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AP Calculus Unit 1 Project - Coggle Diagram
AP Calculus Unit 1 Project
Lines
Increments
If a particle moves from a point (x1, y1), to the points (x2, y2)
--> ∆x = x2 - x1 , ∆y = y2 - y1.
Slope of a line
Point slope equation of lines: if a line has a slope m and passes through the point P(x1, x2), then the equation of the line is (y-y1) = m (x-x1)
Slope intercept equation of lines: y=mx+b - most common way to express a function, where m is the slope and b is the y-intercept
Parallel and perpendicular lines
Parallel - from equal angles with the x axis - non-vertical lines with same slope and never touch → m1 = m2
Perpendicular - two non-vertical lines which intersect each other at a 90 degree angle → m1 m2 = -1
Equation of lines
Vertical lines: writrten in the form x=a where all the points one the line have coordinates (a,y)
Horizontal lines: written in the form y=b, where all of the points one the line have coordinates (x,b)
General linear equation
Ax+By=C
Helps convert the linear function among different forms. These forms of linear function can help us calculate slope, y intercept and a variety of other info.
Functions and Graphs
Vertical Line Test
A method that is used to determine whether a given relation is a function or not. Draw a vertical line cutting through the graph of the relation, and then observe the points of intersection.
Domain & Range
The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of values that the function assumes. This set is the values that the function shoots out after we plug an x value in.
Piecewise functions
Intervals may be open, closed or half open. The end points of an interval make up the intervals boundary and are called boundary points. The remaining points make up the interval interior and are called interior points. Closed intervals (contain their boundary points. Open intervals contain no boundary points, every point of an open internal is an interior point of the interval
Open interval: (...). Closed Interval: [...] 1/2 Open interval: [...), (...]
Even & Odd Functions
Odd Function of x = f(-x) = -f(x). Even Function of x = f(-x) = f(x). Graph of Odd function - symmetric with the origin Graph of Even Function - symmetric with the y-axis
Same as Trig. Functions Even & Odd
Piecewise Functions
A function built from pieces of different functions over different intervals, including domains
Intervals (open, closed)
Composite Functions
A function whose values are found from two given functions by applying one function to an IV and then applying the second function to the result and whose domain consists of those values of the IV for which the result yielded by the first function lies in the domain of the second. --> g(x) --> f(g(x))
(g o f)(x) = g(f(x)) , (f o g)(x) = f(x) * g(x) --> (f o g)(x) = f(g(x))
Connects back to when graphing trig.
Exponential Functions
Exponential Growth
A pattern of data that shows greater increases with passing time, creating the curve of an exponential function. Example: Bacteria Growth & Compound Interest
Applications
Compound Interest: A = Ao (1 + i/n) ^n*t.
Bacteria Growth:A = Ao (B) ^T/P
Number e: an important mathematical constant, approximately equal to 2.71828. When used as the base for a logarithm, we call that logarithm the natural logarithm (ln)
Exponential Graphing --> Functions and Graphs
Exponential Decay
Opposite of Exponential Growth, describes the process of reducing an amount by a consistent percentage rate over a period of time, until it reaches a limit then it cannot go further and has reached an asymptote
Parametric Equations
Relations
Set of ordered pairs (x,y) of real numbers. Graph of a relation is the set of points in the plane that corresponds to the ordered pairs of the relation. If x and y are functions of a third variable t, called a parameter. -->. x=cost , y=sint
Circles
Equation for all circles: x^2 + y^2 = r^2
Trig Functions --> Unit circle
Parametric Curve
A curve in which one variable defines the x and y variables (T). In contrast to the cartesian equation, which only utilizes two variables, parametric equations explain curves represented on a plane and are highly helpful in our daily lives.
Elipses
Very similar to a circle, but its stretched either horizontally or vertically. Standard form of an elipses: Centered at (0,0) is
x^2 / a^2 + y^2/b^2 = 1
Functions & Logarithms
One to one Functions
A horizontal line test to show no two ordered pairs with a different first coordinate and the same second coordinate. When performing the horizontal line test, it can intersect any horizontal line maximum once. A equation is a one to one function when if f(x1) = f(x2)
Similar to Vertical Line test - Functions & Graphs
Inverses
To find/graph an inverse you first must switch the x and y values in the function then solve for y after.
Logarithmic Functions
y=loga(x) - inverse and exponential function of y=a^x , where a > 0, a ≠ 1. Which determines how many times a specific number, called the base, is multiplied by itself to reach another number.
Properties of Logs
Change of Base:
Trigonometric Functions
Radian Measure
The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle --> to find arc length:
Trigonometric Ratios
Unit circle: x^2 + y^2 = r^2
Parametric Equations - Circles Equations
Periodicity
When an angle of measure
and an angle of measure
are in standard position their terminal rays coincide. The two angles therefore have the same trigonometric function values
Period
The length of a wave of a function. Can be calculated with the formula 2pi/b
Even & Odd Trig Functions
Odd Function of x = f(-x) = -f(x). Even Function of x = f(-x) = f(x). Graph of Odd function - symmetric with the origin Graph of Even Function - symmetric with the y-axis. Cos(x) & Sec(x) are both even functions while the other 4 are odd
Same with Functions & Graphs
Transformations of Trig. Graphs
Would connect to a vertical shift, horizontal shift, reflection in x/y axis, vertical/horizontal compression/expansion
Very similar to regular transformations for graphs in 1.3 or even in general
A = the Amplitude, B = Period, C = Phase Shift, &.
D =. Vertical Displacement
Inverse Trig. Functions
Inverse trig. functions are the arc functions, since none of the 6 trig ratios have inverses, in each case the domain can be restricted to produce a new function that does have an inverse.
