Homework MH1810
This homework contributes 10% of your final mark. Please submit your solutions to
Yin Daiying by Email: YIND0004@e.ntu.edu.sg before 11:59pm, Friday, April 1, 2022.
1. Find the limits if exist.
(a) limπ
t→ 4
(2 + h)3 − 8
h→0
h
cos 2t
cos t − sin t
(c) lim x2020 1 + sin2 (2020x) .
(b) lim
x→0
tan−1 x
dx.
1 + x2
(b) Find the first-order derivative of
Z
2. (a) Find the value of
f (x) = esin(
√
3
x)
.
3. Use the Intermediate Value Theorem to show that f (x) = x3 + 2x − 4 has a root
in (1, 2). Then use Newton’s Method to find the root to five decimal places.
4. Use the definition to prove that f (x) = x − sin x is an increasing function.
5. Find the global maximum and minimum values of f (x) = ex − 2x on [0, 1].
.