Dosing a child’s medicine
The leaflet gives 15 mg of paracetamol per kilogram of body weight — a function f(m) = 15m whose argument is the weight and whose value is the dose. A child of 18 kg gets 270 mg at a time and 1,080 mg across four doses in a day. The domain matters just as much: the formula holds from roughly 4 to 50 kg, and above that a flat adult dose of 1,000 mg takes over, so substituting into the formula stops meaning anything.
What a function is
The electricity bill
A bill comes in two parts: standing charges that do not depend on usage, and a price per kilowatt-hour. With 38 of standing charges and 0.92 per kWh the bill is f(x) = 0.92x + 38, so 210 kWh costs 231.20. The intercept explains the thing that surprises people most: twice the usage is not twice the bill — 420 kWh comes to 424.40, which is 1.84 times as much, not two.
Linear function Convert it here: Energy
Braking distance
Braking distance grows with the square of the speed: s = v²/(2a). On dry asphalt the deceleration is around 7 m/s², so at 50 km/h (13.9 m/s) a car needs 13.8 m, and at 100 km/h it needs 55.1 m — four times as far for twice the speed. The same parabola is why dropping a limit from 50 to 30 km/h shortens braking to 5.0 m rather than to the 8.3 m a proportion would predict.
Quadratic function Convert it here: Speed
Tuning an instrument
A tuning fork sounds A4 at 440 Hz, that is the wave y = a·sin(2π·440·t). The same string pulled tighter sounds 466 Hz — the coefficient beside t grows, so the period drops from 1/440 ≈ 2.27 ms to 1/466 ≈ 2.15 ms. The tuner changes no shape, only one coefficient in the formula; plucking harder changes the amplitude alone, which is the loudness.
Transforming a graph
The weight chart in an app
The app plots 12 weeks of readings. The extreme points on their own say little: a maximum of 84.2 kg in week 3, a minimum of 79.6 kg in week 11. It is the intervals of monotonicity that form a sentence — the curve falls from week 3 to week 11, that is 4.6 kg over 8 weeks, 0.58 kg a week. The last week climbs again, which reading the extremes alone would never have shown.
Reading properties from a graph
Travel time
The Warsaw–Kraków route is about 295 km, so the driving time is t = 295/v — inverse proportion in its purest form. At 90 km/h that is 3 h 17 min, at 120 km/h only 2 h 28 min. Doubling the speed from 100 to 200 km/h would cut the drive from 2 h 57 min to 1 h 29 min, exactly in half — and that is the whole content of the word "inverse".
Inverse proportion and the homographic function
Carbon-14 dating
The half-life of carbon-14 is 5730 years, so N(t) = N₀·(1/2)^(t/5730). An archaeologist measures 25% of the original isotope content in a find and solves (1/2)^(t/5730) = 0.25, that is t/5730 = 2 — the sample is 11,460 years old. At 12.5% it would come out at 17,190 years: each further halving is one more full period.
Exponential and logarithmic functions
Training plan
A ten-week running plan: 20 minutes of easy running in week one, 5 minutes more in every week after that. That is a sequence with the general term aₙ = 15 + 5n, so week ten calls for 15 + 50 = 65 minutes. The difference between consecutive terms is a constant 5 minutes — positive, so the load rises and no week is lighter than the one before it.
Number sequences Convert it here: Time
A savings account
You deposit 10,000 into an account paying 4% a year, compounded annually. The balance after each year is a geometric sequence with ratio q = 1.04: after five years you hold 10,000 · 1.04⁵ = 12,166.53, so 2,166.53 in interest — 166.53 more than plain addition of 400 a year would give. That surplus is exactly the interest earned on interest.
Arithmetic and geometric sequences See the lesson: Compound interest
A bouncing ball
A ball dropped from 2 m returns to 60% of the previous height after every bounce. The bounce heights form a geometric sequence with q = 0.6, and the whole distance travelled is 2 + 2 · (1.2 + 0.72 + 0.432 + …) = 2 + 2 · 1.2/(1 − 0.6) = 8 metres. There are infinitely many bounces and the distance is finite — which is exactly the situation a convergent series describes.
Limit of a sequence and the geometric series
A currency desk with a commission
