Math

Functions

Equations with an input (x) and an output (f(x))

Domain

The list of Possible inputs for a function

Range

The list of Possible Outputs for a function

Natural Base Rules

Important Properties such as e^a+b = e^a * e^b and e^a-b = e^a/e^b.

Natural Logarithm Rules

Important properties such as ln(1) = 0 and ln(ab) = ln(a) + ln(b).

Basic Trigonometry

Mnemonic SohCahToa for sine, cosine, and tangent definitions.

Unit Circle

A circle with a radius of 1 centered at (0,0) used to find points and angles.

Limit

The value (y) that is approached as the x-value gets closer to a desired input.

Existence of Limit

A limit exists only if the values approached from both the negative and positive directions are the same.

Graphical Method

One way to find limits by analyzing the behavior of a function on a graph.

Numerical Method

A way to find limits by plugging in values closer to the desired input to observe the number being approached.

Analytical Method

A strategy that saves time by using intuition to find limits instead of trial and error numerical method.

Piecewise Function

A function composed of multiple equations, each applicable to a specific segment of the domain.

Continuity

A property of a function where it can be drawn without lifting the pencil, indicating no breaks in the graph.

Discontinuity

A characteristic of a function that has breaks or points where the graph is not connected.

Jump Discontinuity

A type of discontinuity where the function has a sudden jump between two points.

Hole (Removable) Discontinuity

A type of discontinuity where a single point is missing from the graph, but the function could be made continuous by filling in that point.

Vertical Asymptotes Discontinuity

A type of discontinuity that occurs when the function approaches infinity at certain points, creating a vertical line that the graph cannot cross.

Derivative

Represents the rate at which a function is changing at any given point.

Tangent Line

A straight line that touches a curve at a single point, representing the instantaneous rate of change at that point.

Power Rule

If ( f(x) = xⁿ), then ( f'(x) = nx^n-1)

Product Rule

(f(x)g(x))’ → f’(x)g(x) + f(x)g’(x)

Quotient Rule

(f(x)/g(x))’ → (g(x)f’(x) - f(x)g’(x))/(g(x))²

f(x) sec(x)

Derivative is f’(x) sec(x)tan(x)

f(x) cot(x)

Derivative is f’(x) - csc²(x)

f(x) csc(x)

Derivative is f’(x) - csc(x)cot(x)

f(x) tan(x)

Derivative is f’(x) sec²(x)

Chain Rule

(g(f(x))’ = g’(f(x))(f’(x))

1/x

(ln(x))’

Implicit Form

A representation of a function where y is isolated (e.g., y = f(x)).

Explicit Form

A representation of a function where y is not isolated (e.g., 1 = y² + x²).

Related Rates

A method in calculus used to find the rate at which one quantity changes in relation to another.