intoduction to Calculus

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