AM

Question 1

What is the 1st FTC?

Answer 1

If f is integrable on [a,b] and continuous at c=[a,b] and F(x) =∫(x,a)f, then F is differentiable at c with F’(c) = f(c)

Question 2

What is the 2nd FTC?

Answer 2

If f is integrable on [a,b] and f=g’ for some function g, then ∫(b,a) f = g(b)-g(a).

Question 3

What is the definition of the maximum of X?

Answer 3

Let X denote a subset of |R:

We call m∈|R the maximum of X (max X) if m∈X and x≤m ∀x∈X

Question 4

What is the definition of an upper bound of X?

Answer 4

Let X denote a subset of |R:

We call M∈|R an upper bound for X if x≤M ∀x∈X

Question 5

What is the definition of the least upper bound/supremum?

Answer 5

Let X denote a subset of |R:
We call ~M∈|R the least upper bound or supremum of X (sup X) if
a) x≤~M ∀x∈X
b) ~M≤M ∀upper bounds M.

Question 6

sin^2 (x)

Answer 6

1/2(1-cos2x)

Question 7

cos^2(x)

Answer 7

1/2(1+cos2x)

Question 8

sinxcosx

Answer 8

1/2(sin2x)

Question 9

tan^2(θ) + 1

Answer 9

sec^2(θ)

Question 10

d/dx(tanx)

Answer 10

sec^2x

Question 11

d/dx(secx)

Answer 11

secxtanx

Question 12

What is the ε-P definition?

Answer 12

If f is bounded, then f is integrable on [a,b] iff for each ε>0, there exists a parition P of [a,b] s.t
U(f,P) - L(f,P)≤ε