Index Index
Algorithms
Euler’s Method, Algorithm
Derivative, Section
defined as a differential ratio, Paragraph
higher order derivatives, Paragraph
of \(\inverse\sin(x){}\), Example
prime notation for, Paragraph
Differential Equations
defining functions via, Example
Drills
amplitude, period, sinusoidal axis, and phase shift, Drill
approximation, Drill
asymptote
horizontal, Drill
binary digits, Drill
curvature of a circle, Drill
Definitions
Near, Drill
derivative
doesn’t exist, Drill
of sine, Drill
derivative changing signs, Drill
of an exponential, Drill
of logarithms, Drill
of the natural logarithm, Drill
differentiation, Drill Drill Drill Drill Drill Drill Drill Drill Drill Drill Drill Drill Drill Drill Drill Drill
brute force, Drill
Constant, Sum, and Product rules are fundamental, Drill
logarithmic, Drill
Euler’s Identity, Drill
functions
natural logarithms, Drill
graphing, Drill
without a formula, Drill
hanging chain, Drill
intervals, Drill
intervals, open and closed, Drill
limit
infinite, Drill
graphs, Drill
indeterminate forms, Drill
L’Hôpital’s Rule, Drill
one–sided, infinite, Drill
line
lines
perpendicular, Drill
L’Hôpital’s Rule, Drill
natural exponential, Drill
Newton’ Method, Drill
definition of global maximum, Drill
definition of local maximum, Drill
maximal parallelogram, Drill
Quotient Rule, Drill
refraction, Drill
Similar graphs due to limited knowlege, Drill
sine and cosine as line segments, Drill
sine to cosine phase shift, Drill
slope, Drill
Squeeze Theorem, Drill
the Second Derivative Test, Drill
thrust, Drill
variable names don’t matter, Drill
Euler
Euler’s Identity, Drill
evaluation notation, Example
Examples, Example
approximating \(\sqrt{2}\) via the Bisection Method, Example
curvature, Example
Derivative
of \(\inverse\sin(x){}\), Example
of the Heaviside Function, Example
of trigonometric functions, Example
via limits, Example
Differential Equations
defining functions via, Example
brute force, Example
the Quotient Rule, Example
Double Angle Formulas, Example
equation vs. parametric form, Example
evaluation notation, Example
exponential decay, Example
half–life, Example
exponential growth, Example
flight path of an airliner, Example
\(\pm \infty \), Example
at \(\infty \), Example
at a discontinuity, Example
formal proof, Example
infinite, Example
of composed functions, Example
vertical asymptotes, Example
logarithmic differentiation, Example
logarithms, Example
maximal triangle, Example
Newton’s Law of Cooling, Example
Newton’s Method, Example
subtle failure of, Example
normal line, Example
constructing a square on a circle, Example
constructing a square on a line, Example
constructing a square on a parabola, Example
constructing a square on a parabola, redux, Example
minimal perimeter, Example
transition points, Example
pitch of a sound wave, (Hertz), Example
Population Growth, Example
Product Rule, Example
quadratic in form, Example
speed, Example
the Absolute Value Function, Example
the Principle of Local Linearity, Example
the unit circle, Example
Limit
infinite, Definition
Natural Logarithm
properties, Paragraph
Newton, Isaac
and the Law of Cooling, Example
Second Law of Motion, Paragraph
Newton-Raphson Method. See Newton’s Method
Newton’s Method, Section
Principle of Local Linearity, Paragraph Example Paragraph Example Paragraph Definition Example Paragraph Paragraph Subsection Paragraph Paragraph Paragraph
failure of, non-differentiability, Paragraph
failure of, non-differentiability of the Absolute Value function, Paragraph
Product Rule, Example
Proof by Contradiction. See Sherlock Holmes’ Maxim
Proof by Elimination. See Sherlock Holmes’ Maxim
Reductio Ad Absurdum. See Sherlock Holmes’ Maxim
Sherlock Holmes’ Maxim, Digression Paragraph Paragraph
largest triangle with a fixed perimeter, Problem
literary source, Digression
statement of, Principle
Snell’s Law
Derivation via differentials, Subsection
Snell’s Window, Paragraph
Table
Differentiation Rules
General, Table
