Introduction to Calculus - Differential Calculus

Introduction to Calculus

• Calculus is a branch of mathematics that deals with continuous change.
• It comprises two major branches: differential calculus and integral calculus.
• Differential calculus focuses on rates of change, slopes of curves, and tangents.
• Integral calculus focuses on accumulating quantities, areas under curves, and volumes.

Differential Calculus

• Concept of Limits:
  • The foundation of differential calculus is the concept of a limit.
  • A limit describes the behavior of a function as its input approaches a particular value.
  • Understanding limits is crucial for defining derivatives and continuity.
• Derivatives:
  • The derivative of a function at a point represents the instantaneous rate of change of the function at that point.
  • It measures the slope of the tangent line to the graph of the function at that point.
  • The derivative can be calculated using various rules, such as the power rule, product rule, quotient rule, chain rule, and more.
• Applications of Derivatives:
  • Finding maximum and minimum values of functions (optimization problems).
  • Determining the concavity and points of inflection of functions (curve sketching).
  • Solving related rates problems.
  • Estimating values of functions using linear approximations.
• Common Derivatives:
  • The derivative of xⁿ is nx^n-1.
  • The derivative of e^x is e^x.
  • The derivative of sin(x) is cos(x).
  • The derivative of cos(x) is -sin(x).

Integral Calculus

• Indefinite Integrals:
  • An indefinite integral represents a family of antiderivatives of a function.
  • It's essentially the reverse process of differentiation.
  • An antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x).
  • The indefinite integral of f(x) is written as ∫f(x) dx.
• Definite Integrals:
  • A definite integral represents the area under a curve between two specific points.
  • It is used to calculate areas, volumes, and other quantities that involve accumulation.
  • The definite integral of f(x) from a to b is written as ∫_a^b f(x) dx.
• Fundamental Theorem of Calculus:
  • The fundamental theorem of calculus establishes a connection between differentiation and integration.
  • Part 1 relates the definite integral to antiderivatives.
  • Part 2 provides a method for evaluating definite integrals.
• Applications of Integrals:
  • Calculating areas bounded by curves.
  • Determining volumes of solids of revolution.
  • Calculating work done by a variable force.
  • Solving differential equations.

Further Concepts

• Implicit Differentiation:
  • Finding derivatives when the relationship between the variables isn't explicitly given in the form y = f(x).
• Related Rates:
  • Problems involving rates of change of different quantities related to each other by an equation.
• Optimization Problems:
  • Finding maximum or minimum values of functions under certain constraints.
• Differential Equations:
  • Equations that involve rates of change, derivatives, and solutions.
• Sequences and Series:
  • These often appear in applications of calculus, especially in infinite sums or convergence of functions.

Techniques

• Integration by Substitution:- A method for evaluating integrals that involve a complex function by substituting a suitable variable.
• Integration by Parts:- A method to evaluate certain integration integrals, used when the integrand can be split into differentiated and integrated parts.
• Partial Fractions:- This technique decomposes a rational function into simpler fractions for integration.
• Trigonometric Substitution:- A technique to evaluate integrals that involve trigonometric functions or expressions involving square roots of expressions.