NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Homework Assignment 4 Solution
x
1
+ sin x
cos x
sin x sin x
(1 x2 )3/2
1 x2
1. (a) lim
= lim
= lim
x 0
x 0
x 0
x3
3x2
6x
11
1
sin x
1
= lim
=+=.
+
2 )3/2
x0 6(1 x
6x
66
3
1
x2
(b) l
LESSON 3: DIFFERENTIATION AND PROCEDURE
1. Implicit Differentiation
We have introduced the command diff in Lesson 2, which is used to dierentiate a
function in the form y = f (x). However, sometimes we are given an equation f (x, y ) = 0
instead. For exam
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
MAPLE Quiz
Instruction
1. The Maple Quiz is set up in IVLE as an online assessment. There are altogether 10
questions with 2 marks for each question, and it constitutes 5% of the nal
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Solution to Tutorial 11
Tutorial Part I (Partial)
1
dy
dy
= 1, x, y > 0. Then y
= . Integrate with respect to
1. (a) Suppose 2 xy
dx
dx
2x
x:
2
1
dx y 3/2 = x + C.
y dy =
3
2x
That
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Solution to Tutorial 10
Tutorial Part I (Partial)
1. Solving x4 = 8x, we have x = 0 and x = 2. Note that 8x x4 on [0, 2]. Then the area
enclosed by y = x4 and y = 8x is given by
2
x5
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Solution to Tutorial 9
Tutorial Part I (Partial)
1
ln | sin 3x|.
3
1 dy
2
1 3 cos 3x
Dierentiate with respect to x:
= + ln 2
.
y dx
x
3 sin 3x
2
dy
x2 2x
2
Then
=y
+ ln 2 cot 3x =
+
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Solution to Tutorial 8
Tutorial Part I (Partial)
du
. Then
= 2 . So
x
dx
x
cos(/x)
1
1
1
cos u du = sin u + C = sin
+ C.
dx =
2
x
x
1. (a) Let u =
(2 + tan2 ) d =
(b)
(1 + sec2 ) d =
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Solution to Tutorial 6
Tutorial Part I (Partial)
1. (a) f (x) = 3 3x2 . Then f (x) = 0 x = 1.
(, 1) (1, 1) (1, )
f (x)
f (x)
+
Then f is increasing on (1, 1), and decreasing on (, 1)
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Solution to Tutorial 7
Tutorial Part I (Partial)
(t3 4t + 15)
3t2 4
3(3)2 4
23
t3 4t + 15
= lim 2
= lim
=
= .
t3 (t t 12)
t3 2t 1
t3 t2 t 12
2(3) 1
7
1. (a) lim
8x2
(8x2 )
16x
x
= lim
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Solution to Tutorial 5
Tutorial Part I (Partial)
1.
i) Existence:
ii) Uniqueness:
x = 0 is a zero to f (x) = 2x sin x. So f (x) has at least one zero.
Suppose f (x) has two zeros at x
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Solution to Tutorial 2
Review
Recall the intuitive denition of limit: We write lim f (x) = L, or
x
xa
(x = a)
f (x) L
if by taking x suciently close to a (but not equal to a), the va
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Solution to Tutorial 3
Tutorial Part I (Partial)
1.
(i) It is given by denition that f (1) = 1. Since
lim f (x) = lim 2x = 2 and
x1
lim f (x) = lim+ (2x + 4) = 2,
x1+
x 1
x 1
lim f (x
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Solution to Tutorial 1
Tutorial Part I (Partial)
1. Let A and B be the domains of f (x) = 1 x3 and g (x) =
and B = R \ cfw_0.
1
(i) f g (x) = f (g (x) = f ( x ) = 1
1
,
x3
cfw_x | x
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 6 (07/1011/10)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provided
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 11 (11/1115/11)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provide
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 10 (04/1108/11)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provide
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 8 (21/1025/10)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provided
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 9 (28/1001/11)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provided
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 7 (14/1018/10)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provided
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 5 (30/0904/10)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provided
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 4 (16/0920/09)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provided
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Homework Assignment 5 Solution
4
ln(2x + e3x )
x
1. (a) lim (2x + e3x )4/x = lim exp
x
x
4
= exp lim
1
x
= exp
(b) lim
x0
2+3e3x
2x+e3x
36
lim
x 2e3x
+3
x 0
1
ln a + ln b + ln c
3
= e
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 3 (09/0913/09 E-Learning)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 2 (02/0906/09)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provided
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
Tutorial 1 (26/0830/08)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provided
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2013/2014
MA1102R Calculus
Homework Assignment 3 Solution
1. Let f (x) = x5 5x + c. Then f is dierentiable on R.
Assume that f (x) = 0 has two roots c1 < c2 on the interval [0, 1]. Apply Rolles
Theorem to f on