Tables
Differentiation Rule
Inverse Trigonomentric, Table
Differentiation Rules
Trigonometric, Table
The First Derivative Test Analysis, Table
The Second Derivative Test Analysis, Table
trigonometric functions
cosine, Paragraph
TRIUMPHS, Primary Source Project
Fermat’s Method for Finding Maxima and Minima, TRIUMPHS Project
Fourier’s Infinite Series Proof of the Irrationality of \(e\), TRIUMPHS Project
How to Calculate Pi: Machin’s Inverse Tangents, TRIUMPHS Project
Hôpital’s Rule, TRIUMPHS Project
The Derivatives of the Sine and Cosine Functions, TRIUMPHS Project
The Radius of Curvature According to Christiaan Huygens, TRIUMPHS Project
Vocabulary
acceleration, Paragraph
amplitude, Paragraph
analytic functions, Paragraph
angular speed, Paragraph
angular velocity, Paragraph
antiderivative, Problem
arctangent, Paragraph
bound, Example
bounded, Example
bounded above, Paragraph
carrying capacity, Paragraph
Catenary
weighted catenary, Problem
complex numbers, Paragraph
compound interest, Paragraph
nominal yield, Paragraph
concave downward, Paragraph
concave upward, Paragraph
constant, Item
of acceleration, Paragraph
constraints, Paragraph
continuity, Paragraph
continuous, Paragraph
coordinates
Cartesian, Paragraph
polar, Paragraph
rectangular, Paragraph
cycloid, Paragraph
Definition
Near, Definition
derivative, Digression Digression Digression Paragraph
deviation angle, Paragraph
differentiable, Definition Paragraph Section Problem
dynamical system, Paragraph
equilibrium points, Paragraph
Euler’s Identity, Paragraph
exponential functions, Paragraph
fluent, Paragraph
fluxion, Paragraph
global maximum, Problem
global minimum, Paragraph Definition Problem
half–life, Paragraph
Hertz, Example
Hooke’s Law, Paragraph
horizontal asymptote, Paragraph
Hyberbolic Functions
hyberbolic cosine, Paragraph
hyberbolic sine, Paragraph
Hyperbolic Trigonometric Functions, Paragraph
Hôpital’s Rule, Paragraph
Indeterminate Form, Paragraph
index of refraction, Paragraph
infinite series, Paragraph
Initial Value, Definition
Initial Value Problem, Paragraph
Initial Value Problem (IVP), Definition
injective, Paragraph
interval, Digression
closed, Digression
open, Digression
intrinsic growth rate, Paragraph
left–hand, Paragraph
one–sided, Paragraph
right–hand, Paragraph
linear speed, Paragraph
linear velocity, Paragraph
local extrema, Paragraph
local minimum, Definition
local properties, Paragraph
logarithm
common logarithm (base 10), Paragraph
Logarithmic Differentiation, Paragraph
minimum viable population, Problem
multifunction, Paragraph
multifunction branch, Paragraph
natural exponential function, Definition Paragraph Paragraph Digression Paragraph
near, Paragraph
objective function, Paragraph
one–to–one function, Paragraph
operator notaton, Digression
optimal transition points, Paragraph
osculating, or “kissing” circle, Problem
parameter (see Vocabulary/constant)
parametric function, Digression
Possible Inflective Transition Points (PITPs), Paragraph
Possible Optimal Transition Points (POTPs), Paragraph Definition
Principal of Local Linearity, Paragraph
radian, Paragraph
radian measure, Paragraph
radioactive decay, Paragraph
radiocarbon dating, Paragraph
rainbow angle, Paragraph
velocity, Paragraph
recovery rate, Paragraph
Reductio Ad Absurdum (see Sherlock Holmes’ Maxim)
relative growth rate, Paragraph
relative rate of growth, Example
secant line, Paragraph
second order differential equation, Paragraph
spring constant, Paragraph
tangent line, Definition
tangential velocity, Paragraph
Taylor Polynomial, Paragraph
trajectory, Paragraph
transmission rate, Paragraph
Triangle Inequality, Paragraph
trigonometric functions, Paragraph
\(\arctan\), Paragraph
\(\inverse\cot\), Paragraph
\(\inverse\tan\), Paragraph
arccotangent, Paragraph
undefined, Digression Digression
undefined derivative, Example
variable, Item
vectors, Example
Parallelogram Rule, Example
Vector Addition, Example
with respect to, Paragraph
“in the limit”, Paragraph
