Introduction to Calculus

• Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.
• There are two major branches of calculus: differential calculus and integral calculus.

Differential Calculus

• Concerns instantaneous rates of change and slopes of curves.
• The derivative is a central tool in differential calculus, representing the instantaneous rate of change of a function.
• Differentiation is the process of finding the derivative of a function.
• Derivatives have applications in optimization problems (finding maxima and minima), curve sketching, and related rates problems.

Integral Calculus

• Deals with the accumulation of quantities and the areas under and between curves.
• The integral is a central tool, representing the accumulation of a function over an interval.
• Integration is the process of finding the integral of a function.
• Integrals have applications in finding areas, volumes, work, and average values of functions.
• The definite integral calculates the net signed area between a function and the x-axis over a specific interval.
• The indefinite integral represents the family of antiderivatives of a function.

Limits

• The concept of a limit underlies both differential and integral calculus.
• A limit describes the value that a function approaches as the input approaches some value.
• Limits are essential for defining continuity, derivatives, and integrals rigorously.

Functions

• Calculus operates on functions, which are relationships between inputs and outputs.
• Functions can be represented graphically, algebraically, or numerically.
• Common types of functions in calculus include polynomial, trigonometric, exponential, and logarithmic functions.

Derivatives

• The derivative of a function f(x) at a point x represents the slope of the tangent line to the graph of f(x) at that point.
• The formal definition of the derivative is: f'(x) = lim (h->0) [f(x+h) - f(x)] / h
• Notation for derivatives includes f'(x), dy/dx, and d/dx [f(x)].
• Rules of differentiation:
  • Power Rule: d/dx (x^n) = nx^(n-1)
  • Constant Multiple Rule: d/dx [cf(x)] = c * d/dx [f(x)]
  • Sum/Difference Rule: d/dx [f(x) ± g(x)] = d/dx [f(x)] ± d/dx [g(x)]
  • Product Rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
  • Quotient Rule: d/dx [f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]^2
  • Chain Rule: d/dx [f(g(x))] = f'(g(x)) * g'(x)
• Higher-order derivatives: The second derivative f''(x) is the derivative of the first derivative f'(x); the third derivative f'''(x) is the derivative of the second derivative, and so on.

Integrals

• The integral of a function f(x) represents the area under the curve of f(x).
• The definite integral of f(x) from a to b is denoted as ∫ab f(x) dx.
• The indefinite integral of f(x) is denoted as ∫ f(x) dx and represents the family of antiderivatives of f(x).
• The Fundamental Theorem of Calculus connects differentiation and integration:
  • Part 1: If F(x) = ∫ax f(t) dt, then F'(x) = f(x).
  • Part 2: ∫ab f(x) dx = F(b) - F(a), where F(x) is any antiderivative of f(x).
• Rules of integration:
  • Power Rule: ∫ x^n dx = (x^(n+1)) / (n+1) + C (where n ≠ -1)
  • Constant Multiple Rule: ∫ cf(x) dx = c ∫ f(x) dx
  • Sum/Difference Rule: ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
• Integration techniques:
  • Substitution: Used to simplify integrals by substituting a part of the integrand with a new variable.
  • Integration by parts: Used to integrate products of functions: ∫ u dv = uv - ∫ v du
  • Partial fractions: Used to integrate rational functions by decomposing them into simpler fractions.

Applications

• Physics: Calculus is used extensively in mechanics (motion, velocity, acceleration), electromagnetism, and other areas.
• Engineering: Calculus is used in designing structures, circuits, and control systems.
• Economics: Calculus is used in modeling economic growth, optimization, and marginal analysis.
• Computer science: Calculus is used in computer graphics, machine learning, and optimization algorithms.
• Statistics: Calculus is used in probability theory and statistical modeling.

Infinite Series

• An infinite series is an expression of the form ∑(n=1 to ∞) a_n = a_1 + a_2 + a_3 + ...
• Convergence: A series converges if its sequence of partial sums approaches a finite limit.
• Divergence: A series diverges if its sequence of partial sums does not approach a finite limit.
• Tests for convergence:
  • Integral Test
  • Comparison Test
  • Limit Comparison Test
  • Ratio Test
  • Root Test
  • Alternating Series Test
• Power Series: A power series is a series of the form ∑(n=0 to ∞) c_n(x - a)^n, where c_n are coefficients and a is a constant.
• Taylor Series: A Taylor series is a power series representation of a function f(x) centered at x = a: f(x) = ∑(n=0 to ∞) [f^(n)(a) / n!] (x - a)^n
• Maclaurin Series: A Maclaurin series is a Taylor series centered at x = 0: f(x) = ∑(n=0 to ∞) [f^(n)(0) / n!] x^n

Multivariable Calculus

• Extends the concepts of calculus to functions of multiple variables.
• Partial Derivatives: The derivative of a function with respect to one variable, holding other variables constant.
• Gradients: A vector containing the partial derivatives of a function.
• Multiple Integrals: Integrals of functions over regions in two or more dimensions.
• Applications in optimization, physics, and engineering.