Part I
(1) Calculus
Generally should have college freshman and
sophomore calculus.
(1.1) Functions
So, there can understand better what a
function is. E.g., function
f(x) = 3x^2 + 1.
(1.2) Derivatives
Then will learn how to find the slope of
the graph of a function. That is the
derivative of the function. E.g., for
function f with f(x) = 3x + 2, as in high
school algebra, the slope is 3. Then for
each x, the derivative of f at x is just
3.
The derivative of function f is denoted by
either of
f'(x) = d/dx f(x)
E.g., for function f(x) = 3x^2 + 1 it turns
out that
f'(x) = 6x.
(1.3) Integration
For function
g(x) = 6 x
maybe we want to know what function f(x)
will give us
f'(x) = g(x)
Finding such a function f is
anti-differentiation, that is, undoes
differentiation. So, sure,
f(x) = 3x^2 + C
for any constant C.
Such anti-differentiation is also the way
to find the area under a curve. So, can
use that to find the area of a circle,
volume of a cylinder, etc. Doing that the
anti-differentiation is integration.
The fundamental theorem of calculus shows
how differentiation and integration are
related.
(1.4) Analytic Geometry
Commonly taught at the beginning of a
calculus course is analytic geometry.
So, take a cone an cut it. Then the cut
surfaces will be one of a circle, an
ellipse, a parabola, a hyperbola, or just
two crossed straight lines. So, those
curves are from a cone and are the conic
sections.
There is some simple associated algebra.
Conic sections are important off and on;
e.g., applied math is awash in circles;
the planets move in ellipses; a baseball
moves in a parabola or nearly so; an
electron moving toward a negative charge
will turn away from that charge in a
hyperbola.
It turns out that in linear algebra
(below) circles and ellipses are important.
(1.5) Role of Calculus
Calculus was invented by Newton as part of
working with force and acceleration for
understanding the motion of the planets.
E.g., if at time t function d(t) gives
distance traveled, then function v(t) =
d'(t) is the velocity at time t and
function a(t) = v'(t) is the acceleration
at time t.
Then Newton's second law is
F(t) = m a(t)
where F(t) is the force at time t applied
to mass m.
Calculus is the first approach to the
analysis of continuous change and is a
pillar of civilization.
Knowledge of calculus will commonly be
assumed in work in ML/AL, data science,
statistics, optimization, applied math,
engineering, etc.
E.g., a lot in ML, AI, and data science is
getting best fits to data; best fitting is
to minimize errors in the fit; such
minimization is mostly a calculus problem;
one of the main steps in ML is steepest
descent, and that is from a derivative.
Probability theory (e.g., evaluating coin
tossing, poker hands, accuracy in ML) will
be important in ML/AI, etc.; two of the
basic notions in probability are
cumulative distributions and density
distributions; the cumulative is from an
integration, and the density is from a
differentiation.
(1) Calculus
Generally should have college freshman and sophomore calculus.
(1.1) Functions
So, there can understand better what a function is. E.g., function
(1.2) DerivativesThen will learn how to find the slope of the graph of a function. That is the derivative of the function. E.g., for function f with f(x) = 3x + 2, as in high school algebra, the slope is 3. Then for each x, the derivative of f at x is just 3.
The derivative of function f is denoted by either of
E.g., for function f(x) = 3x^2 + 1 it turns out that (1.3) IntegrationFor function
maybe we want to know what function f(x) will give us Finding such a function f is anti-differentiation, that is, undoes differentiation. So, sure, for any constant C.Such anti-differentiation is also the way to find the area under a curve. So, can use that to find the area of a circle, volume of a cylinder, etc. Doing that the anti-differentiation is integration.
The fundamental theorem of calculus shows how differentiation and integration are related.
(1.4) Analytic Geometry
Commonly taught at the beginning of a calculus course is analytic geometry.
So, take a cone an cut it. Then the cut surfaces will be one of a circle, an ellipse, a parabola, a hyperbola, or just two crossed straight lines. So, those curves are from a cone and are the conic sections.
There is some simple associated algebra.
Conic sections are important off and on; e.g., applied math is awash in circles; the planets move in ellipses; a baseball moves in a parabola or nearly so; an electron moving toward a negative charge will turn away from that charge in a hyperbola.
It turns out that in linear algebra (below) circles and ellipses are important.
(1.5) Role of Calculus
Calculus was invented by Newton as part of working with force and acceleration for understanding the motion of the planets.
E.g., if at time t function d(t) gives distance traveled, then function v(t) = d'(t) is the velocity at time t and function a(t) = v'(t) is the acceleration at time t.
Then Newton's second law is
where F(t) is the force at time t applied to mass m.Calculus is the first approach to the analysis of continuous change and is a pillar of civilization.
Knowledge of calculus will commonly be assumed in work in ML/AL, data science, statistics, optimization, applied math, engineering, etc.
E.g., a lot in ML, AI, and data science is getting best fits to data; best fitting is to minimize errors in the fit; such minimization is mostly a calculus problem; one of the main steps in ML is steepest descent, and that is from a derivative.
Probability theory (e.g., evaluating coin tossing, poker hands, accuracy in ML) will be important in ML/AI, etc.; two of the basic notions in probability are cumulative distributions and density distributions; the cumulative is from an integration, and the density is from a differentiation.