An exchange office converts euros to zloty at g(x) = 4.30x and then takes a 2% commission, f(y) = 0.98y. A customer changing 500 € receives f(g(500)) = 0.98 · 2150 = 2107 zloty. The reverse operation answers "how many euros must I sell to get 3000 zloty": from 4.214x = 3000 we get 711.91 €, and that is precisely the inverse of the composition.
Composition and inverse functions
Unit cost in manufacturing
Setting up a production line costs 40,000 zloty and each item costs 12, so the cost per item is k(n) = (40000 + 12n)/n. At 500 items that is 92 zloty, at 5000 it is 20, at 50,000 it is 12.80. The horizontal asymptote y = 12 says where the curve stops: it will never drop below the material cost, however far production grows.
Rational functions and asymptotes
A coffee machine instead of the café
The machine costs 1,200 and a cup made on it costs 1.10, so the cost per cup after n cups is f(n) = (1200 + 1.1n)/n = 1.1 + 1200/n. After a hundred cups that is 13.10, after a thousand 2.30, and the limit as n grows without bound is 1.10 — a floor you will never get under, not if you drink coffee daily for twenty years. The limit is what answers "how little can this possibly cost", and set against a café at 14 it puts the break-even at the 93rd cup.
Limit of a function
Pace on a running watch
The watch has no speed sensor — all it knows is a run of GPS positions. It gets instantaneous speed from a difference quotient: over one second the position moved 3.1 m, which is 3.1 m/s, or 11.2 km/h, a pace of 5:23 min/km. The shorter the interval, the closer the figure gets to the value right now, and that limit as h goes to zero is precisely the definition of the derivative — displayed on your wrist once a second.
The derivative Convert it here: Speed
Designing a drinks can
A can has to hold 330 ml on as little aluminium as possible. At radius r the height is h = 330/(πr²), so the surface area is S(r) = 2πr² + 660/r. The derivative S′(r) = 4πr − 660/r² vanishes at r³ = 660/(4π), that is r = 3.74 cm and h = 7.5 cm — the optimal can is squat, exactly as tall as it is wide. A real one has r = 3.3 cm and spends 268 cm² of metal instead of 264, because the lid must be thicker; across a billion cans that 1.5% is a budget line of its own.
What the derivative is for
Fuel used on a climb
The trip computer shows instantaneous consumption in litres per hour, while what you want to know is how much fuel actually went. On a two-hour climb consumption rises from 6 to 12 l/h, that is f(t) = 6 + 3t. The amount used is the integral: ∫ (6 + 3t) dt from 0 to 2 = [6t + 1.5t²] = 12 + 6 = 18 litres. Multiplying by the opening reading would say 12 l, by the closing one 24 l — both wrong, and the car’s own gauge does exactly what the integral does: it sums hundreds of short stretches.
The integral Convert it here: Flow rate
Work done stretching a resistance band
The force needed to stretch a rubber band is not constant — it builds up and levels off: F(x) = 150(1 − e^(−4x)) newtons. Work is the integral of force along the distance, so stretching it by 0.5 m takes 150·(0.5 + 0.25·e^(−2) − 0.25) ≈ 42.6 joules. The exponential term has no row in the table of integrals — the linear substitution u = −4x is what handles it.
Integration by parts and substitution
How much further the bike will roll
Once you stop pedalling, the speed decays exponentially: v(t) = 6·e^(−t/12) metres per second. The distance to a standstill is the integral of speed over all remaining time — an improper integral over an infinite interval — and it comes out as a finite 6·12 = 72 metres. In theory the bike never quite stops, and it still travels exactly 72 metres.
Improper integrals
A stack of blocks leaning off the table
Blocks piled one on another can hang over the edge of a table further the more of them there are: with n blocks the maximum overhang is half the n-th partial sum of the harmonic series. Four blocks give 1.04 block lengths, and to pass two lengths you need as many as 31. The overhang grows without bound, because the harmonic series diverges — it just grows desperately slowly.
Number series
How a calculator evaluates a sine
A processor holds no table of sines — it sums a few terms of a series. For x = 0.5 the polynomial x − x³/6 + x⁵/120 alone gives 0.4794255, and the true sine is 0.4794255 — seven digits of agreement from three terms. The next term would change the result by less than 10⁻⁹, so it is simply never computed.
Power and Taylor series

