Calculus (IB)

Calculus

The branch of mathematics that studies continuous change, divided into Differential and Integral Calculus.

Differential Calculus

Concerns the study of rates of change and slopes of curves.

Integral Calculus

Involves accumulation of quantities and areas under and between curves.

Limit

The value that a function approaches as the input approaches a certain point.

Continuous Function

A function is continuous if it is defined and the limit exists at a point.

Removable Discontinuity

A discontinuity that can be fixed by redefining the function at a point.

Jump Discontinuity

A discontinuity where there are sudden jumps in function values.

Infinite Discontinuity

A discontinuity characterized by asymptotic behavior.

Derivative

A measure of how a function changes as its input changes; represents the instantaneous rate of change.

Power Rule

A basic rule for differentiation: if f(x) = x^n, then f'(x) = n*x^(n-1).

Product Rule

A rule for differentiating products of functions: (uv)' = u'v + uv'.

Quotient Rule

A rule for differentiating quotients of functions: (u/v)' = (u'v - uv')/v^2.

Chain Rule

A rule for differentiating composite functions: if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Higher-Order Derivatives

Derivatives of derivatives, such as the second derivative which represents acceleration.

Tangent Line

The line that touches a curve at a single point without crossing it.

Normal Line

A line perpendicular to the tangent line at a given point on a curve.

Concavity

Refers to the direction of the curvature of a function. Concave up if it curves up, concave down if it curves down.

Critical Point

A point in the domain of a function where the derivative is either zero or undefined.

Integral

The reverse process of differentiation; used to find areas under curves.

Indefinite Integral

Integral without specified bounds, representing a family of functions plus a constant of integration.

Definite Integral

Integral with specified limits, representing the signed area under the curve between those limits.

Area Under a Curve

The definite integral of a function from a to b gives the area between the curve and the x-axis.

Volume of Revolution

Calculated using integrals to find the volume of a solid obtained by rotating a function around an axis.

Kinematics

The branch of mechanics that deals with the motion of objects; integration can be used to find displacement.

Differential Equation

An equation that relates a function with its derivatives.

Ordinary Differential Equations (ODEs)

Differential equations involving one independent variable.

Partial Differential Equations (PDEs)

Differential equations involving multiple independent variables.

Separable Equations

Differential equations that can be expressed as the product of a function of x and a function of y.

Newton's Law of Cooling

A formula describing the rate of cooling of an object to the ambient temperature.

Logistic Growth

A model describing population growth that is initially exponential but slows as it approaches a maximum limit.

Applications of Derivatives

Includes finding tangents, identifying maxima and minima, and analyzing rates of change.

Applications of Integration

Includes finding areas, volumes, and solving problems in physics and engineering.

Mastery of Calculus

Crucial for solving real-world problems in various disciplines.

Concept of Continuity

A fundamental concept in calculus where small changes in input result in small changes in output.