AP Calc AB Theorems

Theorems for the calc test. 5 BOUND RAHHH

Composition Rule (limits)

If the limit gf g(x) at x= c exists, such that lim(x→c) g(x) = D, and f(x) is continuous at x = D, such that lim x→D f(x) = f(D), then lim(x→c) f(g(x)) = f(lim(x→c) g(x)) = f(D)

Squeeze Theorem

if f(x) <= g(x) <= h(x) and if lim (x→c) f(x) = lim (x→c) h(x) = L, then lim (x→c) g(x) = L

Rational Function theorem for Limits

Rational Function Theorem for Limits states that the limit of a rational function as x approaches a value is the ratio of the leading coefficients of the numerator and denominator.

Extreme Value Theorem

Requirements: f is continuous on the close interval [a.b]

Tells us: f attains a minimum value and a maximum value somewhere in the interval

Intermediate Value Theorem

Requirements: f is continuous on the closed interval [a,b] and M is a number such that f(a) <= M<= f(b), then there is at least one number, c, in the interval [a,b] such that f© = M

L’Hospital’s Rule

Requirements: limits of the following are indeterminate forms 0/0, ∞/∞

Tells us: to find the limit of indeterminate forms, take the derivative of the numerator and denominator functions.

Second Fundamental Theorem of Calculus (2nd FTC)

Requirements: f is continuous on the closed interval [a,b] and F(d/dx) = f),

Tells us: there exists a definite integral from bounds a to b where integral of f at bounds a to b is equal to F(a) - F(b)

Mean Value Theorem for Integrals

Requirements: f is continuous on [a,b]

Tells us: exists at least one number c, a<c<b such that

f(c ) *(b -a) = Integral of f from bounds (a to b)

Fundamental Theorem of Calculus (FTC)

Requirements: f is a continuous function, c is a constant

Tells us: if you define a function F(x), as the accumulation (or integral) of another function f(x), from a fixed point a up to x, then F(x) will be continuous and differentiable, and its derivative at any point x will equal the original function f(x).

Mean Value Theorem

Requirements: f(x) is continuous on [a,b] and differentiable on (a, b)

Tells Us: guarantees the existence of one point where instantaneous rate of change is equal to the average rate of change