6. A Development of R
6.1. (b)
For s, t ∈ Q show that s = t
if and only if s* = t*.
6.2.
Show that if α and β are Dedekind cuts, then so is
α + β =
{r1 + r2 :
r1 ∈ α,
r2 ∈ β}.
6.3. (a) Show that α + 0* = &alpha for all Dedekind cuts α.
6.3. (b) How would you define the Dedekind cut –&alpha such that
&alpha + (–&alpha) = 0*?
14. Series
14.1 Determine which of the following series converge. Justify your answers.
17. Continuous Functions
17.3. Prove that the following functions are continuous.
17.4. Prove that the function √x is continuous on its domain
[0, ∞).
17.5. (a) Prove that if m ∈ N, then the function
f(x) = xm is continuous on R.
17.5. (b) Prove that every polynomial function
p(x) is continuous on R.
18. Properties of Continuous Functions
18.1. Let f be as in Theorem 18.1.
Show that if –f assumes its maximum at
x0 ∈ [a, b],
then f assumes its minimum at x0.
18.2. Reread the proof of Theorem 18.1 with [a, b]
replaced by (a, b).
Where does it break down? Discuss.
18.3. Use calculus to find the maximum and minimum of
f(x) = x3 –
6x2 + 9x + 1 on [0, 5).
18.5. (a) Let f and g be continuous functions on
[a, b] such that f(a) ≥ g(a)
and f(b) ≤ g(b).
Prove that f(x0) = g(x0)
for at least one x0 in [a, b].
18.5. (b) Show that Example 1 can be viewed as a special case of part (a).
18.6. Prove that x = cos x for some x in
(0, π / 2).
18.7. Prove that x2x = 1 for some x
in (0, 1).
18.8. Suppose that f is a real-valued continuous function on R
and that f(a)f(b) < 0 for some
a, b ∈ R. Prove that there exists x between
a and b such that f(x) = 0.
19. Uniform Continuity
19.2. Prove that each of the following functions is uniformly continuous on
the indicated set by directly verifying the ε-δ
property in Definition 19.1.
28. Basic Properties of the Derivative
28.2. Use the definition of derivative to calculate the derivatives
of the following functions at the indicated points.
28.15. Prove Leibniz' rule.
29. The Mean Value Theorem
29.11. Show that sin x ≤ x for all x ≥ 0.