Branch formulas

Branch: Functions and analysis

What a function is

A function from X into Y
$f \colon X \to Y$
assigns exactly one element of Y to every element of X
The value notation
$y = f(x)$
x is the argument, y the value of the function at it
Domain
$D_f = \{x : f(x) \text{ exists}\}$
the set of all arguments
Range
$R_f = \{f(x) : x \in D_f\}$
the set of all values the function takes
Zero of a function
$f(x_0) = 0$
an argument whose value is zero
The graph
$\text{graph of } f = \{(x,\, f(x)) : x \in D_f\}$
the set of points (argument, value)

Linear function

Slope–intercept form
$f(x) = ax + b$
a — the slope, b — the intercept
Slope from two points
$a = \frac{y_2 - y_1}{x_2 - x_1}$
the rise divided by the run
The zero
$x_0 = -\frac{b}{a} \quad (a \neq 0)$
where the graph crosses the x-axis
The value at zero
$f(0) = b$
where the graph crosses the y-axis
General form
$Ax + By + C = 0$
reducible to the slope–intercept form whenever B ≠ 0
Condition for parallel lines
$a_1 = a_2$
parallel lines have equal slopes
Condition for perpendicular lines
$a_1 \cdot a_2 = -1$
hence a₂ = −1/a₁

Quadratic function

General form
$f(x) = ax^2 + bx + c \quad (a \neq 0)$
the graph is a parabola
Coordinates of the vertex
$p = -\frac{b}{2a}, \quad q = -\frac{\Delta}{4a} = f(p)$
(p, q) — the turning point of the parabola
Canonical form
$f(x) = a(x - p)^2 + q$
the vertex is readable straight off it
Factored form
$f(x) = a(x - x_1)(x - x_2)$
exists only when Δ ≥ 0; shows the zeros
Axis of symmetry
$x = p$
the vertical line through the vertex
Range
$R_f = \langle q, +\infty) \ \text{for } a > 0$
for a < 0 it is (−∞, q⟩

Transforming a graph

Shift along the y-axis
$y = f(x) + q$
up for q > 0, down for q < 0
Shift along the x-axis
$y = f(x - p)$
to the right for p > 0 — against the sign in the formula
Translation by a vector
$y = f(x - p) + q$
\vec{u} = [p,\, q]
Reflection in the x-axis
$y = -f(x)$
flips the sign of the value
Reflection in the y-axis
$y = f(-x)$
flips the sign of the argument
Vertical stretch
$y = a \cdot f(x)$
a times further from the x-axis
Horizontal squeeze
$y = f(b x)$
b times closer to the y-axis
The general sinusoid
$y = a \sin(bx + c)$
amplitude |a|, period \tfrac{2\pi}{|b|}, shifted by -\tfrac{c}{b}

Reading properties from a graph

A zero
$f(x_0) = 0$
where the graph crosses the x-axis
The sign of a function
$f(x) > 0 \iff \text{graph above the } x\text{-axis}$
below the axis the values are negative
Increasing function
$x_1 < x_2 \implies f(x_1) < f(x_2)$
on an interval where the graph rises
Decreasing function
$x_1 < x_2 \implies f(x_1) > f(x_2)$
on an interval where the graph falls
Minimum
$f(x) \geq f(x_{\min}) \ \text{for every } x$
the lowest point of the graph; a maximum works the same way
Domain and range
$D_f \to x\text{-axis}, \quad R_f \to y\text{-axis}$
the shadows of the graph on the two axes

Inverse proportion and the homographic function

Inverse proportion
$y = \frac{a}{x}, \qquad a \neq 0$
domain: x \neq 0
The proportionality condition
$x \cdot y = a$
the product is constant — that is how a is found
Homographic function
$y = \frac{a}{x - p} + q$
a hyperbola translated by [p,\, q]
The asymptotes
$x = p, \qquad y = q$
vertical and horizontal — approached, never reached
Vertex form
$\frac{ax + b}{x - p} = \frac{ap + b}{x - p} + a$
a quotient of linear expressions as a translated hyperbola

Exponential and logarithmic functions

The exponential function
$f(x) = b^x, \qquad b > 0,\ b \neq 1$
domain: \mathbb{R}, range: (0,\ \infty)
The logarithmic function
$f(x) = \log_b x, \qquad b > 0,\ b \neq 1$
domain: (0,\ \infty), range: \mathbb{R}
Mutually inverse
$b^{\log_b x} = x, \qquad \log_b b^x = x$
each one undoes what the other does
Exponential equation
$b^{x_1} = b^{x_2} \iff x_1 = x_2$
one-to-one — which is why the exponents may be compared
Logarithmic equation
$\log_b x_1 = \log_b x_2 \iff x_1 = x_2$
given x_1 > 0 and x_2 > 0
Radioactive decay
$N(t) = N_0 \cdot \left(\tfrac{1}{2}\right)^{t/T}$
T — the half-life

