Analysis & Calculus
21 concepts
lim(n→∞) aₙ = LThe Limit of a Sequence
As you push the term number n to infinity, if the terms keep getting closer to some single number L, then L is the limit of the sequence.
lim(x→a) f(x) = LLimits
Push x closer and closer to some value a, and watch which number the function crowds toward.
lim(x→a) f(x) = f(a)Continuity
If the value you expected on approach (the limit) exactly equals the value actually marked at that point, the curve runs through it unbroken.
f'(x) = lim(h→0) [f(x+h) − f(x)] / hThe Derivative — Slope at an Instant
Take the slope between two points (rise ÷ run), then slide the two points infinitely close to get 'the slope at a single point'.
d/dx xⁿ = n·xⁿ⁻¹The Power Rule
The slope function of xⁿ is: multiply out front by the exponent n, and lower the exponent to n−1.
(f·g)' = f'·g + f·g'The Product Rule
To differentiate a product of two functions, differentiate one at a time and add the two results.
(f(g(x)))' = f'(g(x))·g'(x)The Chain Rule
To differentiate a composite (a function inside a function), multiply the outer function's derivative by the inner function's derivative.
f'(x) = 0Critical Points — Where the Slope Is Zero
Set the derivative equal to 0 and solve; the x's you get are where the tangent goes flat — candidates for a max or min.
f''(x) > 0 ⇒ ∪ , f''(x) < 0 ⇒ ∩Concavity and the Second Derivative
The second derivative f'' is the rate of change of the slope. Its sign tells you whether the curve bends like a cup (∪) or a cap (∩).
∫ₐᵇ f(x) dx = lim Σ f(xᵢ)·ΔxThe Integral — Accumulating Area
The area under a curve, filled in with many very narrow rectangles whose areas you add together. Send their width to 0 and you get the exact area.
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)The Definite Integral
The area over [a,b], computed not by a Riemann sum but by finding an antiderivative F and taking F(b)−F(a).
d/dx ∫ₐˣ f(t) dt = f(x)The Fundamental Theorem of Calculus
Differentiate the function that has been accumulating area from a to x, and the original function f(x) pops right back out. Differentiation and integration are inverses.
e = lim(n→∞) (1 + 1/n)ⁿ ≈ 2.71828The Number e
Take 100% interest, chop it into ever-finer compounding steps, and as the number of steps goes to infinity the result approaches e ≈ 2.718.
ln x = y ⇔ eʸ = xThe Natural Logarithm
ln x answers 'to what power must e be raised to get x'. In other words, it's the inverse of the exponential eˣ.
d/dx eˣ = eˣDeriving the Exponential — e^x Is Its Own Slope
The slope function of eˣ is eˣ again. At any point, the height there is exactly the slope there.
f(x) ≈ f(a) + f'(a)(x−a) + ½f''(a)(x−a)²Taylor Approximation — Curves as Polynomials
A complicated function approximated, near one point, by a simple polynomial tuned to share the same value, slope, and bend there.
lim(x→a) f(x) = L ⟺ ∀ε>0, ∃δ>0 : 0 < |x−a| < δ ⟹ |f(x)−L| < εThe ε–δ Definition of a Limit
If, however small an error ε your opponent names, you can shrink the input window δ so that f(x) stays within ε of L, then the limit is L.
S = Σ(n=1..∞) aₙ = lim(N→∞) (a₁ + a₂ + ⋯ + a_N); Σ(n=0..∞) arⁿ = a/(1−r), |r|<1Convergence of Series
The 'sum' of an infinite series is not literally added to the end. You form the partial sum of the first N terms; if that value approaches some number S as N grows, the series 'converges to S'.
z = f(x, y) → ∂f/∂x, ∂f/∂yPartial Derivatives
For a function of two or more variables, you freeze every variable except the one you care about, then take an ordinary derivative.
∬_R f(x, y) dA = ∫∫ f(x, y) dx dyMultiple Integrals
You chop the base region R into tiny rectangles, multiply each patch's area by the function's height above it to make a little column, and add up every column.
f(x) = a₀/2 + Σₙ (aₙ cos nx + bₙ sin nx)Fourier Series
A periodic function can be written exactly as a sum of sines and cosines whose frequencies are integer multiples of a base. You only need to decide how much of each pure tone goes in (the coefficients aₙ, bₙ).
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