Calculus Fundamentals

Calculus

Branches of Calculus

• Differential Calculus: deals 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 behavior of a function as the input (or x-value) approaches a specific value
• Derivatives: a measure of how a function changes as its input changes
• Differentiation Rules:
  • Power Rule
  • Product Rule
  • Quotient Rule
  • Chain Rule
• Applications of Derivatives:
  • Finding maximum and minimum values
  • Determining the rate at which a quantity changes
  • Optimization problems
• Integrals:
  • Definite Integrals: the area between a curve and the x-axis over a specific interval
  • Indefinite Integrals: the antiderivative of a function
• Integration Techniques:
  • Substitution Method
  • Integration by Parts
  • Integration by Partial Fractions
  • Trigonometric Substitution

Important Theorems

• Fundamental Theorem of Calculus: relates the derivative of an antiderivative to the original function
• Mean Value Theorem: guarantees the existence of a point where the derivative of a function equals the average rate of change

Applications of Calculus

• Physics: modeling motion, force, and energy
• Economics: optimizing profit and cost functions
• Computer Science: algorithms and data analysis
• Biology: modeling population growth and chemical reactions

Calculus

• Branch of mathematics that deals with the study of continuous change

Branches of Calculus

• Differential Calculus: studies rates of change and slopes of curves
• Integral Calculus: studies accumulation of quantities

Key Concepts

• Limits: behavior of a function as input approaches a specific value
• Derivatives: measure of how a function changes as input changes
• Differentiation Rules:
  • Power Rule
  • Product Rule
  • Quotient Rule
  • Chain Rule
• Applications of Derivatives:
  • Finding maximum and minimum values
  • Determining the rate at which a quantity changes
  • Optimization problems

Integrals

• Definite Integrals: area between a curve and the x-axis over a specific interval
• Indefinite Integrals: antiderivative of a function
• Integration Techniques:
  • Substitution Method
  • Integration by Parts
  • Integration by Partial Fractions
  • Trigonometric Substitution

Important Theorems

• Fundamental Theorem of Calculus: relates derivative of an antiderivative to the original function
• Mean Value Theorem: guarantees existence of a point where derivative of a function equals average rate of change

Applications of Calculus

• Physics: modeling motion, force, and energy
• Economics: optimizing profit and cost functions
• Computer Science: algorithms and data analysis
• Biology: modeling population growth and chemical reactions

Geometry

• Geometry is a branch of mathematics that deals with the study of shapes, sizes, and positions of objects.

Key Concepts

• A point is a location in space, represented by a set of coordinates (x, y, z).
• A line is a set of points extending infinitely in two directions.
• A ray is a line that extends infinitely in one direction from a single point.
• An angle is formed by two rays sharing a common endpoint.
• A plane is a flat surface that extends infinitely in all directions.

Properties of Shapes

• Collinear points lie on the same line.
• Coplanar points lie on the same plane.
• Parallel lines never intersect.
• Perpendicular lines intersect at a right angle (90°).
• Acute angles are less than 90°.
• Obtuse angles are greater than 90° but less than 180°.
• Right angles are exactly 90°.
• Straight angles are exactly 180°.

Types of Shapes

• An equilateral triangle is a triangle with all sides equal.
• An isosceles triangle is a triangle with two sides equal.
• A scalene triangle is a triangle with all sides unequal.
• A rectangle is a quadrilateral with opposite sides equal and parallel.
• A square is a quadrilateral with all sides equal and perpendicular.
• A rhombus is a quadrilateral with all sides equal, but not perpendicular.
• A regular polygon is a polygon with all sides and angles equal.
• An irregular polygon is a polygon with sides and angles not equal.

Theorems and Formulas

• The Pythagorean Theorem states that a^2 + b^2 = c^2 for right-angled triangles.
• The Distance Formula is √((x2 - x1)^2 + (y2 - y1)^2) for the distance between two points.
• The Midpoint Formula is ((x1 + x2)/2, (y1 + y2)/2) for the midpoint of a line segment.