Calculus: Differential Calculus

• Calculus is a branch of mathematics focused on continuous change
• It has two major branches: differential calculus and integral calculus
• These branches are related by the fundamental theorem of calculus

Differential Calculus

• Deals with the instantaneous rate of change (derivatives) of functions
• A derivative measures how a function changes as its input changes
• Geometrically, the derivative at a point is the slope of the tangent line to the function's graph at that point
• Key concepts include:
  • Limits: The foundation of calculus, describing the value that a function approaches as the input approaches some value
  • Derivatives: Measures the instantaneous rate of change of a function
  • Differentiation rules: Techniques for finding derivatives of various types of functions (power rule, product rule, quotient rule, chain rule)
  • Applications: Optimization (finding maxima and minima), related rates, curve sketching, and analysis of function behavior

Integral Calculus

• Deals with the accumulation of quantities and areas under curves (integrals)
• Integration is the reverse process of differentiation
• Key concepts include:
  • Antiderivatives: A function whose derivative is the given function
  • Definite integrals: Represents the area under a curve between two limits
  • Indefinite integrals: Represents the family of all antiderivatives of a function
  • Integration techniques: Methods for evaluating integrals (substitution, integration by parts, partial fractions)
  • Applications: Finding areas, volumes, average values, work, and solving differential equations

Fundamental Theorem of Calculus

• Connects differentiation and integration
• Part 1: The derivative of the integral of a function is the original function itself
• Part 2: The definite integral of a function can be evaluated by finding the antiderivative of the function at the upper and lower limits of integration and subtracting the values
• This theorem provides a method to compute definite integrals without directly calculating areas

Limits

• The limit of a function f(x) as x approaches a value 'c' is the value that f(x) gets closer and closer to as x gets closer and closer to 'c'
• It is written as lim (x→c) f(x) = L, where L is the limit
• Limits are essential for defining continuity, derivatives, and integrals
• Techniques for evaluating limits include:
  • Direct substitution: Plugging in the value 'c' into the function
  • Factoring and simplifying: Algebraic manipulation to remove indeterminate forms (0/0)
  • L'Hôpital's Rule: If the limit results in an indeterminate form (0/0 or ∞/∞), take the derivative of the numerator and denominator separately and then evaluate the limit.

Derivatives

• The derivative of a function f(x) is denoted as f'(x) or df/dx
• It represents the instantaneous rate of change of f(x) with respect to x
• Definition: f'(x) = lim (h→0) [f(x+h) - f(x)] / h
• Common derivative rules:
  • Power Rule: d/dx (x^n) = nx^(n-1)
  • Constant Multiple Rule: d/dx [cf(x)] = cf'(x)
  • Sum/Difference Rule: d/dx [f(x) ± g(x)] = f'(x) ± 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)

Integrals

• The integral of a function f(x) is denoted as ∫f(x) dx for indefinite integrals and ∫[a to b] f(x) dx for definite integrals
• Indefinite integrals represent the family of all antiderivatives of f(x)
• Definite integrals represent the net area under the curve of f(x) from a to b
• Common integration rules:
  • 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
  • Substitution Rule (Reverse Chain Rule): ∫f(g(x))g'(x) dx = F(g(x)) + C, where F'(x) = f(x)
  • Integration by Parts: ∫u dv = uv - ∫v du

Applications of Differential Calculus

• Optimization: Finding maximum and minimum values of functions
  • Set the derivative equal to zero and solve for critical points
  • Use the first or second derivative test to determine whether the critical points are maxima, minima, or inflection points
• Related Rates: Determining how the rate of change of one variable affects the rate of change of another variable
  • Differentiate an equation relating the variables with respect to time
  • Substitute known rates and solve for the unknown rate
• Curve Sketching: Analyzing the behavior of a function to sketch its graph
  • Find the domain, intercepts, asymptotes, critical points, and inflection points
  • Determine intervals of increasing/decreasing and concavity

Applications of Integral Calculus

• Area between curves: Finding the area between two or more curves
  • Integrate the difference between the functions over the interval of interest
• Volume of solids of revolution: Finding the volume of a solid formed by rotating a region around an axis
  • Use the disk, washer, or shell method
• Average Value of a Function: Calculating the average value of a function over an interval
  • Average Value = (1/(b-a)) ∫[a to b] f(x) dx
• Work: Calculating the work done by a force moving an object
  • Work = ∫F(x) dx, where F(x) is the force function

Sequences and Series

• A sequence is an ordered list of numbers, while a series is the sum of the terms in a sequence
• Convergence: A sequence or series converges if it approaches a finite limit as the number of terms increases
• Divergence: A sequence or series diverges if it does not approach a finite limit
• Tests for convergence and divergence of series include:
  • Integral Test
  • Comparison Test
  • Limit Comparison Test
  • Ratio Test
  • Root Test
  • Alternating Series Test

Multivariable Calculus

• Extends the concepts of calculus to functions of multiple variables
• Partial derivatives: Derivatives of a function with respect to one variable, holding the other variables constant
• Gradients: Vector of partial derivatives, pointing in the direction of the greatest rate of increase
• Multiple integrals: Integrals over regions in two or three dimensions
• Applications: Optimization, surface integrals, volume integrals, vector calculus