1091 Calculus (I)

1091 Calculus (I)

Homework 1 1. Find the domain of the function

(a) g(t) =√

3−t−√ 2 +t (b) h(x) = 1

√4

x²−5x (c) f(u) = u+ 1

1 + _u+1¹

2. Determine whether the curve is the graph of a function of x. If it is , state the domain and range of the function

3. Give an example of an unbounded function f(x) on (0,1] such that f(x) is bounded on [¹_n,1]for all n= 2,3,· · ·.

4. Let A⊂Rbe bounded above and B :={−a| a∈A}. (a) Prove that B is bounded below.

(b) supA=−infB.

(c) Use the “Least Upper Bound Property” to prove that “any nonempty, bounded below set of real numbers has a greatest lower bound”.

5. Let f and g be two bounded functions.

(a) Prove that f +g, f −g and f g are bounded functions.

(b) Give an example that f

g is unbounded.

(c) Prove that if0< c <|g(x)|for allx, then f

g is a bounded function.

6. Prove that the following three triangle inequalities are equivalent.

(i) |a+b| ≤ |a|+|b| (ii) |a| − |b| ≤ |a−b| (iii) |a| − |b|≤ |a−b| 7. Show that

(a) f(x)g(x)−f(a)g(a)≤ |f(x)|g(x)−g(a)+|g(a)|f(x)−f(a). (b) f(x)

g(x) − f(a) g(a)

≤ |f(x)|g(x)−g(a)

|g(x)g(a)| + |g(x)|f(x)−f(a)

|g(x)g(a)| . 8. Determine whether each of the following functions has a maximum. If

yes, find it; if not, explain your reason.

(a) f(x) = sinx onR. (b) f(x) =−1

x on(0,∞).

(c) f(x) =−1

x on(0,1).

(d) f(x) =−1

x on(0,1].