Calculus Parts

• Calculus is a branch of maths dealing with change and motion, providing tools for understanding and analyzing functions and describing relationships between different variables
• Differentiation is the process of finding the derivative of a function, measuring the rate of change at a particular point
• Notation for the derivative: The derivative of a function f(x) is denoted as f'(x) or dy/dx
• The derivative of a constant is 0
• Integration is the reverse process of differentiation, involving finding the antiderivative of a function
• Notation for integration: The integral of a function f(x) is denoted as ∫f(x) dx
• Basic rules of differentiation include the power rule, product rule, quotient rule, and chain rule
• Basic rules of integration include the power rule, constant rule, and the fundamental theorem of calculus
• An integral is the inverse operation of differentiation, used to find the area under a curve, the volume of a solid, and other related quantities
• Limits: A limit is the value that a function approaches as the input approaches a certain value
• Notation for limits: The limit of a function f(x) as x approaches a is written as lim(x→a) f(x)
• Basic limit rules include the sum rule, quotient rule, and the limit of a constant
• Calculus has applications in physics, engineering, economics, and biology, used to analyze rates of change, find maximum and minimum values, solve optimization problems, and model real-world phenomena
• The fundamental theorem of calculus establishes a connection between differentiation and integration
• The first fundamental theorem of calculus states that if a function f(x) is continuous on an interval [a, b] and F(x) is its antiderivative, then ∫[a, b] f(x) dx = F(b) - F(a)
• The second fundamental theorem of calculus states that if F(x) is an antiderivative of f(x), then d/dx ∫[a, x] f(t) dt = f(x), where a is a constant