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Prerequisites of Calculus in the Six Main Units, Source: University of…
Prerequisites of Calculus in the
Six Main Units
Lines
Parallel Lines
If M1 = M2, the lines in the function are parallel
Perpendicular Lines
If M1 * M2 = -1, the lines in the function are perpendicular
Forms for Graphing
y = mx + b is the Slope-Intercept form
(y - y1) = m(x - x1) is the Point-Slope form
Co-ordinates are kept in (x,y) form
Functions And Graphs
Test if it is a function
Try Vertical Line test (More than one value in a vertical line --> not a function)
One X value per Y value
Intervals
Closed interval uses [ Symbol
Open interval uses ( Symbol
Even/Odd Functions
An even function is symmetrical over the Y axis
An odd function remains the same when the graph is Rotated two quadrants Clockwise/Counterclockwise
Piecewise Functions
Piecewise functions are multiple seperate functions to be plotted on the same graph
Normally are formatted as so with their OWN SEPERATE DOMAINS:
Exponential Functions
Exponential Growth/Decay Graphs
y = 2^x --> Exponential GROWTH
y = (1/2)^x --> Exponential DECAY
Translations
Horizontal Translation --> A number added/subtracted directly from x within the f(x) (e.g. y = f(x-3) )
Vertical Translation --> A number is added/subtracted from to the f(x) (e.g. y = f(x) - 3 )
Transformations
Vertical Expansion/Compression --> When a number is multiplied to the f(x) (e.g. y = Af(x) )
Horizontal Expansion/Compression --> When a number is multiplied to the x within f(x) (e.g. y = f(Ax) )
Three Main Formulas
Compound Interest Formula
Exponential Growth/Decay Formula
Compounding Continually Formula
Inverse functions
Swap X and Y values
f(x)^-1
Trigonometric Functions
Radians/Degrees
Note:
you see π --> You are 99% of the time dealing with Radians
Conversion rate
π = 180 degrees
Arc Length
S = r * Θ where S = Side length, r = Radius, Θ = Center Angle
Calculations are GENERALLY done in radians
Unit Circle & Special Triangles
Important Formulas/Concepts
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
CoTangent = Adjacent / Opposite
Secant = Hypotenuse / Adjacent
CoSecant = Hypotenuse / Opposite
If +180°or + 2π to the value of any trig function, it remains the same (e.g. sin(100 + 180°) = sin(100), sin(2π/3 + 2π) = sin(2π/3)
Pythagorean Theorem
In a right angle triangle, a^2 + b^2 = c^2
Sine law (works in triangles)
Cosine Law (works in triangles)
Graphing
y = af(b(x+c)) + d
a --> Vertical Expansion/Compression
b --> Horizontal Expansion/Compression
c --> Horizontal Translation
d --> Vertical Translation
Calculate for Period with 2π/b
Parametric Equations
Graphing Circles
Can use the Cartesian Equation x^2 + y^2 = r^2
Oftentimes come in forms of 2π in questions
Note:
When in doubt, PLUG THE EQUATION INTO GRAPHING CALCULATOR
Parametric Curve
Definitions (source: Khan Academy):
If "x" and "y" are given as functions x = f(t), y = f(t) over an interval of t-values, then set the points (x, y) = (f(t), g(t)) defined by these equations are known as a parametric curve.
Simply put, a parametric curve is a normal curve where we choose to define the curve's x and y values in terms of another variable for simplicity or elegance
Graphing Elipses
Can use the cartesian equation x^2/a^2 + y^2/b^2 = 1
Oftentimes seen in a format where there is a sine/cosine pair with different numbers being multiplied to each
Functions and Logarithms
Log and Ln
A logarithm (log) is the opposite of an exponent --> can be used to cancel out exponents
A natural logarithm (ln) is the opposite of the constant 'e' --> can be used to cancel out said constant
Graphically
A log graph is an exponential graph with the X and Y values reversed
Rules of Log
One-to-one function
A function that also passes the HORIZONTAL line test (Each X value gives a DIFFERENT y value)
Source: University of Florida
IMPORTANT TO REMEMBER THESE VALUES
Source: Chlilimath.com
Source: Google
Source: Google
Related
None of these are one to one unless you restrict the domain (Often done with inverses)
