Definitions 2

Question 1

The exponential

Answer 1

exp(x)=Σ(x^n)/n!

Question 2

Powers

Answer 2

x^p=exp(plogx)

Question 3

Limits of functions

Answer 3

Let I be an open interval, c ∈ I and f a real valued
function defined on I except possibly at c. We say that
limx→c f(x) = L
if for every ε > 0 there is a number δ > 0 so that if 0 < |x − c| < δ then
|f(x) − L| < ε.

Question 4

One sided limits

Answer 4

Let f a real valued function defined on the open interval
(c, d). We say that
limx→c+ f(x) = L
if for every ε > 0 there is a number δ > 0 so that if c < x < c + δ then
|f(x) − L| < ε.

Question 5

Infinite limits

Answer 5

If f : I{c} → R is defined on an interval I except perhaps
at c ∈ I we write
limx→c f(x) = ∞
if for every M > 0 there is a δ > 0 so that if 0 < |x − c| < δ then f(x) > M.

Question 6

Limits at infinity

Answer 6

If f : R → R we write
limx→∞ f(x) = L
if for every ε > 0 there is an N so that if x > N then |f(x) − L| < ε

Question 7

The derivative

Answer 7

Suppose f : I → R is defined on the open interval I and
c ∈ I. We say that f is differentiable at c if limh→0 (f(c + h) − f(c))/h
exists. If so we call the limit f’(c)

Question 8

The trig functions

Answer 8

cos(x) = Σ(-1)^k * (x^2k)/(2k)!
sin(x) = Σ(-1)^k * (x^2k+1)/(2k+1)!

Question 9

Upper and lower sums

Answer 9

Let f : [a, b] → R be bounded and P={x0, x1, . . . , xn} be a partition of [a, b]. The upper and lower Riemann sums of the function
f with respect to P are
U(f, P) = ΣMi (xi − xi−1) and L(f,P)=Σmi (xi − xi−1)
respectively, where for each i
mi = inf{f(x) : xi−1 ≤ x ≤ xi}
and
Mi = sup{f(x) : xi−1 ≤ x ≤ xi}.

Question 10

Upper and lower integrals

Answer 10

Let f : [a, b] → R be bounded. The upper
and lower Riemann integrals of the function f are
upper integral f = inf U(f, P)
lower integral f = sup L(f, P)
where the sup and inf are taken over all partitions of the interval [a, b].

Question 11

Riemann integral

Answer 11

Let f : [a, b] → R be bounded. Then f is said to
be Riemann integrable if
upper integral f = lower integral f
and in this case we write
integral from a to b f(x) dx
for the common value.

Question 12

Refinements

Answer 12

If P and Q are partitions of an interval [a, b] then Q is said
to be a refinement of P if every point of P belongs to Q.

Question 13

Improper integrals

Answer 13

Suppose f : [a, b] → R is integrable on each subinterval [c, b]. We say that f is improperly Riemann integrable on [a, b] if
lim
c→a+ integral from c to b f(x) dx
exists and in that case we call the limit
integral from a to b f(x) dx
and say that the latter improper integral converges

Question 14

Improper integrals with infinity

Answer 14

Suppose f : [a,∞) → R is integrable on each subinterval [a, b]. We say that f is improperly Riemann integrable on [a, ∞) if
lim b→∞ intergal from a to b f(x)dx
exists and in that case we call the limit integral from a to ∞ f(x) dx
and say that the latter improper integral converges.