Analytic Geometry: Formulas, Lines, Circles & Conics
Analytic geometry is the branch of mathematics that describes geometric shapes with algebra by placing them on a coordinate grid, so a point becomes a pair of numbers and a curve becomes an equation. Its core tools are the distance formula d = √((x₂ - x₁)² + (y₂ - y₁)²), the slope m = (y₂ - y₁)/(x₂ - x₁), the equation of a line, and the circle (x - h)² + (y - k)² = r². Because it turns curves into equations, it is the bridge that made calculus possible.
Alternating Series Test: Leibniz Rule & Examples
The Alternating Series Test, also called the Leibniz Test, says that a series whose signs flip, ∑ (-1)ⁿ bₙ or ∑ (-1)ⁿ⁺¹ bₙ with every bₙ > 0, converges when two things hold: the terms bₙ are eventually decreasing, and lim n→∞ bₙ = 0. When both are true, the sum settles on a finite value, and the error from stopping at the Nth term is never larger than the first term you left out, S - S N ≤ b N+1.
Alternating Series: Test, Convergence & Error Bound
An alternating series is a sum whose terms flip sign, written ∑ (-1)ⁿ bₙ or ∑ (-1)ⁿ⁺¹ bₙ with bₙ ≥ 0. The Alternating Series Test says such a sum converges when the sizes bₙ decrease and shrink to zero: b n+1 ≤ bₙ and lim n → ∞ bₙ = 0. The alternating harmonic series 1 - 12 + 13 - 14 + … converges to ln 2, and if you stop after N terms the error is no bigger than the first term you dropped, R N ≤ b N+1.
Absolute Value Functions: Graph, Derivative & Integral
Absolute Value Functions measure distance from zero, so f(x) = |x| equals x when x ≥ 0 and -x when x < 0. The graph is a V with its vertex at the origin, domain all real numbers, and range y ≥ 0. In calculus the surprise is that |x| is continuous everywhere yet not differentiable at x = 0, where the V has a sharp corner; away from that corner its derivative is sgn(x) and its integral is (x|x|)/2 + C.
Trigonometric Substitution: Rules & Examples
Trigonometric substitution is an integration method that replaces a square root of a quadratic with a trig function, so a Pythagorean identity can clear the root. Match the radical to its substitution: √(a²-x²) takes x=asinθ, √(a²+x²) takes x=atanθ, and √(x²-a²) takes x=asecθ. After integrating in θ, a reference triangle converts the answer back to x.
What Is Calculus? Definition, Branches & Examples
What is calculus? It is the mathematics of continuous change. It has two branches that mirror each other: differential calculus, which measures how fast something changes (the derivative, the slope of a curve), and integral calculus, which measures how much accumulates (the integral, the area under a curve). Both rest on one idea, the limit, and the Fundamental Theorem of Calculus proves the two branches are inverses of each other.
Taylor Series: Formula, Examples And Convergence
A Taylor series rewrites a function as an infinite sum of power terms built from the function's derivatives at a single point a: f(x) = ∑ n=0 ∞ (f (n) (a))/(n factorial)(x-a)ⁿ. When the centre is a = 0 the same formula is called a Maclaurin series. Truncating the sum after a few terms is how calculators evaluate e x, sin x, and cos x.
Trapezoidal Rule: Formula, Examples & Error
The Trapezoidal Rule estimates a definite integral ∫ₐ b f(x),dx by slicing [a,b] into n strips of width h=(b-a)/n and treating each strip as a trapezoid, giving Tₙ=h/2[f(x₀)+2f(x₁)+2f(x₂)+…+2f(x n-1)+f(xₙ)]. It is the simplest reliable way to find an area under a curve when the integral has no clean antiderivative, and its error shrinks like 1/(n²) as you add strips.
Taylor Polynomials: Formula, Examples & Error Term
Taylor polynomials approximate a function f near a chosen point a by matching the function's value and its first n derivatives at that point. The degree-n Taylor polynomial is Pₙ(x)=∑ k=0 ⁿ (f (k) (a))/(k factorial)(x-a) k; when the center is a=0 it is called a Maclaurin polynomial. Near a the polynomial hugs the curve, and the Lagrange remainder measures how far off it can be.
