Fundamental Thm O Calculus

Question 1

What is linearity of integration

Answer 1

Linearity of integration is:
If f,g are integrable on [a,b] then f+g is integrable and integral (f+g) = integral f + integral g
Integral b down to a Af = A integral b down to a f

Question 2

If f(x) <= g(x) for all a,b in interval, what does this mean for the integrals

Answer 2

If f(x) <= g(x) for all a,b in interval, then integral b down to a f(x) <= integral b down to a g(x)

Question 3

If f is integrable on [a,b], then what does it mean for mod(f)

Answer 3

If f is integrable on [a,b], it means mod(f) is integrable on [a,b] and mod( integrable b down to a f) <= integral b down to a mod(f)

Question 4

What is 1st version of fundamental theorem of calculus

Answer 4

1st version of fundamental theorem of calculus is:
There is a function I : [a,b] to R s.t
F is continuous and differentiable on [a,b]
F’(x) = f(x) F is primitive/derivative of f
Then integral b down to a f = F(b) - F(a)

Question 5

What does the mean value theorem state

Answer 5

Mean value theorem states:

F(Xj+1) - F(Xj) = F’(Cj)(Xj+1 - Xj)

Question 6

If F(x) = integral x down to a f, then what is F

Answer 6

If F(x) = integral x down to a f, then F is:
Continuous on [a,b]
If f is continuous on (a,b), then F’(x) = f(x) for all x in (a,b)

Question 7

What does 2nd version of fundamental theorem of calculus state

Answer 7

2nd version of fundamental theorem of calculus states:
Define F = integral x down to a f
Then F: [a,b] to R is continuous on [a,b]
F’(c) = f(c) if f is continuous at c in (a,b) (then F is differentiable at c )

Question 8

What happens if f is not continuous at c

Answer 8

If f is not continuous at c then:

In general F may not be differentiable at c, but if f is increasing it is integrable