Number sequences

A sequence is a function
$a_n = f(n), \qquad n \in \mathbb{N}_+$
the domain is the natural numbers, so the graph is a set of dots
General term
$a_n = 2n + 1$
substitute the index and the term is there
Recursive formula
$a_1 = 3, \qquad a_{n+1} = 2a_n - 1$
each term from the previous one — you have to walk through them
Increasing sequence
$a_{n+1} - a_n > 0$
the difference of consecutive terms is positive for every n
Decreasing sequence
$a_{n+1} - a_n < 0$
the same difference, negative for every n
Constant sequence
$a_{n+1} - a_n = 0$
every term equal

Arithmetic and geometric sequences

Common difference
$r = a_{n+1} - a_n$
the same for every pair of neighbouring terms
nth term, arithmetic
$a_n = a_1 + (n - 1)r$
the first term plus n − 1 steps
Arithmetic mean
$a_n = \frac{a_{n-1} + a_{n+1}}{2}$
every term is the mean of its neighbours
Sum of n terms, arithmetic
$S_n = \frac{a_1 + a_n}{2} \cdot n$
the mean of the outer terms times how many there are
Common ratio
$q = \frac{a_{n+1}}{a_n}$
the same for every pair of neighbouring terms
nth term, geometric
$a_n = a_1 \cdot q^{n-1}$
the first term times the ratio raised to n − 1
Geometric mean
$a_n^2 = a_{n-1} \cdot a_{n+1}$
a term squared equals the product of its neighbours
Sum of n terms, geometric
$S_n = a_1 \cdot \frac{q^n - 1}{q - 1}, \qquad q \neq 1$
for q = 1 the sum is simply n times a₁

Limit of a sequence and the geometric series

Limit of a sequence
$\lim_{n \to \infty} a_n = g$
a sequence convergent to the number g
Definition of the limit
$\forall \varepsilon > 0 \;\; \exists N \;\; \forall n > N : \; |a_n - g| < \varepsilon$
from some point on, every term lies closer to g than epsilon
The basic limit
$\lim_{n \to \infty} \frac{1}{n} = 0$
every quotient limit is derived from it
Quotient of polynomials
$\lim_{n \to \infty} \frac{a n^k + \ldots}{b n^k + \ldots} = \frac{a}{b}$
for equal degrees — the ratio of the leading coefficients
The number e
$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828$
a limit that is the definition of a constant
Sum of a geometric series
$S = \frac{a_1}{1 - q}, \qquad |q| < 1$
only for a convergent one — otherwise there is no sum

Composition and inverse functions

Composition of functions
$(f \circ g)(x) = f\big(g(x)\big)$
g first, then f — read from the inside out
Order matters
$f \circ g \neq g \circ f$
composition is not commutative
One-to-one
$x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$
the condition for an inverse to exist
The inverse function
$f^{-1}\big(f(x)\big) = x, \qquad f\big(f^{-1}(y)\big) = y$
undoes exactly what f did
Roles swapped
$y = f(x) \iff x = f^{-1}(y)$
domain and range change places
The graph of the inverse
$(a,\, b) \in f \iff (b,\, a) \in f^{-1}$
a reflection in the line y = x

Rational functions and asymptotes

A rational function
$f(x) = \frac{P(x)}{Q(x)}, \qquad Q(x) \neq 0$
a quotient of two polynomials
The domain
$D_f = \{x \in \mathbb{R} : Q(x) \neq 0\}$
the zeros of the denominator are excluded
A pole
$x = x_0 \ \text{— a vertical asymptote}$
when Q(x_0)=0 and P(x_0) \neq 0
Horizontal asymptote
$\deg P < \deg Q \implies y = 0$
the denominator grows faster
Horizontal asymptote
$\deg P = \deg Q \implies y = \frac{a_P}{a_Q}$
the ratio of the leading coefficients
Oblique asymptote
$\deg P = \deg Q + 1 \implies y = ax + b$
the quotient of the polynomial division

