Summary of functions
| class of function ƒ
| ƒ'
| ƒ''
| increasing/decreasing/constant
| extrema
| concavity
| inflection points
|
| constant ƒ(x)= h
| 0
| 0
| constant
| —
| —
| —
|
| linear ƒ(x)= mx
| m
| 0
| m>0: increasing, m<0: decreasing
| —
| —
| —
|
| quadratic ƒ(x)= ax2+bx+c
| 2ax+b
| 2a
| a>0: decreasing, then increasing
a<0: increasing, then decreasing
| Global:
a>0: min, a<0: max
(-b/2a,ƒ(-b/2a))
| a>0: up, a<0: down
| —
|
| cubic ƒ(x)= ax3+bx2+cx+d
| 3ax2+2bx+c
| 6ax+2b
| all inc or all dec or inc/dec or dec/inc
| None (if all dec or all inc) or
Local min and max
| up/down or down/up
| (-b/(3a),ƒ(-b/(3a))
m=ƒ'(-b/(3a))
|
| exponential ƒ(x)= ex
| ex
| ex
| increasing
| —
| up
| —
|
| logarithmic ƒ(x)= ln x
| 1/x
| -1/x2
| increasing
| —
| down
| —
|
| Gaussian ƒ(x)= e-x2
| -2xe-x2
| (4x2-2)e-x2
| increasing on (-∞,0)
decreasing on (0,∞)
| max (0,1)
| up/down/up
| two: (±1/√2,1/√e)
m=±√2/√e≈.858
|
| Standard normal ƒ(x)= e-x2/2/√(2π)
| -xe-x2/2/√(2π)
| (x2-1)e-x2/2/√(2π)
| increasing on (-∞,0)
decreasing on (0,∞)
| max (0,1/√(2π)) ≈.3989
| up/down/up
| two: (±1,1/√(2πe)) ≈.242
m=±1/√(2πe)≈±.242
|
| Trig ƒ(x)= sin x
| cos x
| -sin x
| increasing π, then decreasing π
| maxima ((4n+1)π/2,1)
minima ((4n-1)π/2,-1)
| up for π, down for π
| (nπ,0) n∈ℤ
m=±1
|
| Trig ƒ(x)= tan x
| sec2 x
| 2sec2tan x
| increasing. Vertical asymptote every π
| —
| up for π/2, down for &pi/2;
| (nπ,0) n∈ℤ
m=1 min
|
| Hyperbolic ƒ(x)= cosh x
| sinh x
| cosh x
| decreasing then increasing
| min (0,1)
| up
| —
|
| Hyperbolic ƒ(x)= sinh x
| cosh x
| sinh x
| increasing
| —
| up then down
| (0,0)
m=1 min
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