UNIVERSITY OF ZIMBABWE
DEPARTMENT OF MATHEMATICS AND COMPUTATIONAL SCIENCE
HAI,HCC,HCS,HDS-103 CALCULUS
Mr D Mamutse
(Applications of the Derivative)
1. Use√ differentials to compute approximate value
√ of √
◦
3
(i) 25 (ii) sin 31 (iii) ln(1.12) (iv) 36 (v) 5 31 (vi)
5
√ cos 62◦
1
(vii) sin 48◦ (viii) (1005) 3 (ix) tan 31◦ (x) ln(1.2) (xi) 26 (xii) tan−1 (1.1)
1
(xiii) (1.1)2 + 6(1.1)2 (xiv) sin 1◦ (xv) √ .
96
2. Explain why Rolle’s theorem is not applicable for the function f (x) = |x| on the interval
[−1, 1].
3. Verify Rolle’s Theorem for f (x) = x2 (1 − x)2 , 0 ≤ x ≤ 1.
4. Prove that if an an−1 a1
+ + ··· + + a0 ,
n+1 n 2
then the equation an xn + an−1 xn−1 + · · · + a1 x + a0 = 0 has at least one real root between 0
and 1.
5. Find the value of c in Rolle’s Theorem when f (x) = (x − a)m (x − b)n where m and n are
positive integers.
6. If f 0 (x) ≤ 0 at all points of (a, b), prove that f (x) is monotonic decreasing in (a, b). Under
what conditions is f (x) strictly decreasing in (a, b)?
sin x
7. Use the mean value theorem to show that sin x ≤ x and tan x ≥ x. Hence show that is
x
strictly decreasing on (0, π2 ).
a
b b
8. Use the mean value theorem, to prove that, if 0 < a < b, then 1 − < ln < −1 .
b a a
1 1
Hence show that < ln(1.2) < .
6 5
b−a b−a
9. Show that if 0 < a < b, then √ < sin−1 (b) − sin−1 (a) < √ . Hence show that
1 − a 2 1 − b 2
√ !
π 3 −1 π 1
+ < sin (0.6) < + .
6 15 6 8
10. Use the Mean Value Theorem to prove the following inequalities
(i) ln(1 + x) < x if x > 0.
1
√ 1
(ii) 1 + x < 1 + x if x > 0.
2
x−1
(iii) < ln x < x − 1 for x > 1.
x
11. Find the relative extrema of the given functions.
2
(i) f (x) = 2x3 + 3x2 − 36x (ii) f (x) = x5 − 53 x3 + 2 (iii) 4x − 6x 3 + 2
x2 − 2x + 2
(iv) f (x) = .
x−1
12. Find the relative extrema and the points of inflection of the following functions.
1
(i) f (x) = x4 + 8x3 + 18x2 (ii) f (x) = x6 − 3x4 + 5 (iii) f (x) = 10 − (x − 3) 3
5
(iv) f (x) = x(x − 1) 2 .
