Basics to Learn Flashcards
derivative of cos
-sinx
derivative of sin
cosx
an integral of cos
sinx
an integral of sin
-cosx
sin(π/6)
1/2
sin(π/4)
√2/2
sin(π/3)
√3/2
cos(π/6)
√3/2
cos(π/4)
√2/2
cos(π/3)
1/2
tan(π/6)
√3/3
tan(π/4)
1
tan(π/3)
√3
chain rule f(g(x))
g’(x) f’(g(x))
product rule
(uv)’ = u’v + uv’
Quotient rule
(u/v)’ = u’v - uv’ / v^2
finding the area between two curves
∫ᵇₐ(function of top curve - function of bottom curve)dx
integral of sec²x
tanx
integral of eᵏˣ
1/keᵏˣ
integral of 1/x
ln|x|
integral of f’(x)/f(x)
ln|f(x)|+c
sin cos identity
sin²x+cox²x=1
sec tan identity
sec²x=1+tan²x
cot cosec identity
1 + cot²x = cosec²x
sin(2x)
2sin(x)cos(x)
cos(2x) =
cos²x-sin²x
tan(a+b)
tana+tanb/1-tanatanb
y = f(x) + a
translates a in the y direction
y = f(x+a)
translates -a in the x direction
y = af(x)
stretch SF a in y direction
y = f(ax)
stretch of 1/a in x direction
y = -f(x)
reflection in x axis
y = f(-x)
reflection in y axis
limit as x tend to 0 of sinx/x
1
limit as x tends to 0 of (1-cosx)/x²
1/2
limit as x tends to infinity of (1+1/x)²
e
limit as x tends to minus infinity of (1+1/x)²
e
limit as x tends to 0 of (1+x)¹/ˣ
e
limit as x tends to 0 of (log(1+x))/x
1
limit as x tends to 0 of (eˣ-1)/x
1
limit as x tends to infinity eˣ/xᵇ
infinity (for all b>0)
limits as x tends to infinity logx/xᵇ
0
limit as x tends to 0 from the right of xᵇlogx
0
L’Hopitals Rule
Let a,b,c∈ℝ such that a≤b≤c and a!=b. Let Ω = (a,b){c}. Let f,g: Ω -> ℝ be differentiable functions such that g’(x)!=0 for all x∈Ω. Assume that one of the following conditions holds:
(i) limit as x tends to c of f(x) = 0 and limit as x tends to c of g(x) = 0
(ii) limits as x tends to c of f(x) = +- infinity and limit as x tends to c of g(x) = +- infinity
