Calculus Study Guide

Limit

The value that a function approaches as the input variable gets closer to a specific value.

Direct Substitution

A method for finding limits of continuous functions by evaluating the function at a point.

Left-Hand Limit

The value a function approaches as the input approaches from the left side.

Right-Hand Limit

The value a function approaches as the input approaches from the right side.

Indeterminate Form

A mathematical expression that does not initially yield a clear limit, such as 0/0.

Removable Discontinuity

A point in a function where there is a hole, which can be 'removed' by redefining the function at that point.

Vertical Asymptote

A vertical line where a function approaches positive or negative infinity.

Horizontal Asymptote

A horizontal line that the graph of a function approaches as the input goes to positive or negative infinity.

Mean Value Theorem (MVT)

States that a continuous function that is differentiable on an interval must have at least one point where the derivative equals the average rate of change.

First Derivative Test

A method to determine where a function is increasing or decreasing based on the sign of its derivative.

Definite Integral

Represents the total accumulation or the geometric area under a curve bounded by the x-axis.

Riemann Sums

Approximations of the area under a curve using shapes like rectangles or trapezoids.

Integrating by U-Substitution

A technique used to simplify integrals by substituting a part of the integral with a new variable.

Slope Fields

Visual representations showing the direction of solutions of a differential equation at various points.

Average Value of a Function

The average height of a function over an interval, calculated using the definite integral.

Volume by Cross-Sections

The method of finding volume by integrating the area of cross-sections over a given interval.

Volume of Revolution

Volume created by rotating a two-dimensional area around an axis, calculated using the disc method.