UNIVERSITY OF ZIMBABWE

DEPARTMENT OF MATHEMATICS AND COMPUTATIONAL SCIENCES

HMTH101 CALCULUS 1 : AUG – DEC 2020 TUTORIAL 1

Dr J. Mushanyu

(Number Systems)

1. Express the following recurring decimals in the form p/q where p and q are integers

(i) 2, 1434˙3 (ii) 0 ˙

, 2˙1˙3.˙

2. Express 0, mnmnmnmn . . . = 0, m˙ n˙ , where m and n are distinct integers, in the form p/q

where p and q are integers.

3. State, giving a reason, whether each of the following numbers is rational or irrational.

(i) 0.20200200020 . . . (ii) 537.137137137 . . . .

4. Does the decimal 0, 1234567891011121314151617181920 . . . whose digits are natural numbers

strung end-to-end represent a rational or an irrational number? Give a reason for your answer.

5. You are told π =

22

7

. Is this true or false? Give a reason.

6. Show that x

2 + y

2 + z

2 ≥ xy + yz + zx, for all x, y, z ∈ R.

7. Show that x

2 +

1

x

2

≥ 2 for all real numbers x 6= 0.

8. Show that

x +

1

x

≥ 2 for all real numbers x 6= 0.

9. Show that if a, b, c, d ∈ R, a2 + b

2 = 1 and c

2 + d

2 = 1, then ac + bd ≤ 1.

10. Show that if 0 < a < b then a

2 < b2

. If a

2 < b2

, is it necessarily true that a < b? Give an

example to illustrate your answer.

11. If a ≥ 0 and b ≥ 0, prove that 1

2

(a + b) ≥

√

ab.

12. Solve the following inequalities.

(i) x

2 + x − 2 > 0 (ii) 2x + 3

x − 5

> 3 (iii) 2|x| > 3x − 10 (iv) |x + 1| ≥ 3

(v) 2|x| − 1 > 3|x| − 3 (vi) 4|x| < 7x − 6 (vii) |x| + 2

x − 2

< 4.

13. Prove that |ab| = |a||b| for all a, b ∈ R.

14. Show that |a + b| ≤ |a| + |b| for all a, b ∈ R.

15. Prove that |x|

2 = x

2

for any real number x.

1

16. If x and y are real numbers, prove that |x| − |y| ≤ |x − y|.

17. Prove that 1

2

+

1

4

+

1

8

+ · · · +

1

2

n−1

< 1 for all positive integers n > 1.

18. Prove the following by induction.

(a) n! > 2

n

for all n ≥ 4.

(b) For all n ≥ 1 and f(x) = x

n

, then f

0

(x) = nxn−1

.

(c) n

2 ≤ 2

n

for all n ≥ 5.

(d) The sum of terms in a geometric series is Xn

i=0

r

i =

r

n+1 − 1

r − 1

, if r 6= 0, r 6= 1, n ∈ N.

(e) 1 + 2 + · · · + n =

n(n + 1)

2

.

(f) For any positive integer n, n

3 + 2n is divisible by 3.

(g) 3n > n2

for n > 2.

(h) 72n − 48n − 1 is divisible by 2304.

(i) All numbers of the form 8n − 2

n are divisible by 6.

(j) 11n − 4

n

is divisible by 7 for all n ∈ Z

+.

(k) |sin nx| ≤ n|sin x| for all x ∈ R, n ∈ Z

+.

(l) 13 + 23 + 33 + · · · + n

3 =

n

2

(n + 1)2

4

.

(m) Prove Bernoulli’s inequality, (1 + x)

n > 1 + nx, for n = 2, 3, . . . , if x > −1, x 6= 0.

(n) 12 + 22 + 32 + · · · + n

2 =

1

6

n(n + 1)(2n + 1).

(o) Prove that if a1, a2, a3, . . . , an are real numbers, then

|a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an|.

(p) Prove that, if sin x 6= 0 and n is a natural number, then

cos x · cos 2x · · · · cos 2n−1x =

sin 2nx

2

n sin x

.

2

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