fundamental theorem of calculus: derivative of an integral form
if g(x) = ∫xa f(t)dt where a stands for a constant, and f is continuos in the neighbourhood of a, then g’(x) = f(x)
the natural logarithm function
ln x = ∫x1 1/t dt
where x is a positive number
derivative of lnx
d/dx (lnx) = 1/x
integral of the reciprocal function
∫ 1/u du = ln |u| + C
the uniqueness theorem for derivatives
If: 1. f’(x) = g’(x) for all values of x in the domain, and
f(a) = g(a) for one value, x = a, in the domain,
then f(x) = g(x) for all values of x in the domain
the uniqueness theorem verbally
if two functions have the same derivative everywhere and they also have a point in common, then they are the same function
logarithm properties of ln - product
ln (ab) = lna + lnb
logarithm properties of ln - quotient
ln(a/b) = lna - lnb
logarithm properties of ln - power
ln(ar) = r lna
logarithm properties of ln - intercept
ln 1 = 0
algebraic definition: logarithm
a = logbc if and only if ba = c where b > 0, b ≠ 1, and c > 0
change-of-base property for logarithms
logbx = logax/logab
in particular: logbx = logex/logeb = lnx/lnb = 1/lnb * lnx
e (functionally)
e = lim n→0 (1+n)1/n = lim n→∞ (1 + 1/n)n
e (numerically)
e = 2.7182818284…
indeterminate forms
0/0, 00, 1^∞, ∞/∞, 0(∞), ∞0, ∞-∞
determinate forms
∞+∞=∞, -∞-∞=-∞, 0^∞=0, 0^-∞=∞, ∞*∞=∞
transcendental numbers
like e or π, numbers that are not only irrational, but that are not able to be expressed using only a finite number of the operations of algebra performed on integers, it goes beyond these operations
inverse relationship between ex and lnx
ln(ex) = x, and elnx = x
exponential with base b
bx = exlnb
derivative and integral of a base-b exponential function
for any positive constant b≠1 (to avoid division by zero)
d/dx (bx) = bxlnb ∫budu = bu/lnb + C
logarithmic differentiation
take ln of both sides
ln of a power
differentiate implicitly with respect to x
solve for y’
substitute for y
l’hopitals rule
if f(x) = g(x)/h(x) and if lim g(x0 as x—>c = lim h(x) as x—>c = 0
then lim f(x) as x—>c = lim g’(x)/h’(x) as x—>c provided the latter limit exists
limit-function interchange for continuous functions
for the function f(x) = g(h(x)), if h(x) has a limit, L, as x approaches c and if g is continuous at L, then lim g(h(x)) as x—>c = g(lim h(x) as x—>c)
∫sinxdx
= -cosx + c
∫cosxdx
= sinx + c
∫tanxdx
= -ln|cosx| + C = ln|secx| + C
∫cotxdx
= ln|sinx| + C = -ln|cscx| + C
∫secxdx
= ln|secx + tanx| + C
∫cscxdx
= -ln|cscx + cotx| + C = ln|cscx - cotx| + C