C7-11 Flashcards
what is an arithmetic sequence
when the difference between two successive terms in a sequence is always the same
what is the formula for arithmetic sequence
Sn = n/2 [2a + (n-1) d]
where Sn, n is subscript
a= first term
d= common difference
what is a geometric sequence
a sequence in which each term is a constant multiple of the preceding term
what is the formula for geometric sequence
Sn= a. (1 - r^n) / (1 - r)
where a= first term
r= common ratio
what is the definition which explains when a sequence converges to the limit 0
A sequence converges to a limit ℓ for every ε >0 there is a natural number N such that for all natural numbers n, if n is equal to or greater than N then |un - ℓ| < ε
we write
lim (as x tends to infinity) un = ℓ
when adding limits
lim (un + vn)
lim (un) + lim (vn)
lim (aun) where aER
a lim (un)
lim (un.vn)
(lim (un)) (lim (vn))
lim (un/vn)
lim un / lim vn
binomial coefficient
(n choose r) n!/ r!(n-r)!
(x + y)^n =?
= sum[r=0,n][(n choose r)x^(n-r)b^r]
if f approaches the limit l near a if ∀ ε > 0 ∃ δ > 0 such that 0 < |x − a| < δ then what
f approaches the limit l near a if ∀ ε > 0 ∃ δ > 0 such that 0 < |x − a| < δ
then =⇒ |f(x)−l| < ε
if f approaches both m and l near a then ?
then l=m
lim[f(x) + g(x)]
x→a
= l + m
lim αf(x) =?
as x tends to a
= αl
lim[αf(x) + βg(x)] = ?
αl + βm
lim f (x)g(x) = ?
= lm
if m doesn’t equal 0
lim 1/g(x) = ?
=1/m
lim f(x) /g(x) = ?
=l/m
lim (x tends to 0) sin/x =?
1
lim (x tends to 0) sin (mx) / nx =?
m/n
when is a function f continuous at a
if lim (x tends to a) f(x) = f(a)
if f and g are continuous at a then
f+g is continuous at a
f.g is continuous at a
if g(a) does not equal 0, then 1/g is continuous at a
if g is continuous at a and f is continuous at g(a) then?
then fog is continuous at a
when is a function f continuous in an open interval (a,b)?
if it is continuous at all points x ∈ (a, b)
when is a function f continuous in a closed interval [a, b]?
if it is continuous in (a, b) and if lim f(x) = f(a) and lim f(x) = f(b)
when is the function f differentiable at a?
if lim (f(a + h) − f(a))/h exists
when f(x) = c (constant function)
f′(a) = lim (c − c)/h = lim 0 = 0
h tends to 0
when f(x)=cx+d
f′(a) = lim [c(a + h) + d − (ca + d)] /h = lim ch/h = lim c = c
h tends to 0
when f(x) = x^2
2a (more working)
when f(x) = x^3
3a^2
if f is differentiable at a
then f is continuous at a
