Recover university calculus · Calculus × Computing

01
Functions
A rule $f$ that assigns to each input $x$ of a set $A$ exactly one output $f(x)$ in a set $B$ . Calculus studies how outputs change when inputs change; computing is, quite literally, evaluating functions.
FundamentalFoundations
Mark as studied
02
Limit of a function
$\lim_{x\to a} f(x) = L$ : the values $f(x)$ can be made as close to $L$ as we like by taking $x$ close enough to $a$ (but not equal). Derivatives, integrals and continuity are all defined as limits.
FundamentalSequences and limits
Mark as studied
03
Continuity
$f$ is continuous at $a$ if $\lim_{x\to a} f(x) = f(a)$ : small changes in the input produce small changes in the output. Continuous on an interval means you can draw the graph without lifting the pen.
FundamentalContinuity
Mark as studied
04
Derivative
$f'(a)$ is the instantaneous rate of change of $f$ at $a$ : the slope of the tangent line to the graph, defined as the limit of slopes of secant lines.
FundamentalDerivatives
Mark as studied
05
Chain rule
The derivative of a composition is the product of the derivatives: $(g\circ f)'(x) = g'(f(x))\,f'(x)$ . Rates of change multiply along a chain — and backpropagation is this rule applied, very efficiently, to a neural network.
FundamentalDerivatives
Mark as studied
06
Mean value theorem
Somewhere on $(a, b)$ the instantaneous rate of change equals the average rate: $f'(c) = \frac{f(b) - f(a)}{b - a}$ . It is the bridge from derivatives to inequalities — and therefore to every error bound in numerical analysis.
UniversityFundamental theorems
Mark as studied
07
Taylor polynomial
The polynomial of degree $n$ that matches $f$ and its first $n$ derivatives at a point $a$ . Degree 1 is the tangent line, degree 2 adds curvature; the higher the degree, the wider the region where it is a good approximation. At $a = 0$ it is called a Maclaurin polynomial.
UniversityTaylor series
Mark as studied
08
Antiderivatives and indefinite integrals
$F$ is an antiderivative of $f$ if $F' = f$ ; all of them differ by a constant, written $\int f(x)\,\dd x = F(x) + C$ . Many elementary functions, such as $e^{-x^2}$ , have no elementary antiderivative.
FundamentalIntegrals
Mark as studied
09
Definite integral
$\int_a^b f(x)\,\dd x$ is the limit of Riemann sums: the signed area between the graph and the axis, or, more generally, the total accumulated by a rate $f$ over $[a, b]$ .
FundamentalIntegrals
Mark as studied
10
Fundamental theorem of calculus
Differentiation and integration are inverse operations. Accumulating a rate of change recovers the total change: $\int_a^b f'(x)\,\dd x = f(b) - f(a)$ , and the derivative of an accumulated quantity is the rate.
FundamentalFundamental theorems
Mark as studied
11
Numerical series
$\sum_{k=1}^\infty a_k$ is the limit of the partial sums $S_n = a_1 + \dots + a_n$ . Terms going to zero is necessary but not sufficient: the harmonic series $\sum 1/k$ diverges, while $\sum 1/k^2 = \pi^2/6$ .
FundamentalSeries
Mark as studied
12
Taylor and Maclaurin series
Let the order go to infinity: $f(x) = \sum_{k\ge0}\frac{f^{(k)}(a)}{k!}(x - a)^k$ whenever the remainder tends to 0. Functions equal to their Taylor series are called analytic; $e^x$ , $\sin$ , $\cos$ are, on all of $\R$ .
UniversityTaylor series
Mark as studied
13
Functions of several variables
$f : \R^n \to \R$ (or $\R^m$ ): many inputs, one (or many) outputs. With two inputs the graph is a surface over the plane; with a million inputs — the weights of a model — we reason with its level sets and its gradient.
UniversityMultivariable calculus
Mark as studied
14
Partial derivatives
$\frac{\partial f}{\partial x_i}$ : the derivative with respect to one variable, holding the others fixed. Each answers "how sensitive is the output to this input?" — for a neural network, to this weight.
UniversityMultivariable calculus
Mark as studied
15
Gradient
$\nabla f = \left(\frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n}\right)$ : the vector of all partial derivatives. It points in the direction of steepest ascent, its length is that steepest slope, and it is perpendicular to the level sets. Walk against it and you go downhill fastest.
UniversityMultivariable calculus
Mark as studied