Squeeze Theorem: Statement, Proof & Examples
The Squeeze Theorem says that if a function f is trapped between two functions g and h near a point a, and both g and h approach the same limit L there, then f must approach L too. It is the standard tool for limits you cannot compute directly, and it is what proves the two cornerstone results lim x → 0 x² sin1/x = 0 and lim x → 0 (sin x)/x = 1.
Rolle's Theorem: Statement, Proof & Examples
Rolle's Theorem states that if a function f is continuous on the closed interval [a,b], differentiable on the open interval (a,b), and f(a) = f(b), then there is at least one point c in (a,b) where f'(c) = 0. Geometrically, a smooth curve that starts and ends at the same height must have a horizontal tangent somewhere in between. It is the special case of the Mean Value Theorem where the two endpoints sit at equal height.
Riemann Sums: Left, Right And Midpoint Explained
Riemann sums approximate the area under a curve by slicing it into n thin rectangles and adding their areas: ∑ i=1 ⁿ f(xᵢ^*),Δ x, where Δ x = (b-a)/n. The left, right, and midpoint versions differ only in where you sample the height of each rectangle, and as n → ∞ every version converges to the exact definite integral ∫ₐ b f(x),dx.
Rational Functions: Asymptotes, Holes & Limits
Rational functions are ratios of two polynomials, f(x) = (P(x))/(Q(x)) with Q(x) ≠ 0. Their whole behaviour follows three checks: the domain excludes the zeros of Q; a zero of Q that does not cancel gives a vertical asymptote, while one that does cancel gives a hole; and comparing the degrees of P and Q gives the horizontal asymptote (or a slant one when P is exactly one degree higher). Limits describe each asymptote precisely, and the quotient rule differentiates the function.
Rate Of Change: Average vs Instantaneous
Rate of change measures how fast one quantity changes with respect to another. The average rate of change over an interval [a, b] is (f(b) - f(a))/(b - a), the slope of the secant line. The instantaneous rate of change at a point is the derivative f'(x) = lim h → 0 (f(x+h) - f(x))/h, the slope of the tangent line. Units always carry meaning: metres per second, dollars per unit, or square centimetres per second.
Quotient Rule: Formula, Proof & Examples
The quotient rule differentiates a ratio of two functions. For f(x) = (u(x))/(v(x)) with v(x) ≠ 0, the derivative is f'(x) = (u'v - uv')/(v²): the low (denominator) times the derivative of the high (numerator), minus the high times the derivative of the low, all over the low squared. The subtraction sign and the order of the two terms are what most students get wrong.
Power Series: Radius & Interval Of Convergence
A power series centered at a is an infinite sum ∑ n=0 ∞ cₙ (x-a)ⁿ, a polynomial with endlessly many terms. It converges to a finite value only on an interval around a, whose half-width is the radius of convergence R, found with the ratio test. The plainest example is the geometric power series ∑ n=0 ∞ xⁿ = 1/(1-x), which converges for |x| < 1 (so R = 1).
Product Rule: Formula, Proof, And Examples
The product rule gives the derivative of two functions multiplied together: if f(x) = u(x),v(x), then f'(x) = u'(x),v(x) + u(x),v'(x), often written (uv)' = u'v + uv'. The derivative of a product is not the product of the derivatives, and this article proves why from the limit definition and from the area of an expanding rectangle.
Power Rule in Calculus: Formula, Proof, Examples
The Power Rule is the calculus shortcut for differentiating a variable raised to a fixed power: d/dx[xⁿ] = n,xⁿ⁻¹. In words, multiply by the exponent, then drop the exponent by one. It works for every real n (positive, negative, and fractional), and it is a rule about the derivative of xⁿ, not the algebra rule for simplifying exponents.
Polynomial Functions: Degree, End Behavior & Calculus
Polynomial functions have the form f(x)=aₙ xⁿ + … + a₁ x + a₀, and their whole shape is fixed by two numbers: the degree n and the leading coefficient aₙ. Every polynomial is continuous and differentiable across all real numbers, so calculus reads it cleanly: the derivative f'(x) is a polynomial of degree n-1, its sign tells you where the graph rises or falls, and the second derivative tells you where it curves.
Piecewise Functions: Graphs, Continuity & Examples
Piecewise functions use different sub-rules on different parts of the domain, written with a single brace as f(x) = cases … cases. To evaluate one, pick the rule whose condition matches the input. At a breakpoint x = c the function is continuous when the left limit, the right limit, and f(c) all agree, and differentiable when the one-sided derivatives also agree. Continuity does not guarantee differentiability: |x| is continuous at 0 but has a corner there.
