Please enable JavaScript.

Coggle requires JavaScript to display documents.

Calculus(The art of) (The new logic (limits (tanscendental functions…

- - - - if f is conious on [a, b] and L is between f(a) and f(b), then there must be at least one c in (a,b) where f(c) = L
    - - intervals of continuity
        
        f(x) = lin(x) is continuous on (0,inf)
        
        [a,b] = both of the above
        
        (a,b] => f is left conious at b=> lim x-> b- f(x) = f(b)
        
        [a,b) => function is right continous at a=> lim x-> a+ f(x) = f(a)
        
        if f is continuous on (a,b), then it is continuous on all poitns between a & b
      - if f and g are continuous at a, are all continous at point a
        
        c = constant, n = post integer
        
        f(g())
        
        (f)^n
        
        f/g, g(a)!=0
        
        fg
        
        cf
        
        g+-g
      - a function is contontinous at a point a if
        
        3) lim x->a f(x) = f(a)
        
        2) lim x-> a f(x) exists [ lim x->a- f(x) = lim x->a+ f(x)
        
        1) f(a) is defined
    - - continous on their domains
      - y=tan^-1(x) HA@ pi/2, -pi/2
      - y=sin(x) DNE
      - y = ln(x) no HA
      - y=e^x HA@ y=0
    - - if the degree of numerator is 1 more than the degree of denominator the the function still approaches +- inf, but does so linearly (oblique or slant asymptote)
      - if the degree of the numerator is greater than the degree of the denominator (top heavy) the function approaches +- inf
      - if the degree of numerator is equal to the degree of denominator the function approaches the ratio of the coefficients
    - - polynomials are unboinded as x -> =- inf => no HA
    - - if lim x -> inf f(x) = L, y=L is a HA of f
    - - lim x->c [f(x)g(x)] = LK
    - - lim x->c [f(x)+-g(x)] = L+-K
    - - lim x->c [bf(x)] = bL
    - - If lim x-> a f(x) = L and lim-> a h(x) = L then limx-> a g(x) = L
      - assume f,g,h are functions satisfying f(x) <= g(x) <= h(x) for all near a
      - Assume f,g,h are functions