Limit of a function

Limit of a function
$\lim_{x \to x_0} f(x) = g$
the values f(x) close in on g as x closes in on x₀
Continuous function
$\lim_{x \to x_0} f(x) = f(x_0)$
a polynomial is continuous, so the limit is plain substitution
The 0/0 indeterminate form
$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} (x + 2) = 4$
cancel the common factor first, substitute afterwards
Equal degrees
$\lim_{x \to \infty} \frac{a_n x^n + \ldots}{b_n x^n + \ldots} = \frac{a_n}{b_n}$
the ratio of the leading coefficients
Numerator of lower degree
$\lim_{x \to \infty} \frac{a_m x^m + \ldots}{b_n x^n + \ldots} = 0 \quad (m < n)$
the denominator grows faster, so the quotient dies away
One-sided limits
$\lim_{x \to x_0^-} f(x) = \lim_{x \to x_0^+} f(x) = g \iff \lim_{x \to x_0} f(x) = g$
a limit exists only when both sides agree
Continuity at a point
$\lim_{x \to x_0^-} f(x) = \lim_{x \to x_0^+} f(x) = f(x_0)$
the value exists, the limit exists and the two agree
The intermediate value property
$f(a) \cdot f(b) < 0 \implies \exists\, c \in (a; b): f(c) = 0$
a continuous function with opposite signs at the ends has a root

The derivative

Definition of the derivative
$f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$
the limit of the difference quotient as the increment shrinks to zero
Power rule
$(x^n)' = n \cdot x^{n-1}$
the exponent steps in front of x and drops by one
Constants and sums
$(c \cdot f)' = c \cdot f', \qquad (f + g)' = f' + g'$
a constant factor passes through; a sum differentiates term by term
Product rule
$(f \cdot g)' = f' g + f g'$
the derivative of a product is NOT the product of the derivatives
Quotient rule
$\left(\frac{f}{g}\right)' = \frac{f' g - f g'}{g^2}$
a difference on top — the order matters
Chain rule
$\big(f(g(x))\big)' = f'(g(x)) \cdot g'(x)$
the outer derivative times the derivative of the inside
Root and reciprocal
$(\sqrt{x})' = \frac{1}{2\sqrt{x}}, \qquad \left(\frac{1}{x}\right)' = -\frac{1}{x^2}$
the same power rule with exponents ½ and −1
Trigonometric functions
$(\sin x)' = \cos x, \qquad (\cos x)' = -\sin x$
the cosine picks up a minus sign

What the derivative is for

Equation of the tangent
$y = f'(x_0)(x - x_0) + f(x_0)$
a line of slope f′(x₀) through the point of tangency
Increasing function
$f'(x) > 0 \ \text{on an interval} \implies f \ \text{increasing}$
a positive derivative is a tangent leaning upwards
Decreasing function
$f'(x) < 0 \ \text{on an interval} \implies f \ \text{decreasing}$
a negative derivative is a tangent leaning downwards
Necessary condition for an extremum
$f'(x_0) = 0$
a critical point — a candidate, not yet an extremum
Sufficient condition (maximum)
$f' : + \to - \ \text{at} \ x_0 \implies \text{maximum at} \ x_0$
the derivative must CHANGE sign from plus to minus
Sufficient condition (minimum)
$f' : - \to + \ \text{at} \ x_0 \implies \text{minimum at} \ x_0$
minus to plus — a trough instead of a crest

The integral

Antiderivative
$F'(x) = f(x)$
F is an antiderivative of f when differentiating F gives f
Indefinite integral
$\int f(x)\,dx = F(x) + C$
the result is a whole FAMILY of functions, hence the constant C
Power rule
$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
the exponent grows by one, then divide by the new exponent
Newton–Leibniz formula
$\int_a^b f(x)\,dx = F(b) - F(a)$
the constant C cancels in the subtraction, so it never matters
Area under a curve
$f(x) \geq 0 \ \text{on} \ \langle a; b \rangle \implies A = \int_a^b f(x)\,dx$
the definite integral equals the area only above the axis
Area in general
$A = \int_a^b |f(x)|\,dx$
where the curve dips below the axis, the area is computed piece by piece
Trigonometric functions
$\int \sin x\,dx = -\cos x + C, \qquad \int \cos x\,dx = \sin x + C$
the minus sign changes sides compared with differentiation
The integral of 1/x
$\int \frac{1}{x}\,dx = \ln\lvert x \rvert + C$
the one exponent the power rule cannot handle
The exponential function
$\int e^x\,dx = e^x + C$
its own derivative, so its own antiderivative too
Area between two curves
$f(x) \geq g(x) \ \text{on} \ \langle a; b \rangle \implies A = \int_a^b \big(f(x) - g(x)\big)\,dx$
integrate the difference: upper curve minus lower