Parametric Equations: Slope, Arc Length & Examples
Parametric equations describe a curve by giving both coordinates as functions of a third variable, a parameter t: x = f(t) and y = g(t). You can often eliminate t to recover a Cartesian equation, and calculus works directly on the parameter: the slope is dy/dx = (dy/dt)/(dx/dt) (whenever dx/dt ≠ 0), and arc length is L = ∫ₐ b √((dx/dt)² + (dy/dt)²), dt.
Natural Logarithm: Definition, Derivative & Rules
The natural logarithm, written ln x, is the logarithm with base e ≈ 2.71828, so ln x = log e x. In calculus it is defined as the area under the curve y = 1/t from t=1 to t=x, which is why ln 1 = 0 and ln e = 1. Its derivative is the cleanest in all of calculus, d/dx[ln x] = 1/x, and it is the antiderivative that fills the gap left by the power rule: ∫ 1/x,dx = ln|x| + C.
Newtons Method: Formula, Steps & Examples
Newtons Method is an iterative root-finding technique: from a starting guess x₀, each step follows the tangent line of f down to the x-axis to produce a better estimate, using x n+1 = xₙ - (f(xₙ))/(f'(xₙ)). When it works it converges quadratically, roughly doubling the number of correct digits every step. It can fail when f'(xₙ) = 0, when the starting guess is poor, or when the estimates cycle or run off to infinity.
Monotonic Functions: Increasing And Decreasing
Monotonic Functions move in one direction across an interval: they are either entirely non-decreasing or entirely non-increasing, never both. You test monotonicity with the first derivative, where f'(x) > 0 means increasing, f'(x) < 0 means decreasing, and f'(x) = 0 marks a stationary candidate. A strictly monotonic function is one-to-one, which is exactly what makes it invertible.
Mean Value Theorem: Statement, Proof & Examples
The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there is at least one point c in (a,b) where the instantaneous rate of change equals the average rate of change: f'(c) = (f(b) - f(a))/(b - a). Geometrically, some tangent line is parallel to the secant line joining the endpoints, and Rolle's theorem is the special case where f(a) = f(b).
Moments And Center Of Mass In Calculus
Moments measure how much a mass, or a spread-out region, tends to turn about an axis: the moment of a point mass about an axis is its mass times its distance from that axis. For a flat region under a curve f(x) on [a,b] with constant density, the moment about the y-axis is M y=∫ₐ b x,f(x),dx, the moment about the x-axis is Mₓ=∫ₐ b 1/2[f(x)]²,dx, the mass is m=∫ₐ b f(x),dx, and the center of mass sits at x=M y/m, y=Mₓ/m.
Mathematical Induction: Proof Method & Examples
Mathematical induction is a method for proving that a statement P(n) is true for every integer n ≥ n₀. You prove two things: the base case P(n₀) is true, and the inductive step, that whenever P(k) is true, P(k+1) is true as well. Those two facts together force the statement to hold for the whole infinite chain, the way one falling domino topples an endless line.
Maxima and Minima: Tests, Examples & Rules
Maxima and minima are the highest and lowest values a function reaches, either near a point (local) or across its whole domain (absolute). You find them at critical points, where the derivative f'(x) equals zero or is undefined, then classify each one with the first derivative test (sign change of f') or the second derivative test (f''(c)<0 gives a maximum, f''(c)>0 gives a minimum). For an absolute maximum or minimum on a closed interval, compare the function's values at every critical point against its values at the two endpoints.
Maclaurin Series: Formula And Key Expansions
The Maclaurin Series rewrites a function as an infinite polynomial built from its derivatives at 0: f(x)=∑ n=0 ∞ (f (n) (0))/(n factorial)xⁿ. It is the special case of the Taylor series centred at zero, and it is how a calculator turns sin x, cos x, and e x into additions it can actually perform, as long as x stays inside the interval of convergence.
Logarithmic Derivative: Formula & Examples
The logarithmic derivative of a function f is (f'(x))/(f(x)), which equals d/dx[ln f(x)] and measures the relative (percentage) rate of change of f. Logarithmic differentiation is the technique that puts this to work: take the natural log of both sides, use log laws to turn products, quotients, and powers into sums, differentiate implicitly, then solve for f'(x). It is the cleanest way to differentiate messy products and any function with a variable in both the base and the exponent, such as x x.
