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PREREQUISITES OF CALCULUS - Coggle Diagram
PREREQUISITES OF CALCULUS
FUNCTIONS
Types Of Functions
Functions Learned During Class
Piece-Wise Function:
Is a function composed of multiple functions, that are all restricted to their specific domains.
Polynomial Function:
A polynomial function is a function that can only have positive integers as its power.
Line Function:
A line functions is most commonly found in the form of y=mx+b. Within this function there is a independent variable "x", and dependant variable "y".
Point Slope Equation:
If a line has a slope (m) and passes through the point of P(x1,y1), then the equation of the line could be represented as (y-y1) = m(x-x1)
Slope:
The slope of an equation can be seen as Rise/Run of a graph/function. The slope can be found by completing the formula y2-y1/x2-x1. These points can be found by using the x and y coordinates of two points along the graph/function.
Perpendicular
: The slopes of two equations are equal to -1 when multiplied together. In other words one slope can be described as the negative reciprocal of the other.
Parallel
: When the slopes or "m" values of of two equations are the same (m1 = m2)
Rational Function:
A rational functions can be seen as a fraction, that has a polynomial as its numerator and denominator.
Exponential Function:
An exponential function is a function that has the variable "x" as it's exponent. (If the base of the function is between one and zero, the function is decay. If the base is greater than one, then it is growth)
Compound Interest Formulas
A=Ao(1+i/n)^(n*t)
A=Ao(e)^(it)
The value "e" represents the approximate number of 2.71828183. It can also be seen as the base value for the term ln (natural log).
Application of the growth and decay formula
A=Ao(B)^(t/p)
Radical Function:
A radical function is a function that has an "n" value. It includes an n'th value of "x", in which n is a positive integer.
Logarithmic Function:
A logarithmic function can be described as an inverse of an exponential function. Logarithms are also forced to abide by the manipulation rules listed bellow.
Asymptote:
Each graph has an asymptotes, which is a line that a graph approaches infinity but will never reach (infinitely smaller values).
Logarithms are often used throughout calculus to isolate exponents and find variables that would otherwise be tedious and difficult to prove. Logarithms make it simpler to isolate exponents and must also follow by the log rules listed bellow.
One to One Function:
There is only one "x" value for every "y" value. In other words the equation passes horizontal line test.
Table of Values method:
If when listing out the points on a table of values, there are any repeats of "x" values for the same "y" value, then it is not a one to one function.
Horizontal line test:
If any horizontal line is drawn on the graph and it ever intercepts the graph at 2 points, it is not a one to one function.
Surjective
: The value that discuses how that for ever "y" element, there exists a corresponding "x" element
Even and Odd Functions
Odd Functions:
When a function is odd it means that its graph is symmetric of the origin.
Determining an Odd Function:
By using the formula of -f(x) = f(-x). If the values change (negative or positive) then the function is odd.
If the variables within a function
all
have a negative exponent, then the function is indeed odd.
Example
:
f(x) = x^5
Neither even, or odd functions:
There is no consistent pattern to describe a function that is not even or odd.
Determining a nor Even or Odd Function:
If a the variables within a function have both negative and positive exponents, then the function is not even or odd.
Example
:
f(x)= x^4 + x^3 + x^2 + x
Even Functions:
When a function is even it means that its graph will appear as a reflection in the y-axis. In other words it will be the same on both sides of the y-axis
Determining an Even Function:
By using the formula f(x) = f(-x). If the values remain the same (negative or positive) then it is an even function.
If the variables within a function are
all
negative then the function is even.
Example
:
f(x) = x^4
Determining if Different Graphs are Functions
Table of values method (listing points)
If when listing out every point on a graph, if there are multiple "x" values for the same "y" value, then the graph cannot be considered a function
Vertical Line Test
Draw a completely vertical line on the graph at any "x" value. If the graph ever shows two "y" values or intercepts the graph twice on the same line, then the graph is not a function.
Interval Notation
Closed Intervals:
When using normal brackets it is implied that the end values are note included in the graph/equation (they do not exist). Example: (x,y)
Open Intervals:
When using square brackets it is implied that the end values are included in the graph/equation (they do exist). Example: [x,y]
Hole:
A point of discontinuity is a value on a graph that does not actually exist. It is visually represented by a circle/hole between the lines which signifies the point at which the function doesn't exist.
Inverse Function:
An inverse function is often represented by using f'(x) instead of the standard f(x) (signifying the inverse of the function). In an inverse function, all the original "x" and "y" values of a function are swapped. Only one to one functions can be inverted correctly. The most clear example of an inverse function would be logarithmic functions (inversed), and exponential functions (original).
Graphing Basic Functions
Graphing Basic Functions.
Absolute function:
y = |x|
Trigonometric functions:
y = Cos x
y = Tan x
y = Sin x
Exponential function:
y = a^x
Square root function:
y = √x
Logarithmic function:
y = logx
Polynomial function:
y = mx (could be expanded to have more variables)
Hyperbola function:
y = 1/x
Basic Graphing Form:
y = af(b(x+c))-d
b
: Horizontal expansion or compression, Reflection in y-axis
c:
Horizontal translation
a:
Vertical expansion or compression, Reflection in x-axis
d
: Vertical translation
Domain, Range
: The domain and range can be seen as values that restrict the graph/function in the x and y axis respectively.
TRIGONOMETRY
Trigonometric Ratios
Basic Trigonometric Functions
Cosine: cos(θ) = Adjacent/Hypotenuse
Tangent: tan(θ) = Opposite/Adjacent
Sine: sin(θ) = Opposite/Hypotenuse
Reciprocal Functions
Secant: 1/cos(θ) = Hypotenuse/Adjacent
Cotangent: 1/tan(θ) = Adjacent/Opposite
Cosecant: 1/sin(θ) = Hypotenuse/Opposite
Pythagorean Theorem
The pythagorean theorem is a mathematical equation used to solve for missing sides on a triangle that contains one inner angle of 90º. By using the formula a^2 + b^2 = c^2 we can find a missing side (assuming the length of the other two sides is given).
Determining Arc Length
S = θ * r
- S: Arc Length - θ: Central angle - r: Radius
Special triangles
Special triangles are triangles that you may find within a unit circle. These triangles only have 3 possible inner angles (excluding 90º) which are 30º, 45º, and 60º. Knowing these triangles could help speed up mathematical processes as it reduces the need for extensive calculations.
Unit Circle
A unit circle can be seen as a circle with a radius of one that is sat right at the centre of the origin of a cartesian plane. This circle allows for simpler identification of right angle triangle trends (Sin, Cos, Tan)
All values are positive in quadrant 1, and sin is positive in quadrant 2. While tan is positive in quadrant 3 and cos is positive in quadrant 4. This of course can be seen in the image bellow.
PARAMETRIC EQUATIONS
A parametric function is a function that introduces a third variable alongside variables x and y, this variable being labelled as "t".
Circles
Cartesian equation: x^2 + y^2 = r^2
Parametric curves
When the "x" and "y" values are all given equations (Ex. y = f(t), and x = f(t)), and the many "t" values are processed through these two equations, the result is a list of (x,y) points that when graphed display a parametric curve.
Ellipse
Cartesian equation: x^2/a^2 + y^2/b^2 = 1
