Differential Calculus: An Overview

• Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.
• It has two major branches: differential calculus and integral calculus, which are related by the fundamental theorem of calculus.
• Calculus is used extensively in science, engineering, and economics.

Differential Calculus

• Focuses on the rate of change of functions and the slope of curves.
• Key concepts include derivatives, which measure the instantaneous rate of change of a function.
• Techniques like finding limits are essential for defining derivatives.
• Derivatives are used to find maxima and minima of functions, analyze motion, and model rates of change.
• The derivative of a function f(x) is denoted as f'(x) or df/dx.
• Common rules for differentiation include the power rule, product rule, quotient rule, and chain rule.
• The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).
• The product rule states that if h(x) = f(x)g(x), then h'(x) = f'(x)g(x) + f(x)g'(x).
• The quotient rule states that if h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2.
• The chain rule states that if h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).
• Higher-order derivatives (second derivative, third derivative, etc.) describe the rate of change of the rate of change.
• Critical points of a function are points where the derivative is zero or undefined; these points are candidates for local maxima or minima.

Integral Calculus

• Deals with the accumulation of quantities and the areas under and between curves.
• Involves finding integrals, also known as antiderivatives.
• The integral of a function f(x) is denoted as ∫f(x) dx.
• The fundamental theorem of calculus links differentiation and integration, stating that the derivative of the integral of a function is the original function.
• Techniques of integration include substitution, integration by parts, partial fractions, and trigonometric substitution.
• Substitution involves changing the variable of integration to simplify the integral.
• Integration by parts is used to integrate products of functions and is based on the formula ∫u dv = uv - ∫v du.
• Definite integrals have upper and lower limits of integration and represent the net area under a curve between those limits.
• Indefinite integrals do not have limits of integration and result in a family of functions that differ by a constant.
• Applications of integration include finding areas, volumes, arc lengths, and surface areas of solids of revolution.
• Improper integrals involve integrating over an unbounded interval or integrating a function with a singularity.

Limits

• A limit describes the value that a function approaches as the input approaches some value.
• Formally, the limit of f(x) as x approaches c is L if, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.
• Limits are essential for defining continuity, derivatives, and integrals.
• Techniques for evaluating limits include direct substitution, factoring, rationalizing, and using L'Hôpital's rule.
• L'Hôpital's rule states that if the limit of f(x)/g(x) as x approaches c is of the form 0/0 or ∞/∞, then the limit is equal to the limit of f'(x)/g'(x).
• One-sided limits consider the behavior of a function as x approaches a value from the left or the right.
• Infinite limits occur when the value of a function grows without bound as x approaches a certain value.

Functions

• A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• Functions can be represented graphically, algebraically, or numerically.
• Common types of functions include polynomial functions, trigonometric functions, exponential functions, and logarithmic functions.
• Polynomial functions have the form f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0.
• Trigonometric functions include sine, cosine, tangent, cotangent, secant, and cosecant, which relate angles of a triangle to the ratios of its sides.
• Exponential functions have the form f(x) = a^x, where a is a constant base.
• Logarithmic functions are the inverse of exponential functions and have the form f(x) = log_a(x).
• The domain of a function is the set of all possible input values, and the range is the set of all possible output values.
• Composition of functions involves combining two functions such that the output of one function becomes the input of the other.
• Inverse functions "undo" the action of a function; if f(a) = b, then f^{-1}(b) = a.

Infinite Series

• An infinite series is the sum of an infinite sequence of numbers.
• Series can be convergent (the sum approaches a finite value) or divergent (the sum does not approach a finite value).
• Tests for convergence include the ratio test, root test, integral test, comparison test, and alternating series test.
• The ratio test considers the limit of the ratio of consecutive terms; if the limit is less than 1, the series converges.
• The root test considers the limit of the nth root of the absolute value of the terms; if the limit is less than 1, the series converges.
• The integral test compares the series to an integral; if the integral converges, the series converges.
• The comparison test compares the series to another series whose convergence is known.
• The alternating series test applies to series with alternating signs; the series converges if the terms decrease in magnitude and approach zero.
• Taylor series represent functions as an infinite sum of terms involving derivatives evaluated at a single point.
• Maclaurin series are a special case of Taylor series centered at zero.
• Power series have the form ∑ c_n (x - a)^n, where c_n are coefficients and a is the center.
• The interval of convergence is the set of x-values for which the power series converges.