Please enable JavaScript.
Coggle requires JavaScript to display documents.
AP Calculus AB Unit 1: Calculus Prerequisites - Coggle Diagram
AP Calculus AB Unit 1:
Calculus Prerequisites
1.3 - Exponential Functions
Applications
Some real life applications could be the compound interest formula which is A=Ao(1+i/n)^nt or the bacterial growth formula which is A=Ao(B)^(t/P)
Exponential Decay
This is the opposite of exponential growth. The rate of change continues to decrease until it reaches a limit where it does not go further anymore and has reached an asymptote.
The Number e
The number e is a mathematical constant that estimates to 2.718282. It can be expressed as the sum of n equals 0 through infinity of 1/n!. It is also the base for the natural logarithm of ln.
Exponential Growth
Exponential growth is a process that increases quantity over time. It is a geometric growth that grows faster and faster. Examples of this could be bacteria growth or compound interest.
1.1 - Lines
Parallel and Perpendicular Lines
Parallel lines are lines that have the same slope but a different y-intercept. Parallel lines will never touch each other. Perpendicular lines are lines that intersect each other at a 90 degrees angle. You would know this when the slopes of the two lines multiple to negative 1.
Slope of a Line
The slope of a line is rise/run which is also (y2-y1)/(x2-x1), where x1, x2, y1, y2 are points on the line. The slope is also known as delta x, or change in x as y changes.
Equations of Lines
All lines have an equation, in most linear lines, they are expressed using the general form, point slope form, or slope intercept form. Vertical lines are written in the form of x=() while horizontal lines are written in the form of y=().
Applications
General form: Ax+By=C
Point slope form: y-y1 = m(x-x1). This is used to find the equation when slope and one point is known.
Slope-intercept form: y=mx+b. This is the most common way to express a function and the most useful in most cases with m expressing slope and b expressive the y-interceept.
1.4 - Parametric Equations
Lines and Other Curves
Parametric equation graph curves are not actually functions. It is a curve where the x and y variables are defined by another variable T. Parametric equations describe curves represented on a plane and is very useful in our daily lives. This is also a contrast to the Cartesian Equation that we are familiar with since the Cartesian Equation would only use 2 variables.
Circles
The equation that applies to all circles and could be used to graph all circles would be the equation x squared plus y squared equals to R squared, where R represents the radius. This could be further seen and shown on a unit circle.
Ellipses
An ellipse would be very similar to a circle, but it is stretched either vertically or horizontally. The standard form of an ellipse is also very similar to a circle with a few changes. For example, the standard form of an ellipse centered at 0,0 would be (x^2)/(a^2) + (y^2)/(b^2) = 1, whichis similar to the equation of a circle.
Relations
A set of paired values such as x and y that corresponds with a T value which is called a parameter such that we can graph that function into a parametric eqation.
1.2 - Functions and Graphs
Domains/Ranges
A domain shows where all possible values of x can be possible while the range shows where all possible values of y.
Viewing and Interpreting Graphs
A vertical line test can be used to test if an equation is a function or not. If it fails the vertical line test, it is not a function.
Even/Odd Functions
A function is even if the graph is symmetric with respect to the y-axis. A function is odd if the graph is symmetric with the origin.
Absolute Value Functions
An absolute value function contains an algebraic expression within absolute value symbols. The absolute value of a number is its distance from 0 on the number line.
Composite Functions
When two functions is written inside one another, this is a composite function. For example, if you have f(x) and g(x), a composite function would be f(g(x)) where you would substitute the answer you obtain from g(x) into f(x).
1.5 - Functions and Logarithms
Inverses
To graph an inverse function, you would switch all the x and y values in the function and then solve for y after.
Properties of Logarithms
There are many properties pf logarithms. Some include: logb(xk)=klogb(x), logb(x)=loga(x)/loga(b), log(xy)=log(x)+log(y), log(x/y)=log(x)-log(y)
One to One Functions
A one to one function is when in addition to a vertical line test, the function also pass a horizontal line test to show no ordered pairs with different 1st coordinate can have the same 2nd coordinate
Logarithmic Functions
y=loga(x) is the logarithmic functions, It determines how many times a certain number, called the base, is multiplied by itself to reach another number.
Applications
Some real world applications would be the pH scale in chemistry, the richter scale to measure earthquakes, and sound in decibel.
1.6 - Trigonometric Functions
Graphs of Trigonometric Functions
Sine and Cosine functions in the form of y=Asin(B(x-C))+D and y=Acos(B(x-C))+D can be graphed.
Inverse Trigonometric Functions
The inverse trigonometric functions are simply the arc functions. This occurs you are given the value of sin(theta) to find theta etc.
Periodicity
The length of a wave of a function is known as the period. The period can be calculated with the formula of 2pi/b.
Transformation of Trigonometric Graphs
A transformation of sinusoidal functions would connect to a vertical shift, horizontal shift, reflection in x/y axis, horizontal or vertical compression/expansion. This transformation will occur by changing the values of A, B, C and D in y=Asin(B(x-C))+D.
Even and Odd Trigonometric Functions
Since a function is even if the graph is symmetric with respect to the y-axis while a function is odd if the graph is symmetric with the origin, we know that cos(x) and sec(x) is even functions while the other four are odd functions.