Calculus: Functions and Limits

• 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.
• Differential calculus concerns instantaneous rates of change and slopes of curves.
• Integral calculus concerns accumulation of quantities and the areas under or between curves.
• Calculus is used in science, engineering, and economics.

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 input to a function is known as an argument.
• Functions are typically denoted by letters such as f, g, and h.
• If x is the input to the function f, then the output is denoted by f(x).
• The set of all permissible inputs to a function is called its domain.
• The set of all possible outputs of a function is called its range.

Limits

• A limit describes the behavior of a function near a point, rather than at the point itself.
• The limit of a function f(x) as x approaches c is L, written as lim (x→c) f(x) = L, means that the values of f(x) can be made arbitrarily close to L by taking x to be sufficiently close to c, but not necessarily equal to c.
• Limits are essential for defining continuity, derivatives, and integrals.
• Limits can be evaluated numerically, graphically, or algebraically.
• Indeterminate forms such as 0/0 or ∞/∞ often require L'Hôpital's Rule to evaluate limits.

Derivatives

• The derivative of a function measures the instantaneous rate of change of the function with respect to one of its variables.
• Geometrically, the derivative at a point represents the slope of the tangent line to the graph of the function at that point.
• If y = f(x), the derivative of f with respect to x is written as f'(x), dy/dx, or Df(x).
• The derivative is defined as the limit of the difference quotient: f'(x) = lim (h→0) [f(x+h) - f(x)] / h.
• Common derivative rules include the power rule (d/dx x^n = nx^(n-1)), the product rule (d/dx (uv) = u'v + uv'), the quotient rule (d/dx (u/v) = (u'v - uv') / v^2), and the chain rule (d/dx f(g(x)) = f'(g(x)) * g'(x)).
• Higher-order derivatives can be found by differentiating the derivative. The second derivative, f''(x), represents the rate of change of the slope of the original function.
• Derivatives are used to find critical points (where f'(x) = 0 or is undefined), which can be local maxima, local minima, or saddle points.
• The first derivative test uses the sign of f'(x) to determine if a critical point is a local maximum or minimum.
• The second derivative test uses the sign of f''(x) to determine if a critical point is a local maximum (f''(x) < 0) or minimum (f''(x) > 0).
• Derivatives are used in optimization problems to find the maximum or minimum values of a function subject to constraints.

Integrals

• Integration is the inverse operation of differentiation.
• It involves finding a function whose derivative is the given function.
• If F'(x) = f(x), then F(x) is an antiderivative of f(x).
• The indefinite integral of f(x) is denoted as ∫f(x) dx = F(x) + C, where C is the constant of integration.
• Common integration rules include the power rule (∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1), the integral of 1/x (∫(1/x) dx = ln|x| + C), and integrals of trigonometric functions (e.g., ∫sin(x) dx = -cos(x) + C).
• Techniques of integration include substitution (u-substitution), integration by parts (∫u dv = uv - ∫v du), partial fractions, and trigonometric substitution.
• Definite integrals have limits of integration, denoted as ∫(a to b) f(x) dx, and represent the area under the curve of f(x) from x = a to x = b.
• The Fundamental Theorem of Calculus relates differentiation and integration: ∫(a to b) f(x) dx = F(b) - F(a), where F(x) is an antiderivative of f(x).
• Integrals are used to calculate areas, volumes, arc lengths, surface areas, and work.

Applications of Derivatives

• Related Rates: Using derivatives to find the rate at which a quantity is changing by relating it to other quantities whose rates of change are known.
• Optimization: Finding the maximum or minimum value of a function, often subject to constraints.
• Curve Sketching: Using the first and second derivatives to analyze the behavior of a function and sketch its graph.
• Linear Approximation: Using the tangent line to approximate the value of a function near a point.

Applications of Integrals

• Area Between Curves: Calculating the area between two curves by integrating the difference of the functions.
• Volume of Solids of Revolution: Finding the volume of a solid generated by rotating a region about an axis. Methods include the disk method, the washer method, and the cylindrical shell method.
• Arc Length: Calculating the length of a curve using integration.
• Surface Area: Finding the surface area of a solid of revolution.
• Average Value of a Function: Calculating the average value of a function over an interval.
• Work: Calculating the work done by a force over a distance.

Infinite Series

• An infinite series is the sum of an infinite sequence of terms.
• It is represented as Σ (n=1 to ∞) a_n = a_1 + a_2 + a_3 + ....
• The sequence of partial sums (S_n) is defined as S_n = Σ (k=1 to n) a_k.
• If the sequence of partial sums converges to a finite limit S, then the series converges and its sum is S. Otherwise, the series diverges.
• Tests for convergence include the divergence test, the integral test, the comparison test, the limit comparison test, the ratio test, the root test, and the alternating series test.
• A power series is a series of the form Σ (n=0 to ∞) c_n(x-a)^n, where c_n are constants and a is the center of the series.
• The interval of convergence of a power series is the set of all x values for which the series converges.
• Taylor series and Maclaurin series are power series representations of functions.
• The Taylor series of a function f(x) about x = a is Σ (n=0 to ∞) f^(n)(a) / n!^n, where f^(n)(a) is the nth derivative of f evaluated at a.
• The Maclaurin series is the Taylor series centered at a = 0.

Multivariable Calculus

• Extends the concepts of calculus to functions of multiple variables.
• Includes partial derivatives, multiple integrals, gradient, divergence, curl, and line integrals.
• Partial derivatives measure the rate of change of a function with respect to one variable, holding others constant.
• The gradient is a vector of partial derivatives and points in the direction of the greatest rate of increase of a function.
• Multiple integrals are used to calculate volumes, masses, and other quantities in higher dimensions.
• Line integrals are used to integrate functions along curves in space.
• Key theorems include Green's theorem, Stokes' theorem, and the divergence theorem, which relate integrals over regions to integrals over their boundaries.