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Prerequisites of Calculus - Coggle Diagram
Prerequisites of Calculus
Identifying Functions
Definition:
A expression that defines the relationship between an independent variable (usually "x") and a dependent variable (usually "y").
Determine whether a graph is a function
Vertical line test:
Draw a vertical line through the graph and if the vertical line intercepts the graph more than once the graph is not a function.
Listing points on the graph:
If any "x" values repeat for any of the points then the graph will not be a function.
Types of Functions
One to One:
Each "x" value in the domain corresponds with exactly one "y" value in the range.
Horizontal Line Test:
If the horizontal line intersects more than one point the graph is not a one to one function.
Listing Points:
If any "y" values repeat for any of the points then the graph will not be one to one.
Surjective:
A value for each y element there is at least one x element corresponding to it
Odd and Even Functions
Even Function Definition:
If the graph is symmetric about the y-axis then the function will be even. This means the graph is mirroring itself on each side of the y-axis.
Ways of Determining if a Function is Even
If the exponents are all even within all the terms (the exponent on the constant term is 0) this can suggest that the function is indeed even.
Example:
f(x) = x^2, f(-x) = (-x)^2, f(-x) = x^2
Using the formula: f(-x) = f(x) by plugging "-x "into all values of "x" in the function and then simplifying the function. If the function is the exact same then it will be even.
Odd Function Definition:
If the graph is symmetric about the origin then the function will be odd. This means half the graph on one side of the y-axis will be the inverted version of the other half.
Ways of Determining if a Function is Odd
If the exponents are all odd within all the terms this can suggest that the function is indeed odd.
Using the formula: f(-x) = -f(x) by plugging "-x" into all the values of "x" in the function then simplifying the function. If all signs are switched in the function (positive to negative, negative to positive) then the function is odd.
Example:
f(x) = x^3, f(-x) = (-x)^3, f(-x) = -x^3
Neither Even or Odd Functions:
Ways of Determining if a Function is Neither Even or Odd
If none of the two equations f(-x) = f(x), and f(-x) = -f(x) are satisfied by plugging "-x" into all "x" values within the function this means the function is neither
Example:
f(x) = x^2+x^3, f(-x) = (-x)^2 + (-x)^3, f(-x) = x^2 - x^3
If the exponents of all the terms are a mix of even and odd numbers (the exponent on the constant term is 0) this suggest that the function is neither odd or even.
Functions We Learned In Class:
Rational Function:
A rational function is an algebraic function that has polynomials for both the numerator and denomenator.
Polynomial Function:
A function which as only positive integers as its exponential power.
Line Function:
The line function has one independent variable which is "x" and one dependent variable which is "y". Usually in the form y = mx + b
Slope:
Let P1(x1, y1) and P2(x2, y2) be points on a non vertical line "L". m (slope) = Δrise/Δrun = y2-y1/x2-x1
Parallel
: The slopes of the two lines are identical m1=m2
Perpendicular
: The slopes equal -1 when multiplied together. m1*m2 = -1
Point-Slope Equation of Lines:
If a line has a slope "m" passes through the point P(x1, y1), then the equation of the line is: (y-y1) = m(x-x1)
Piece-Wise Function:
A piece-wise function is a function built from pieces of other functions with different domains.
Exponential Function:
A function that has a base with the independent "x" raised to its power.
Asymptotes:
A line that a graph approaches but never touches which acts as a limit to a line or a curve.
Real Life Applications of Exponential Growth and Decay Formula
A = Ao(B)^(T/P)
Compound Interest Formulas
A = Ao(e)^(it)
"e" is a mathematical constant also known as natural log. "ln" is the natural logarithm with the base mathematical constant of "e". "ln" can be used to simplify equations involving "e".
A = Ao(1+i/n)^(n*t)
When the base is smaller than 1 it is exponential decay if it is bigger than 1 it is exponential growth
Logarithmic Function:
This function is the inverse function of the exponential function.
Log Laws:
Radical Function:
A radical function includes a nth root of the independent value of "x" where n is a positive integer.
Trigonometric Function:
Relates an angle of a right triangle to ratios of its two side lengths creating a function
The Greatest Integer Function:
y = |x| or y = int(x), int(x) = the greatest integer that is less than. or equal to x.
Inverse Function:
This type of function is represented by using f'(x) instead of f(x) to indicate its inverse function. This means all of the "x" and "y" values of the original function switch places. Only one to one functions have inverse functions. An example of this is logarithmic functions and exponential functions.
Interval Notation
Open Intervals:
[ x, y] , using square brackets means the end values will be included.
Holes:
A circle on a graph which is not filled in represents a hole meaning the value is not included. This can happen if the domain of a function does not include a certain value.
Close Intervals:
( x, y) , using normal brackets meaning the end values are not included
Graphing Functions
Transformation
b = Horizontal stretch or shrink; reflection about y-axis
c = Horizontal shift
a = Vertical stretch or shrink; reflection about the x-axis
d = vertical shift
Trigonometry:
Finding Arc Length
S = r * θ, S = arc length, r = radius, θ = central angle measurement
Unit Circle:
A circle with a radius of 1 that helps identify different right triangle relationships such as sine cosine and tangent.
Trigonometric Ratios
Sine: sin(θ) = Opposite/Hypotenuse = y/r
Reciprocal Function
Cosine: cos(θ) = Adjacent/Hypotenuse = x/r
Reciprocal Function
Tangent: tan(θ) = Opposite/Adjacent = y/x
Reciprocal Function
Secant = sec(θ) = Hypotenuse/Adjacent = r/x
Cosecant: csc(θ) = Hypotenuse/Opposite = r/y
Cotangent: cot(θ) = Adjacent/Opposite = x/y
Periodicity:
A function f(x) is periodic if there is a positive number "p" such that f(x+p) = f(x) for every value of x. The smallest such value of "p" is the period of f
cos(θ+2ℼ) = cosθ sin(θ+2ℼ) = sin θ tan(θ+2ℼ) = tanθ sec(θ+2ℼ) = secθ csc(θ+2ℼ) = cscθ cot(θ+2ℼ) = cotθ cos(θ-2ℼ) = cosθ sin(θ-2ℼ) = sin θ
Special Triangles:
This can help you visualize and remember trigonometric rations of 30-agree, 45-degree, and 60-degree angles within a unit circle.
Types of Special Triangles
Reciprocal Trigonometric Functions:
Trig functions do not have inverses; however the domain of the function can be restricted thus allowing it to have inverses.
Pythagorean Theorem:
This is a mathematical law that works for right triangles which state the two shorter lengths of the triangle squared and added together will equal to the hypotenuse squared. Equation: a^2 +b^2 = c^2, this can be useful in determining the unknown for solving trig ratios.
Graphing Trigonometric Functions
y = af(b(x+c))+d
d = vertical shift
a (amplitude) = Vertical stretch or shrink; reflection about the x-axis
c = Horizontal shift
b = Horizontal stretch or shrink; reflection about y-axis
Equation for calculating period:
2ℼ/|b| for sine, cosine, and their reciprocals. ℼ/|b| for tangent and cotangent.
Parametric Equations
Definition:
A type of equation that has an independent variable called a parameter usually defined by "t" with "t" being the third variable of the "x" and "y" functions.
Parametric Curve:
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.
Graphing Circles:
The cartesian equation for a circle is
x^2 + y^2 = r^2
Graphing Ellipse:
The cartesian equation for an ellipse is x^2/a^2 +y^2/b^2 =1
