9 topics, 26 short and super clear lessons

This Precalculus course includes 26 short and super clear lessons that take you through 9 topics and help you navigate the bumpy roads of Precalculus. The casual style makes you feel like you are discussing some simple issue, such as cooking scrambled eggs.

Table of contents: 

The course consists of 9 sections: Trigonometric functions and the unit circle, Exponents and logarithms, Graphing and transforming functions, Inverse functions, Probability and Combinatorics, Vectors, Matrices, Complex numbers, Polar coordinates

GRAPHING AND TRANSFORMING FUNCTIONS

  • Domain and range - Those lucky x values for which the function assigns something, are called the Domain, and the part of the y axis that is assigned to the x values is called the Image, or the Range.
  • Transformations - Internal and external transformations.
  • Shifting and reflecting -  Internal and external transformations.

EXPONENTS AND LOGARITHMS

TRIGONOMETRIC FUNCTIONS AND THE UNIT CIRCLE

  • The unit circle - The unit circle is a circle, centered at the origin and with a radius of 1.
  • Initial ray - In a unit circle, the ray on the x-axis is called the initial ray.
  • Terminal ray - In a unit circle, the ray going through terminal point P is called the terminal ray. 
  • Rotation angle - The rotation angle between these two rays can be positive, or negative. 
  • Cosine - The x coordinate of terminal point P is called cosine.
  • Sine - The y coordinate of terminal point P is called sine.
  • Trigonometric functions - Let’s talk a bit about trigonometric functions, and the definitions of sine and cosine.
  • Periodical functions - This means they repeat themselves at certain intervals.
  • Trigonometric equations - A reminder on how to solve trig equations.
  • Quadratic trigonometric equations - A very typical case is when a quadratic equation disguises itself as a trigonometric equation. 

INVERSE FUNCTIONS

COMBINATORICS

  • Permutation - The number of permutations on a set of n elements is given by n!
  • k-permutation of n - Number of permutations of k items chosen from n different items.
  • Combination -  How many ways can we choose k items out of n items?

 

EVENTS & PROBABILITIES

  • Events - Let's start with a very simple thing. We have a die, roll it once, and see what kind of events could occur. We may roll a 1.It's also possible, that we roll a 2. Then, it is also possible that before the die stops, a meteorite hits Earth and destroys the die, along with the entire human civilization. 
  • Independence - In probability theory, to say that two events are independent means that the occurrence of one does not affect the probability of the other.
  • Mutually exclusive events -  Two events are mutually exclusive if they cannot occur at the same time.
  • Conditional probability - A given B it answers the question of what chance does A have, if B definitely occurs.
  • Total probability theorem - This formula is a fundamental rule relating marginal probabilities to conditional probabilities.
  • Bayes-theorem - We use this theorem if we want to calculate the probability of an earlier event (Bk) in light of a later occurring event (A). 

VECTORS

  • Dot product - The dot product, or sometimes inner product, is an algebraic operation that takes two vectors and turns into a single number.
  • Cross product - The cross product, is an algebraic operation that takes two vectors and turns into anather vector.
  • Dyadic product - The dyadic product, is an algebraic operation that takes two vectors and turns into a matrix.
  • Angle between the two vectors - To calculate the angle between the two vectors, we write their dot product using two formulas. 
  • The equation of the line - We will develop the various forms for the equation of lines in plane and three dimensional space.
  • The equation of the plane - Here we will develop the equation of a plane.

 

MATRICES

  • Matrices - Matrices are really harmless creatures in mathematics. An nXk matrix is simply a rectangular array of numbers, arranged in n rows and k columns.
  • Matrix operations - Scalar multiplication, addition and multiplication.
  • Square and diagonal matrices - It is a square-shaped matrix with the same number of rows and columns.The diagonal matrix is a square matrix where all elements outside the main diagonal are zero.
  • Transpose - The transpose matrix is created by swapping the rows and the columns of the matrix

COMPLEX NUMBERS

  • Real numbers - At every point of the number line there is a real number.
  • Imaginary numbers - The imaginary numbers live on the imaginary axis perpendicular to the real number line.
  • Complex numbers - Numbers that consist of real and imaginary parts are called complex numbers.
  • Operation on complex numbers - Let’s see what kind of operations we can do on complex numbers.
  • Complex cojugate - Geometrically, conjugation is a reflection about the real axis.
  • Fundamental theorem of algebra - One significant benefit of complex numbers is that using complex numbers, all polynomials can be factored into linear factors.  
  • Absolute value of complex numbers - Let’s see what kind of operations we can do on complex numbers.
  • Complex plane - Complex numbers are located on the so called complex plane.
  • Polar form - The polar form makes it surprisingly simple to multiply and divide complex numbers.
  • Trigonometric form - The trigonometric form makes it surprisingly simple to multiply and divide complex numbers. 
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