Stuff You Must Know Backwards In Your Sleep Flashcards
f(x) as x -> a from the right
lim f(x) x -> a+
f(x) as x -> a from the left
lim f(x) x->a-
Situations Limits Fail to Exist
1) LH doesn’t equal RH
2) Increases or decreases w/o bounds
3) Oscillates between two fixed points
lim sinx/x
x->0
1
lim 1-cosx/x
x->0
0
Definition of Continuity
1) f(a) is defined
2) lim f(x) has to exist
x->a
3) lim f(x) = f(a)
x->a
Differentiability implies
Continuity
Continuity does not imply
Differentiability
Formal Definition of the Derivative
d/dx(f(x))=
Lim f(x+deltax)-f(x)/deltax
Alternate form of definition of the derivative
d/dx(f(x)) at x=a
Lim f(x)-f(a)/x-a x->a
Situations Derivatives Fail to Exist
1) Discontinuity
2) Sharp turn (or cusp)
3) Vertical Tangent Line
Chain Rule of f(u)
f’(u)*u’
Product Rule (uv)
uv’+by’
Quotient Rule (u/v)
vu’-uv’/v^2
x^n
nx^n-1
Derivative: sinx
cosx
Derivative: cosx
-sinx
Derivative: tanx
sec^2 x
Derivative: cotx
-csc^2 x
Derivative: secx
secxtanx
Derivative: cscx
-cscxcotx
Derivative: lnx
1/x
Derivative: ln u
u’/u
Derivative: e^x
e^x
Derivative: e^u
e^u•u’
Point Slope Form of a Line
y-y1=m(x-x1)
Area of a Trapezoid
A=1/2h(b1-b2)
The Fundamental Theorem of Calculus
b
$ f(x)dx= f(b)-f(a)
a
*f(x) is antiderivative
2nd FTC
g(x)
d/dx $ f(t)dt=f(g(x))•g’(x)
a