A net change is thought of as an increment and written as:
$$ \Delta x = x_2 - x_1 $$
Which reads delta x is equal to the difference of x at position 2 and position 1.
Slope:
$$ m = \dfrac {rise} {run} = \dfrac {\Delta y} {\Delta x} = \dfrac {y_2 - y_1} {x_2 - x_1}$$
$$ m^\prime = \dfrac {y^\prime_2 - y^\prime_1} {x^\prime_2 - x^\prime_1} = \dfrac {\Delta y^\prime} {\Delta x^\prime} $$
The angle of inclination of a line that crosses the x-axis is the smallest angle when measured counterclockwise. The slope of a line is the tangent of the line’s angle of inclination.
$$ m = tan \phi $$
$$ \phi_1 = 89^\circ59^\prime $$
$$ m_1 = tan \phi_1 \approx 3437.7 $$
…
$$ d = \sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
Euler was the one who invented the language that says “y is a function of x” or “y equals f of x”:
$$ y = f(x) $$
Such as the formula to describe the area of a circle as a function of its radius:
$$ A = \pi r^2 $$
We can also define functions in pieces:
$$ y = f(x) = \begin{cases} -x & \text{ if } x\lt 0, \cr x^2 & \text{ if } 0\le x \le 1, \cr 1 & \text{ if } x \gt 1. \end{cases} $$
We can look at absolute values such as:
$$ |x| = \begin{cases} x & \text{ if } x\ge 0, \cr -x & \text{ if } x\le 0. \end{cases} $$
Or we can look at absolute values:
$$ |a| \le 5 \iff -5 \le a \le 5 $$
A line created by connecting two points on a curve is called a secant to the curve.
Fermat in 1629 came up with how to define slopes and tangent lines:
- Stare with what we can calculate. Slope of the secant through P and a point Q nearby.
- Find the limiting value of the secant slope as Q approaches P along the curve.
- Take this number to be the slop of the curve at P and define the tangent to the curve at P to be the line through P with this slope.
That means work backwards until slope stops increasing.
The derivative of a function (per Fermat):
$$ f^\prime(x) = \lim_{\Delta x\rightarrow 0} \frac {f(x + \Delta x) - f(x)} {\Delta x} $$
Velocity function:
$$ s = 1/2 gt^2 $$
where
$$ g = 980 cm/sec^2 $$
Limit:
$$ \lim_{t\rightarrow c} F(t) = L $$
Which says the limit of F(t) as t approaches c is L.
The notation below is from Leibniz. Turns out that Newton and Leibniz had practical solutions but it wasn’t until Louis Cauchy that there was more rigorous definitions of limits and derivatives. Cauchy worked off of Lagrange’s work on error estimates as well as Taylor series. The epsilon in Cuachy’s work can from the French word erreur. $$ \frac {dy} {dx} $$
A function is continuous at every point at which it has a derivative.
Rule 1: The derivative of a constant is zero.
Rule 2: The power rule for positive integer powers of x: $$ \frac d {dx} (x^n) = nx^{n-1}$$ Rule 3: The constant multiple rule: $$ \frac d {dx} (cu) = c \frac {du} {dx} $$ Rule 4: The sum rule: $$ \frac d {dx} (u + v) = \frac {du} {dx} + \frac {dv} {dx} $$ Rule 5: The product rule: $$ \frac d {dx} (uv) = u \frac {dv} {dx} + v \frac {du} {dx} $$ Rule 6: Positive Integer Powers of a Differentiable Function:
Rule 7: The Quotient Rule: $$ \frac d {dx} (\frac u v) = \frac {v \frac {du} {dx} - u {\frac {dv} {dx}}} {v^2} $$
Rule 8: Negative Integer Powers of a Differentiable Function