Introduction to Differential Calculus

• Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.
• It provides tools for understanding continuous change in mathematics and other sciences.
• Calculus has two major branches: differential calculus and integral calculus.

Differential Calculus

• Focuses on the concept of the derivative.
• The derivative measures the instantaneous rate of change of a function.
• Geometrically, the derivative represents the slope of the tangent line to a function's graph at a specific point.
• Differentiation is the process of finding the derivative of a function.
• Derivatives are used to find maxima and minima of functions.
• They help analyze the behavior of functions, such as where they are increasing or decreasing.
• The derivative of a function f(x) is often denoted as f'(x) or df/dx.
• Key rules in differential calculus 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 is used to find the derivative of the product of two functions: (uv)' = u'v + uv'.
• The quotient rule is used to find the derivative of the quotient of two functions: (u/v)' = (u'v - uv')/v^2.
• The chain rule is used to find the derivative of a composite function: [f(g(x))]' = f'(g(x)) * g'(x).
• Higher-order derivatives can be computed by differentiating the derivative: f''(x), f'''(x), etc.
• Second derivatives are used to determine the concavity of a function's graph.

Integral Calculus

• Focuses on the concept of the integral.
• Integration is the inverse process of differentiation.
• Integrals are used to find the area under a curve.
• Definite integrals compute the area over a specific interval [a, b].
• Indefinite integrals represent a family of functions whose derivative is the integrand.
• The indefinite integral of f(x) is denoted as ∫f(x) dx, and it includes a constant of integration, C.
• The definite integral of f(x) from a to b is denoted as ∫[a to b] f(x) dx.
• The fundamental theorem of calculus links differentiation and integration.
• It states that the derivative of the integral of a function is the original function.
• It also provides a method for evaluating definite integrals using antiderivatives.
• Techniques of integration include substitution, integration by parts, trigonometric substitution, and partial fractions.
• Integration by substitution involves changing the variable of integration to simplify the integral.
• Integration by parts is used to integrate the product of two functions: ∫u dv = uv - ∫v du.
• Integrals have applications in finding volumes of solids, arc lengths of curves, and surface areas of solids of revolution.

Limits

• Limits are foundational to calculus, describing the behavior of a function as its input approaches a certain value.
• The limit of f(x) as x approaches 'a' is written as lim (x→a) f(x) = L.
• This means as x gets arbitrarily close to 'a', f(x) gets arbitrarily close to L.
• Limits can be one-sided, approaching 'a' from the left (a-) or the right (a+).
• For a limit to exist, both one-sided limits must exist and be equal.
• Indeterminate forms, such as 0/0 or ∞/∞, require special techniques like L'Hôpital's Rule to evaluate the limit.
• L'Hôpital's Rule states that if lim (x→a) f(x)/g(x) is in an indeterminate form, then lim (x→a) f(x)/g(x) = lim (x→a) f'(x)/g'(x), provided the latter limit exists.
• Limits are used to define continuity: a function f(x) is continuous at x = a if lim (x→a) f(x) = f(a).

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.
• The domain of a function is the set of all possible input values (x-values).
• The range of a function is the set of all possible output values (y-values).
• Functions can be represented graphically, algebraically, or numerically.
• Common types of functions include polynomial functions, trigonometric functions, exponential functions, and logarithmic functions.
• Polynomial functions are sums of terms involving non-negative integer powers of x (e.g., f(x) = 3x^2 + 2x - 1).
• Trigonometric functions include sine (sin x), cosine (cos x), tangent (tan x), and their reciprocals.
• Exponential functions have the form f(x) = a^x, where 'a' is a constant base.
• Logarithmic functions are the inverse of exponential functions, denoted as f(x) = log_a(x).
• Composite functions are formed by applying one function to the result of another: f(g(x)).

Infinite Series

• An infinite series is the sum of an infinite sequence of terms.
• Series can be convergent (sum approaches a finite value) or divergent (sum does not approach a finite value).
• Geometric series have the form a + ar + ar^2 + ar^3 + ..., where 'a' is the first term and 'r' is the common ratio.
• A geometric series converges if |r| < 1, and its sum is a/(1-r).
• Harmonic series have the form 1 + 1/2 + 1/3 + 1/4 + .... The harmonic series diverges.
• Tests for convergence include the ratio test, root test, comparison test, and alternating series test.
• The ratio test examines the limit of the ratio of consecutive terms to determine convergence.
• The root test examines the limit of the nth root of the absolute value of the terms.
• The comparison test compares a given series to a known convergent or divergent series.
• The alternating series test applies to series with alternating signs and decreasing term magnitudes.
• Power series have the form Σ[n=0 to ∞] c_n(x-a)^n, where c_n are coefficients and 'a' is the center of the series.
• Taylor series represent a function as an infinite sum of terms involving its derivatives at a single point.
• Maclaurin series are a special case of Taylor series centered at x = 0.