MATH - Calc Theorems

Intermediate Value Theorem

If f is continuous on [a,b] then there must be a value “c” on (a,) such that f(c) is between f(a) and f(b)

Requirement for Intermediate Value Theorem

f must be continuous on [a,b]

Guarantee for Intermediate Value Theorem

a c on (a,b) where f(c) is between f(a) and f(b)

Squeeze Theorem

If h(x)≤f(x)≤g(x) for all x in an open interval containing “c” except possibly at c itself, then if lim x→c h(x) = L = lim x→c g(x) then lim x→c f(x) = L

Requirement for Squeeze Theorem

h(x)≤f(x)≤g(x)

lim x→c h(x) = L = lim x→c g(x)

Guarantee for Squeeze Theorem

lim x→c f(x) = L

Extreme Value Theorem

If f is continuous on [a,b], then f must attain at least one maximum and one minimum

Requirement for Extreme Value Theorem

f must be continuous on [a,b]

Guarantee for Extreme Value Theorem

f has one max and one min

Fundamental Theorem of Calculus

If f is continuous on [a,b] and F(x) is an anti-derivative of f(x) then a∫b f(x)dx = F(b) - F(a)

Requirement for Fundamental Theorem of Calculus

f is continuous on [a,b]

F is an anti-derivative of f

Guarantee of for Fundamental Theorem of Calculus

a∫b f(x)dx = F(b) - F(a) or a∫b f’(x)dx = f(b)-f(a)

2nd Fundamental Theorem of Calculus

If f is continuous on an open interval containing “a” then d/dx a∫g(x) f(t)dt - f(g(x)) * g’(x)

Requirement for 2nd Fundamental Theorem of Calculus

f is continuous on an open interval containing “a”

Guarantee for 2nd Fundamental Theorem of Calculus

d/dx a∫g(x) f(t)dt - f(g(x)) * g’(x)

Average Value of a Function

If f is continuous on [a,b] then there is a value c on (a,b) such that (b-a)f(c) = a∫bf(x)dx, where f(c) is the average value of f(x) on [a,b]

Requirement of Average Value Function

f is continuous on [a,b]

Guarantee of Average Value Function

the average value of f(x) on [a,b] = 1/(b-a) a∫bf(x)dx

Mean Value Theorem

if if is continuous on [a,b] and differentiable on (a,b) then there is a c on (a,b) such that f’(c) = f(b) - f(a)/ (b-a)

Requirement for Mean Value Theorem

f is continuous on [a,b]

Guarantee for Mean Value Theorem

a c on (a,b) where f’(c) = f(b) - f(a) / (b-a)