# Calculus – Quick Reference Sheet

Limits and Continuity Differential Calculus
Definitions

Limit: the value of the function as x approaches a certain value
Right-Hand Limit: the limit as x approaches from the positive direction; is noted as Left-Hand Limit: the limit as x approaches from the negative direction; is noted as Two-Sided Limit: the limit is the same whether x approaches from the right or left
Indeterminate Limits: Undefined limits such as 0 divided by 0, infinity divided by infinity, infinity times zero, infinity minus infinity, one to the power infinity, zero to the power zero and infinity to the power zero
Continuity: a function, f(x), is continuous over an interval if there exists a value of the function for every value of x on the interval
Discontinuity: a function, f(x), is discontinuous at a point if the function is not defined at that point
Types of Discontinuity

Removable Discontinuity: a limit of the function exists at the point of discontinuity
Jump Discontinuity: the right-handed and left-handed limits are different
Infinite Discontinuity: the limit of the function as x approaches the discontinuity is infinity
Finding Limits

By Substitution: limits for polynomial and rational functions with non-zero denominators can be found by substituting the value x into the function
By Factoring: limits for rational functions where the denominator is zero can be found by factoring the function and then simplifying
By Estimating: limits can be found for any function by using a graph or by calculating the value of f(x) for different values of that approach the limit
Limits and Asymptotes

Vertical Asymptote: a line, x=a that the graph of a function approaches but never touches Horizontal Asymptote: a line, y=a that the graph of a function approaches but never touches Theorems

Intermediate Value Theorem: if the function is continuous between two points a and b, then there exists a value c between a and b where f(c) will be between f(a) and f(b)
Extreme Value Theorem: if a function is continuous over a closed interval then the function has a maximum and minimum on the interval
Sandwich or Squeezed Theorem: if a function f is between two other functions g and h with a common limit, then the function f has the same limit as g and h
Rules

 Sum Rule Quotient Rule Product Rule Constant Multiple Rule Derivative: the function that describes the rate of change of another function
Geometrically, the derivative is the slope of a curve
Analytically, the derivative is the rate of change of a curve
Numerically, the derivative is the difference in y values on a very small interval of x
Higher Order Derivatives: taking the derivative of a derivative; the order of the derivative represents the number of times the derivative was taken
Partial Derivative: the derivative of only one variable in the function; other variables are treated as constants

Derivative Notation or Second Order Derivative or Theorems

Mean Value Theorem: if a function is continuous and differentiable on an interval [a,b], then there exists at least one point on that interval where the derivative is the slope through a and b.
Differentiability and Continuity

If a function is differentiable at a point then it is continuous at that point; the opposite is not always true
Differentiability and Inverse Functions
If a function is differentiable at every point on an interval then it has an inverse function that is also differential on the interval
Finding Derivatives

 Implicitly-Defined Functions: Differentiate both sides of the equation with respect to x, treating y as a function of x. Collect the terms with dy/dx on one side of the equation. Factor out dy/dx. Solve for dy/dx. Inverse Functions Write the inverse function relationship. Differentiate both sides with respect to x. Solve using implicit differentiation. Inverse Functions: Write the inverse function relationship Differentiate both sides with respect to x. Solve using implicit differentiation.

Rules

 Sum Rule Constant Multiple Rule Quotient Rule Chain Rule Product Rule Derivative Formulas

 Polynomial Function Constant Function 