Calculus Basics

Calculus

Definition: CalcuCalculus studies continuous change in functions and limits.ts.

Two Main Branches:

• Differential Calculus: Concerned with the study of rates of change and slopes of curves, including the use of derivatives and differentials to analyze functions, and the application of optimization techniques to find the maxima and minima of functions.
• Integral Calculus: Deals with the study of accumulation of quantities. It involves finding the area under curves, volumes of solids, and other quantities, using techniques such as substitution, integration by parts, and integration by partial fractions.

Key Concepts

• Limits: The concept of approaching a value as the input gets arbitrarily close, allowing us to understand how a function behaves at a specific point, even if it's not defined at that exact point.
• Derivatives: Measure of how a function changes as its input changes. It calculates the rate of change of a function with respect to one of its variables, allowing us to analyze and understand how the function behaves.
  • Notation: f'(x) or (d/dx)f(x)
  • Geometric Interpretation: The slope of the tangent line to the function at a point.
• Differentiation Rules:
  • Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1)
  • Product Rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
  • Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
• Applications of Derivatives: In calculus, derivatives have numerous practical applications in various fields, including physics, engineering, economics, and computer science, such as optimization problems, motion along a line, curve, or surface, and related rates.
  • Optimization: Finding the maximum or minimum of a function.
  • Motion Along a Line: Finding the velocity and acceleration of an object moving along a line.
• Integrals:
  • Definite Integral: The area between a curve and the x-axis over a specific interval.
  • Indefinite Integral: The antiderivative of a function, denoted as ∫f(x)dx.
• Fundamental Theorem of Calculus: Relates the derivative of an antiderivative to the original function.

Important Theorems and Formulas

• Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
• Integration by Parts: ∫udv = uv - ∫vdu.
• Integration by Substitution: ∫f(x)dx = ∫f(u)du, where u = φ(x) and du/dx = φ'(x).

Calculus Definition

• Calculus is a branch of mathematics that deals with the study of continuous change, particularly in the context of functions and limits.

Two Main Branches

• Differential Calculus: concerned with the study of rates of change and slopes of curves.
• Integral Calculus: deals with the study of accumulation of quantities.

Key Concepts

Limits

• The concept of approaching a value as the input gets arbitrarily close.

Derivatives

• Measure of how a function changes as its input changes.
• Notation: f'(x) or (d/dx)f(x)
• Geometric Interpretation: the slope of the tangent line to the function at a point.

Differentiation Rules

• Power Rule: if f(x) = x^n, then f'(x) = nx^(n-1)
• Product Rule: if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
• Quotient Rule: if f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2

Applications of Derivatives

• Optimization: finding the maximum or minimum of a function.
• Motion Along a Line: finding the velocity and acceleration of an object moving along a line.

Integrals

• Definite Integral: the area between a curve and the x-axis over a specific interval.
• Indefinite Integral: the antiderivative of a function, denoted as ∫f(x)dx.

Fundamental Theorem of Calculus

• Relates the derivative of an antiderivative to the original function.

Important Theorems and Formulas

• Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
• Integration by Parts: ∫udv = uv - ∫vdu.
• Integration by Substitution: ∫f(x)dx = ∫f(u)du, where u = φ(x) and du/dx = φ'(x).

Geometry Definition

• Study of points, lines, angles, and planes, and their relationships

Key Concepts

• Points: Represented by a set of coordinates (x, y, z), locations in space
• Lines: Sets of points extending infinitely in two directions
• Line Segments: Parts of a line with a fixed length and endpoints
• Rays: Parts of a line with a fixed starting point and extending infinitely in one direction
• Angles: Formed by two rays sharing a common endpoint (vertex)
• Planes: Flat surfaces with infinite length and width

Properties of Shapes

• Congruent: Shapes with the same size and shape
• Similar: Shapes with the same shape but not necessarily the same size
• Perimeter: Distance around a shape
• Area: Measure of the amount of space inside a shape
• Volume: Measure of the amount of space inside a 3D shape

Types of Angles

• Acute: Less than 90 degrees
• Right: Exactly 90 degrees
• Obtuse: More than 90 degrees but less than 180 degrees
• Straight: Exactly 180 degrees
• Reflex: More than 180 degrees but less than 360 degrees

Theorems and Postulates

• Ruler Postulate: Points on a line can be paired with real numbers
• Angle Addition Postulate: Measure of an angle is equal to the sum of its parts
• Parallel Lines Theorem: Two lines that never intersect are parallel

Measurement Units

• Length: Measured in units such as meters (m), centimeters (cm), or inches (in)
• Area: Measured in units such as square meters (m²), square centimeters (cm²), or square inches (in²)
• Volume: Measured in units such as cubic meters (m³), cubic centimeters (cm³), or cubic inches (in³)

Geometric Transformations

• Translations: Moving a shape to a new location without changing its size or shape
• Rotations: Turning a shape around a fixed point
• Reflections: Flipping a shape over a line
• Dilations: Enlarging or shrinking a shape while keeping its shape and proportions