Download Calculus 1 – Assignment 2 Some standard notation: ∨ stands for ”or

Calculus 1 – Assignment 2
Some standard notation: ∨ stands for ”or”. ∧ means ”and”.
Also, a redefinition of the star symbols: F stands near a problem of level
higher than required for exam. FF is yet higher level. ~ is an exam level
problem, but might require some thinking.
1. Prove the following inequalities:
(a) ~
√
n
1
1
+ √
≥1
m
m+1
n+1
for natural m, n
Hint: Use Bernoulli’s inequality.
P
√
√
(b) ~ 2( n − 1) < nk=1 √1k < 2 n, for any natural n.
P
√
Hint for left inequality: Denote Xn = 2 n − nk=1 √1k . Notice
1
that X1 = 1. Show that Xn+1 − Xn < √1n − √n+1
. Conclude that
1
Xn + √n is a decreasing sequence of numbers.
2. Decide which of the following statements are correct (no proof required). Find pairs of complementary statements.
(a) ∀ > 0 ∃x, y ∈ R : |x − y| < .
(b) ∀ > 0 ∃x ∈ R: ∀y ∈ R |x − y| < .
(c) ∀ > 0 ∃x, y ∈ R : |x − y| ≥ .
(d) ∃ > 0 : ∀x, y ∈ R |x − y| < .
(e) ∃ > 0 : ∀x, y ∈ R |x − y| ≥ .
3. Same as previous. This problem is for practice only, and will not be
graded.
√
(a) ∃n ∈ N: ∀x < n : x > x ∨ x ≤ 0.
√
(b) ∀n ∈ N ∃x < n : x > x ∨ x ≤ 0.
√
(c) ∀n ∈ N ∃x > n : x > x ∨ x ≤ 0.
√
(d) ∀n ∈ N ∃x < n : x ≤ x ∧ x > 0.
√
(e) ∀n ∈ N ∃x < n : x ≤ x ∨ x > 0.
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4. Prove that if m, n ∈ Z then either |m − n| ≥ 1, or m = n.
Hint: First, prove that 1 is the minimal natural number.
5. Let A 6= ∅ be a finite subset of R. Prove that max A exists.
Hint: Use induction on the number of elements of A.
6. Let A, B be nonempty(6= ∅) subsets of R. Prove or disprove:
(a) If A and B are both bounded and sup A = inf B then A ∩ B
contains precisely one element.
(b) If A ∩ B = ∅ then sup A 6= sup B.
(c) If A ⊂ B then sup A ≤ sup B.
7. Let f, g : [a, b] → R be bounded functions. Prove that inf(f + g) ≥
inf f + inf g. Show that strict inequality can happen.
8. For a nonempty A ⊂ R, denote −A = {−x|x ∈ A}. Prove that
sup(−A) = − inf(A).
9. Compute supremum, infimum, maximum and minimum (whenever they
exist):
(a) A = { n1 + (−1)n |n ∈ N}
(b) A = {x2 + x + 1|x ∈ R}
(c) A = { m1 + n1 |m, n ∈ N}
2πn
cos
|n
∈
N
(d) ~A = n−1
n+1
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10. Let a, b ∈ R and f (x) = a sin x + b cos x for x ∈ R.
(a) Find real numbers A and φ such that f (x) = A sin(x + φ).
(b) Compute supx∈R f and inf x∈R f .
√
11. Let f (x) = n 1 − xn be defined on [0, 1]. Find f −1 (x).
x
12. Assume f x+1
= x2 . Find f (x).
x
13. ~ Let n ∈ N and f (x) = √1+x
2 . Find f ◦ f ◦ ... ◦ f (x) (here f is written
n times). This is called the n−th iteration of f .
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14. ~ Let f : R → R. Prove that there exist g, h : R → R such that g is
an even function, h is an odd function, and f = g + h.
15. For each function, write
(i) Whether it is even, or odd, or neither.
(ii)Is it periodic? If so, what is the minimal period?
√
√
(a) f (x) = 3 tan x √ (b) f (x) = tan 3 x
√
(c) f (x) = sin 3x + 2 cos 6x
(d)F f (x) = sin x + sin( 2x)
16. Let f : A → B, g : B → A. Also, ∀x ∈ A, g ◦ f (x) = x. Prove or
disprove:
(a) g = f −1 . (b) f is onto. (c) f is one-to-one. (d) g is onto.
(e) g is one-to-one.
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