Inverse Functions and Differentiation Rule
Inverse Functions and Differentiation connects the slope of a function to the slope of its inverse: if b = f(a) and f'(a) ≠ 0, then (f⁻¹)'(b) = 1/(f'(a)). Geometrically, reflecting a graph across the line y = x swaps rise and run, so the two slopes are reciprocals. The rule needs f to be one-to-one and differentiable with a non-zero derivative at the matching point.
Intermediate Value Theorem: Statement & Examples
The Intermediate Value Theorem states that if f is continuous on a closed interval [a, b] and N is any value between f(a) and f(b), then there is at least one point c in [a, b] with f(c) = N. Its most-used corollary: if f is continuous and f(a) and f(b) have opposite signs, a root lies somewhere between a and b. The theorem promises a value exists, never where it is or how many there are.
Integration of Trigonometric Functions: Rules & Examples
Integration of trigonometric functions means finding the antiderivative of sin, cos, tan, and their relatives. The six building blocks are ∫ sin x,dx = -cos x + C, ∫ cos x,dx = sin x + C, ∫ sec² x,dx = tan x + C, ∫ csc² x,dx = -cot x + C, ∫ sec x tan x,dx = sec x + C, and ∫ csc x cot x,dx = -csc x + C. For higher powers you either split off one factor and substitute (odd powers) or use a power-reduction identity (even powers).
Integration by Substitution: Steps and Examples
Integration by Substitution is the reverse of the chain rule: when an integral has the shape ∫ f(g(x)),g'(x),dx, you set u = g(x) so that du = g'(x),dx, turning the whole thing into the simpler ∫ f(u),du. Integrate in u, substitute x back, and add +C. For a definite integral, change the limits to u as well, then evaluate.
Integration by Parts: Formula, LIATE & Examples
Integration by parts is the technique for integrating a product of two functions, using the formula ∫ u,dv = uv - ∫ v,du. It comes straight from reversing the product rule of differentiation. The whole skill is choosing which factor to call u (differentiate it) and which to call dv (integrate it), and the LIATE rule tells you how to pick.
Implicit Differentiation: Steps, Examples & Formula
Implicit differentiation finds dy/dx for an equation that is not solved for y, such as x² + y² = 25. You differentiate both sides with respect to x, treat y as a function of x so every y-term picks up a dy/dx through the chain rule, then solve for dy/dx. For the circle this gives dy/dx = -x/y, the slope of the tangent at any point.
Hyperbolic Functions: sinh, cosh, tanh Explained
Hyperbolic Functions are built from the exponential e x: x = (e x - e -x)/2, x = (e x + e -x)/2, and x = ( x)/( x). They obey the fundamental identity ² x - ² x = 1, which parametrizes the hyperbola x² - y² = 1 the way sine and cosine parametrize the circle. Their derivatives are clean: d/dx[ x] = x, d/dx[ x] = x, and d/dx[ x] = sech² x.
Harmonic Progression: Formula, Mean & Examples
A Harmonic Progression (HP) is a sequence of non-zero numbers whose reciprocals form an arithmetic progression. So a₁, a₂, a₃, … is an HP exactly when 1/a₁, 1/a₂, 1/a₃, … is an AP. To find any term, work in the reciprocal AP and flip the result back: aₙ = 1/(1/a₁ + (n-1)d). The harmonic mean of a and b is 2ab/(a+b), and the infinite harmonic series 1 + 12 + 13 + … grows without bound.
Graphing Functions: A Calculus Step-By-Step Guide
Graphing functions with calculus means building an accurate curve from what the equation tells you, in seven steps: domain, intercepts, symmetry, asymptotes, the first derivative f'(x) (where the curve rises or falls and where its peaks and valleys sit), the second derivative f''(x) (where it bends up or down and where it changes bend), then the assembled sketch. The first derivative gives the slope; the second derivative gives the concavity. Together they let you draw a curve you have never plotted, without guessing at points.
Geometric Series: Convergence, Sum & Formula
A geometric series adds the terms of a geometric sequence: a + ar + ar² + ar³ + …, where each term is the one before it times a fixed ratio r. Its finite sum is S N = (a(1 - r N+1))/(1 - r), and the infinite series converges to a/(1 - r) exactly when r < 1; for r ≥ 1 it diverges.
