ap calc bc vocab/theorems

1 sem ap calc bc for fi

4 basic concepts of calc

limits, derivatives, (definite) integral, and (indefinite) integral

4 representations of calc

algebraic, graphical, numerical, verbal

continuity

a function is continuous at x=c if and only if…

1. lim_(x—>c) f(x) exists

2. f(c) exists

3. lim_(x—>c) f(x) =f(c)

Intermediate Value Theorem (IVT)

if a function f is continuous for all x in the closed interval [a,b] and y is a number b/w f(a) and f(b), then there is a number x=c in (a,b) for which f(c)=y

definition of a derivative (c form)

lim_x—>c (f(x) - f(c)/x-c)

definition of a derivative (h form)

lim_x—>h (f(x+h) - f(x)/h)

Mean Value Theorem (MVT)

if:

1. f is differentiable for all x b/w (a,b)

2. f is continuous at x=a and x=b

then there is at least 1 number C b/w (a,b) such that f’(c) = f(b) - f(a) / b-a

fundamental theorem of calculus

if f is an integrable function g(x) = ∫f(x)dx, then ∫_abf(x)dx​ = g(b) - g(a)