f: A to B, (describe domain)
A : domain, B : codomain
(f。g。h)(x) = ((f。g)。h)(x) = (f。(g。h))(x)
= f(g(h(x)))
Translation: 平移
y = f(x)+-c
y = f(x+-c)
Reflection: 鏡射
y = f(+-x)
y = +-f(x)
Symmetry: 對稱性
f(-x) = f(x) even function
f(-x) = -f(x) odd function
Families of functions: parameter's (變數) chaning
power functions
polinomial functions: none-zero
rational functions
Monotonic (單調)
increasing
decreasing
Asymptotes (漸近線)
vertical
horizontal
oblique (斜)
Cusps
y = x^(2/3)
f: A to B
f^(-1)(x) = g(x)
g: B to A
1. y = f(x), A = ?, B = ?
2. Solve x as a function of x (if possible)
3. x = f^(-1)(x)
lim f(x) = lim f(x) = L <-> lim f(x) = L(finite number)
x->a+ x->a- x->a
f(x) = p(x) / q(x)
q(x) != 0 => lim f(x) = f(a)
x->a
q(x) == 0 and p(x) != 0 => DNE (vertical asymptote)
and p(x) == 0 => ?
L +- epsillon
a +- delta ; > or < N
|f(x) - L| < epsillon if 0 < |x - a| < delta
=> |f(x) - L| < epsillon if 0 < |x - a| < delta1 < delta
lim f(x) = f(c)
x->c
---------------
[a,b]
lim f(x) = f(c), x on [a, b]
x->c
lim f(x) = f(c)
x->a+
lim f(x) = f(c)
x->b-
---------------
f and g continues at c
=> f +/* g continues at c
---------------
lim g(x) = L and f(x) continues at L
x->c
lim f(g(x)) = f( lim g(x) ) = f(L)
x->c x->c
---------------
f(g(x)) and g(c) continue at c
=> lim f(g(x)) = f(g(c))
x->c
---------------
f has f^-1
if f is continuous then f^-1 is, too.
f(x) continues at [a, b] and k is [f(a), f(b)]
=> exists at least x in [a,b] that f(x) = k
三角函數：c in natural domain -> they are continues
x->0 sinx ~~ 0 cosx ~~ 1
f(x) >= g(x) >= h(x) (all exist) and lim f(x) = lim h (x) = L
=> lim g(x) = L
Differential is
h = x - x0 , x = x0 + h
mtan = lim (f(x0 + h) - f(x0))/h
h->0
Domain(mtan) 包含於 Domain(f)
limit < continuity < differentiability
d(c * x^n)/dx = c * n * x^n-1 (c for linear)
---------------
d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
---------------
(sinx)' = cosx
(cosx)' = -sinx
(tanx)' = sec^2x
(secx)' = tanx * secx (= 1/cosx)
---------------
Chain rule:
d/dx [f(g(x))] = f'(g(x)) * g'(x)
Implicit differentiation: y = f(x) explicit
Related rates
Local linear approximation:
f(x) is differential
(y - y0) ~~ dy = f'(x)dx
differential form: dy = f'(x) * dx
---------------
f'(x) []
increasing >0
decreasing <0
constant =0
***
critical point: relative extrema at c
f'(c) = 0 stationary point
f'(c) DNE
f''(x) ()
concave up >0
concave down <0
***
inflection points x0 that f''(x0) = 0
可能的，還得看+-以及存在與否
Local/Relative Extrema
critical points
+- f', f''
Global/Absolute Extrema [a,b]
critical points
f(a), f(b), f(critical points)
***
Continues on [a,b] -> exist on [a,b]
and exist a relative extrema -> the extrema is the absolute extrema