Exponential Functions: Derivatives And Integrals
Exponential functions have the form f(x) = a x with a > 0 and a ≠ 1, and the star of calculus is the natural exponential f(x) = e x, the one function that is its own derivative: d/dx(e x) = e x. For a general base, d/dx(a x) = a x ln a and ∫ a x, dx = (a x)/(ln a) + C, while the growth-and-decay model y = y₀ e kt describes anything whose rate of change is proportional to its current size.
Fundamental Theorem of Calculus: Part 1 & Part 2
The Fundamental Theorem of Calculus says that differentiation and integration are inverse operations. Part 1 states that if g(x) = ∫ₐ x f(t),dt and f is continuous, then g'(x) = f(x): the derivative of an accumulated area is the function that built it. Part 2 states that ∫ₐ b f(x),dx = F(b) - F(a) whenever F' = f, which turns finding an area into a single subtraction.
Elasticity Of Demand: Formula, Calculus & Examples
Elasticity of demand measures how sharply the quantity people buy responds to a change in price, and the calculus version is the point formula E = p/q/dp. Classify by size: |E|>1 is elastic, |E|<1 is inelastic, and |E|=1 is unit elastic. Because revenue is R = pq, the derivative dR/dp vanishes exactly at unit elasticity, so revenue is maximized where E = -1.
Domain And Range Of A Function: Rules & Examples
The domain and range of a function are, respectively, the set of allowable inputs x and the set of resulting outputs y. To find a domain you exclude the inputs that break the rule (division by zero, even roots of negatives, logs of non-positive numbers); to find a range you track the outputs algebraically, from the graph, or with calculus using extrema, asymptotes, and end behaviour.
Differentiability Of A Function: Definition & Rules
The differentiability of a function at a point a means the derivative f'(a) = lim h → 0 (f(a+h) - f(a))/h exists as a finite number, which requires the left-hand and right-hand difference-quotient limits to agree. Every differentiable function is continuous, but the reverse is false: a function can be continuous yet fail to be differentiable at a corner, a cusp, a vertical tangent, or a break.
Derivatives of Inverse Trigonometric Functions
The derivatives of inverse trigonometric functions turn six angle-finding functions into plain algebraic fractions: d/dx x=1/(√(1-x²)), d/dx x=1/(1+x²), and their four partners. Each one comes from implicit differentiation, and the three "co-function" pairs differ only by a minus sign. Get the chain-rule inner derivative and the domain right, and every problem in this topic follows the same short recipe.
Definition Of The Derivative: Limit Formula & Examples
The definition of the derivative is f'(x) = lim h → 0 (f(x+h) - f(x))/h: it measures the instantaneous rate of change of f at x, which equals the slope of the tangent line to the graph. You get it by shrinking a secant line between two nearby points until the two points merge, so the average rate of change turns into an instantaneous one. Every differentiation rule you will ever use is built from this single limit.
Definite Integrals: Definition, Properties & Examples
Definite integrals measure the signed area between a curve and the x-axis over a fixed interval [a, b], written ∫ₐ b f(x),dx and defined as the limit of Riemann sums. The Fundamental Theorem of Calculus evaluates one as F(b) - F(a), where F is any antiderivative of f. The result is a single number, with no +C, which is what separates it from the indefinite integral.
Antiderivatives: Rules, Formulas & Examples
An antiderivative of a function f is a function F whose derivative gives f back, that is F'(x) = f(x). Every function has a whole family of antiderivatives that differ only by a constant, written F(x) + C, and the indefinite integral ∫ f(x),dx = F(x) + C is the name for that family. Finding an antiderivative is differentiation run in reverse, so you can always check an answer by differentiating it and seeing the original function reappear.
Critical Points: Definition, How To Find & Examples
Critical Points of a function f are the interior domain points where f'(x) = 0 or where f'(x) does not exist. They matter because every local maximum and local minimum on the inside of a domain must occur at one of them, so they are the shortlist you search for peaks and valleys. A critical point is only a candidate, though: some are maxima, some are minima, and some (like x = 0 for f(x) = x³) are neither.
Chain Rule: Formula, Steps, and Examples
The chain rule differentiates a composite function, one function nested inside another, by the formula d/dxf(g(x)) = f'(g(x)),g'(x). In words, differentiate the outer function while holding the inner one fixed, then multiply by the derivative of the inner function. The single most common error is forgetting that final factor, the inner derivative g'(x).