Cloned from: Calculus



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essential discontinuity
curve has a vertical asymptote
removable discontinuity
hole in curve with no point "off the curve"; rational expression with common factors in numerator, denominator that are cancelled. f(x) undefined at point
secant
A line segment connecting two points on a curve.
difference quotient
f'(x0) = lim as Δx->0 =
 [f(x0 - Δx) - f(x0)] / Δx
 
Line Equation
y - y0 = m(x - x0)
tangent
The slope of a point on a secant line.
power rule
If y = x^n,
then
dy/dx = nx^(n-1)
dy/dx when y = x
1
dy/dx when y = kx
k
dy/dx when y = k
0
product rule
If f(x) b= uv
then
f'(x) = u(dv/dx) + v(du/dx)
Quotient rule
If f(x) = u/v

then f'(x) =

[v(du/dx) - u(dv/dx)] / v^2
Chain rule
y = f(g(x))
If y = f(g(x))

f'(x) =

[(df(g(x)/dg)](gd/dx)
Chain rule (alternate version)
If y = y(v)
and v = v(x)

dy/dx = (dy/dv) (dv/dx)
limit when highest power in numerator and denominator is the same
coefficient of the highest term in numerator / coefficient of highest term in denominator
lim sin(x)/x as x -> 0
1
lim cos(x - 1) / x as x -> 0
0
lim sin ax / x as x -> 0
a
lim sin ax / sin bx as x -> 0
a / b
slope of a line
m = (y2 - y1) / (x2 - x1)
Conditions for continuity
at x = c
f(c) exists
lim f(x) exists as x -> c
lim f(x) = f(c) as x -> c
jump discontinuity
curve breaks and starts somewhere else or
lim f(x)as x approaches -a =/ lim f(x)as x approaches +a
point discontinuity
curve has a hole in it because the point at that hole is "off the curve"
lim f(x) as x-> a =/ f(a)
difference quotient (slope of secant line)
f(x1 + h) - f(x1) / h
definition of the derivative
f'(x) = f(x1 + h) - f(x1) / h as h -> 0
Power rule
if y = x^n
dy /dx = nx^n-1
dy/dx when y = x
1
dy/dx when y = kx
k
dy / dx when y = k
(k is a constant)
0
Addition rule
If y = ax^n + bx^m, then dy/dx = a(nx^n-1) + b(mx^m-1)
dx / dy when y = k / x
k / x^2
dy / dx when y = sqrt(k)
k / 2 sqrt(x)
product rule
If f(x) = uv, then
f'(x) = u (dv/dx) + v (du/dx)
quotient rule
If f(x) = u / v, then
f'(x) =
[ v(du/dx) - u(dv / dx) ]
/ v^2
chain rule
If y = f(g(x)), then
y' = [df(g(x))/dg] / dg/dx
OR
derivative of outside relative to inside x derivative of inside
dy / dx when
y = y(v) and v = v(x)
(dy / dv) x (dv / dx)
[d / dx]sin x
cos(x)
Mean Value Theorem for Derivatives (MVTD)
Given y = f(x) is continuous on interval a, b, there exists at least one number c between a and b such that
f(b) - f(a) / b - a = f'(c)
Rolle's theorem
If y = f(x) is continuous and differentiable between a & b and f(x) = f(b) = 0, there's at least one number c between a and b such that f'(c) = 0
x of y cards