Theorems

Question 1

First fundamental theorem of calculus

Answer 1

For a continuous function, the value of any function is the rate of change of its integral over a given function
∫ba f(x) dx = F(b) − F(a)
h(b) = h(a) + ∫ba h’(x) dx (most common)

Question 2

Second fundamental theorem of calculus

Answer 2

Derivative of the integral function is equal to the integrand
Common form:
d/dx ∫xa f(t) dt = f(x) * x’ - f(a) * a’
General form:
d/dx ∫vu f(t) dt = v’ * f(v) - u’ * f(u)

Question 3

Net distance formula

Answer 3

∫ba v(t) dt

Question 4

Total distance formula

Answer 4

∫ba |v(t)| dt

Question 5

Average rate of change

Answer 5

[ f(b) - f(a) ] / [b - a]

Question 6

Average value

Answer 6

[ ∫ba f(x) dx ] / [b - a]

Question 7

Mean value theorem (derivatives)

Answer 7

If f is continuous and differentiable, then there is some c in (a,b) such that…
f ‘(c) = [ f(b) - f(a) ] / [b - a]
(rate of change)

Question 8

Mean value theorem (integrals)

Answer 8

If f is continuous, then there is some c in (a,b) such that…
f(c) = [1 / (b - c) ] * ∫bc f(x) dx
(average value)

Question 9

Intermediate value theorem

Answer 9

If f is continuous over a closed interval [a, b], it encompasses every value between f(a) and f(b) within that range.

Question 10

Extreme value theorem

Answer 10

If f is continuous on a closed and bounded interval, it is guaranteed to have both a maximum and a minimum value on that interval

Question 11

Rolle’s theorem

Answer 11

If f is continuous and differentiable on [a,b] AND f(a) = f(b) , Rolle’s theorem shows that there is some c in [a,b] where f ‘(c) = 0