Integration by parts and substitution

Integration by parts
$\int u(x)\,v'(x)\,dx = u(x)\,v(x) - \int u'(x)\,v(x)\,dx$
the product rule, reversed
Parts, definite
$\int_a^b u\,v'\,dx = \Big[u\,v\Big]_a^b - \int_a^b u'\,v\,dx$
the uv term is evaluated at the bounds, the rest stays an integral
Integration by substitution
$\int f\big(g(x)\big)\,g'(x)\,dx = F\big(g(x)\big) + C$
the chain rule, reversed
The bounds travel
$\int_a^b f\big(g(x)\big)\,g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$
a substitution moves the limits too
Linear substitution
$\int f(ax+b)\,dx = \frac{1}{a}\,F(ax+b) + C$
the commonest case: divide by a
The exponential integral
$\int e^x\,dx = e^x + C$
needed by the worked examples on parts

Improper integrals

Improper integral, first kind
$\int_a^{\infty} f(x)\,dx = \lim_{T \to \infty} \int_a^{T} f(x)\,dx$
an unbounded interval of integration
Improper integral, second kind
$\int_a^b f(x)\,dx = \lim_{\varepsilon \to 0^+} \int_{a+\varepsilon}^{b} f(x)\,dx$
the function is unbounded at the endpoint a
The p-test at infinity
$\int_1^{\infty} \frac{dx}{x^p} = \frac{1}{p-1} \quad \text{for } p > 1$
divergent for p ≤ 1
The p-test at zero
$\int_0^1 \frac{dx}{x^p} = \frac{1}{1-p} \quad \text{for } p < 1$
divergent for p ≥ 1
An exponential tail
$\int_0^{\infty} e^{-ax}\,dx = \frac{1}{a} \quad (a > 0)$
convergent for every positive a
Finite area, infinite height
$\int_0^1 \frac{dx}{\sqrt{x}} = 2$
the flagship example of the second kind

Number series

Partial sum
$S_n = a_1 + a_2 + \dots + a_n = \sum_{k=1}^{n} a_k$
the finite sum of the first n terms
Sum of a series
$\sum_{n=1}^{\infty} a_n = \lim_{n \to \infty} S_n$
the limit of the sequence of partial sums
Necessary condition
$\sum a_n \ \text{convergent} \implies \lim_{n \to \infty} a_n = 0$
it does not run backwards
Comparison test
$0 \leq a_n \leq b_n, \ \sum b_n \ \text{convergent} \implies \sum a_n \ \text{convergent}$
smaller than a convergent series is convergent
Ratio test
$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = g < 1 \implies \sum a_n \ \text{convergent}$
divergent for g > 1, inconclusive for g = 1
Root test
$\lim_{n \to \infty} \sqrt[n]{|a_n|} = g < 1 \implies \sum a_n \ \text{convergent}$
divergent for g > 1, inconclusive for g = 1
Leibniz test
$a_n \searrow 0 \implies \sum_{n=1}^{\infty} (-1)^{n+1} a_n \ \text{convergent}$
a sequence decreasing to zero, alternating signs
Absolute convergence
$\sum |a_n| \ \text{convergent} \implies \sum a_n \ \text{convergent}$
absolute implies plain, never the reverse

Power and Taylor series

Power series
$\sum_{n=0}^{\infty} a_n (x - x_0)^n$
a function of its argument x
Radius of convergence
$R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right|$
convergent for |x − x_0| < R
Taylor series
$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}\,(x - x_0)^n$
the coefficients are derivatives at x_0
Maclaurin series
$f(x) = f(0) + f'(0)\,x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \dots$
the Taylor series about zero
Expansion of eˣ
$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$
convergent for every x
Expansion of sine
$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$
odd powers only
Expansion of cosine
$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$
even powers only
Expansion of the logarithm
$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$
only for −1 < x ≤ 1
Approximation error
$\big|R_n(x)\big| \leq \frac{M\,|x - x_0|^{n+1}}{(n+1)!}$
M bounds the derivative of order n+1

A luminous parabola and a straight line crossing above a faint coordinate grid, the vertex flaring like a star, against a cosmic nebula in violet and magenta

Functions and analysis

Why a function's graph can never double back

The whole definition of a function comes down to one word: uniqueness. One argument, one value — and not a single one more. That seemingly minor condition decides why a circle is not the graph of a function, why a line and a parabola describe so much of the world, and why four datasets with identical summary statistics can look nothing alike.

12 min