Limits and Continuity
If the limit never reaches anything...
...The limit does not exist!
Derivative formula
f'(x)= lim f(x+h)- f(x)/h as h->0
Mean value theorem
f'(c)= f(b) - f(a)/b - a
The limit formula is
lim f(x)=L as x->c
What are the 3 main rules to find the derivative
Product rule
Quotient rule
Chain rule
Concave up/down
particle moving right and up/left and down
con. up= f''(x)>0
con. down= f''(x)<0
particle moving right/up= v(t)>0
particle moving left/down= v(t)<0
Types of discontinuity
Holes, jumps, vertical asymptotes
Derivative of sin x, cos x, tan x, ln x, and e^x
sin x= cos x
cos x= -sin x
tan x= sec^2 x
ln x= 1/x
e^x= e^x
Critical number
f'(c)= 0 or f'(c) is undefined
Three conditions for continuity at x=c
1: f(c) is defined
2: lim f(x) as x->c exists
3: lim f(x)=f(c) x->c
Altenate derivative formula
f'(c)= lim f(x) - f(c)/x - c as x->c
L'Hôpital's rule formula
lim f(x)/g(x)= lim f'(x)/g'(x)
If a function f(x) is continuous on a closed interval [a,b]
it must take on every value of K between f(a) and f(b) at least once
Differentiability implies
continuity but does not imply differentiability
Linearization/ local linear approximation
A complex curve f(x) near a specific point (a) using its tangent line L(x)= f(a)+ f'(a)(x-a)