking enough math in high school is a gate of getting all kinds of jobs. Math tournaments in Canada are good opportunities for students who are seeking to challenge themselves and advance in math.

In this article, we will introduce the different kinds of math competitions in Canada, explain the benefits of taking part in these scientific tournaments, and finally, tell you about the importance of having a tutor to succeed in math contests.

## List of Canadian mathematics competitions

This math contest is an official contest that runs each April. It is a “full solution” exam on paper. winners get the right to represent Canada at the International Mathematical Olympiad (IMO). To participate in the CMO, students should do well on one of the following contests:

1_Canadian Open Mathematics Challenge (COMC): It runs in November and all students can take part in this competition.

2_CMO Qualifying Repêchage (CMOQR): It runs in February and only the top-scoring students in COMC, are invited.

### The Centre for Education in Mathematics and Computing (CEMC)

It is the most recognized Canadian organization. Some exciting Waterloo math competitions and coding contests are held in the Faculty of Mathematics at the University of Waterloo. The center’s aim is to boost interest, confidence, and ability in math and computer science among students in Canada and internationally.

Math Kangaroo is an international math tournament that runs in March. Any student in grades 1 through 12 is eligible to take part in this competition. Students who got a top performance per grade are awarded.

### William Lowell Putnam Mathematical Competition (Putnam Competition)

It is the most reputable university-level math competition in the world that runs on the first Saturday in December by the Mathematical Association of America. Any undergraduate college students who enroll at institutions of higher learning in the United States and Canada are eligible to participate in this tournament.

## Benefits of Math Competitions

Math contests in Canada such as CEMC and the Canadian Mathematical Olympiad are probably the extracurricular math programs that have many participants. The main aim of these math competitions is clear. They make students interested in mathematics and encourage them to value intellectual pursuits. Children like games very much, and many will turn just about any activity into a competition to get good at. Therefore, math contests motivate them to get good at mathematics. By and by, students put aside the games. Till that time, an interest in the underlying activity may develop.

Besides encouraging in mathematics, contests can make students ready for competition. Actually, life is a kind of competition and any sort of competition educates students to deal with success and failure. Contests train them that if they want to have effective performance, they should practice.

Moreover, almost every worthwhile and interesting progress in life is accompanied by some elements of pressure; competition teaches students how to manage them.

Despite these advantages of math contests, they are not an unmitigated good. Such competitions run the risk of encouraging students to overvalue the skills that aren’t almost as worthwhile as the one value a contest should help them develop — the ability to think about and solve complex problems. Moreover, math competitions may cause students to extend beyond their abilities. It is true that students should certainly  be challenged with problems they can’t do from time to time, but if it happens invariably, the experience goes from challenging to contemptuous and disappointing.

These possible risks are usually neutralized by corporations that is the greatest value of math competition. These contests gather students together who have the same interests and abilities. They allow students to set up their own community in which they will find friendship, inspiration, and encouragement to a far greater degree than most of these students can find in the typical classroom. Whereas a student may be one of only three or four in his school that follows mathematics the way others play football, he won’t find himself so lonely at a math competition, where he’ll find lots of similar people.

In short, math contests are a great social and intellectual opportunity for students, but exposing students to contests must be done wisely, otherwise, they become counterproductive to the purpose of encouraging a lifelong interest in math and another intellectual activates.

## How to get higher scores in math contests?

Hiring a private tutor is one of the best ways to be prepared for math contests. A qualified math tutor helps you to master mathematics competitions and enrich you academically.

He/She also teaches efficient strategies required for contest-based problem solving by covering all concepts or topics that frequently occur on the path exams.

Moreover, an experienced tutor reviews questions drawn from past years’ exams as well as different selected resources. All math skills developed by one-to-one tutoring will be beneficial not only for math contests but also for college math classes as well as comprehensive exams like PSAT or SAT.

If you have any other questions about how we may be able to help you, feel free to text us at 647-249-2491 or Book here.