NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2012/2013
MA1102R Calculus
Extra Problems Set 7
Topics: Chapters 0 to 9 (Revision)
1. For the functions f (x) =
x2 1, g (x) =
(ii) g f , and their domains.
1
, nd the composite functions (i) f g ,
x2
2. For eac
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2012/2013
MA1102R Calculus
1. f (x) =
Solution to Homework Assignment 1
x5 + 1 is dened x5 + 1 0 x 1.
g (x) = 1/x2 is dened x2 = 0 x = 0.
h(x) = (x 1)1/5 is dened for all x R.
1
1
(i) g f (x) = g (f (x) = g ( x
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2012/2013
MA1102R Calculus
Homework Assignment 1
IMPORTANT: Please write your name, matriculation card number and tutorial group
number on the answer script, and submit during either Group 1s lecture on 13th Se
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2009/2010 Sem 1
MA1102R Calculus
Tutorial 3
Partial Solution:
1.
(i) It is given by definition that f (1) = 1. Since
lim f (x) = lim 2x = 2 and
x1
lim f (x) = lim+ (2x + 4) = 2,
x1+
x1
x1
lim f (x
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2009/2010 Sem 1
MA1102R Calculus
Tutorial 7
Partial Solution:
(t3 4t + 15)
3t2 4
3(3)2 4
23
t3 4t + 15
1. (a) lim 2
= lim 2
=
lim
=
=
.
t3 t t 12
t3 (t t 12)
t3 2t 1
2(3) 1
7
8x2
(8x2 )
16x
x
(b)
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1102R Calculus (2009/2010 Sem 1)
Challenge of the Week 3
1. If a and b are positive numbers, prove that the equation
b
a
+ 3
=0
2
+ 2x 1 x + x 2
has at least one solution in the interval (1, 1).
4.4
Applied Optimization Problems
(Reference: TC, 4.5)
Goal: To apply differential calculus to solve problems that require maximizing or minimizing a function.
Strategy for Solving Optimization Problems
1. Read and understand the problem.
2. Draw a pictur
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1102R Calculus (2009/2010 Sem 1)
Homework Assignment 3
IMPORTANT:
Please write your name, matriculation card number and tu-
torial group number on the answer script, and submit during either Gro
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2009/2010 Sem 1
MA1102R Calculus
Tutorial 5
1. Show that the function f (x) = 2x sin x has exactly one zero in (, ).
2. Show that the equation x4 4x + 1 = 0 has exactly two real roots.
3. It is kn
REVISION
Chapter 0: Functions
Domain and range of function.
Composite functions.
Examples of functions: linear functions, polynomials, rational functions, piecewise
defined functions, trigonometric functions, root functions, logarithmic functions,
and
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1102R Calculus (2009/2010 Sem 1)
Extra Problems Set 5
Topics: Chapter 7 (Techniques of Integration), Chapter 8 (Applications of
Integration)
1. Evaluate the following indefinite integrals.
Z
Z
CHAPTER 6: TRANSCENDENTAL FUNCTIONS
Inverse Functions and Their Derivatives
6.1
(Reference: TC, 7.1)
One-to-One Functions and Their Inverses
Definition
A function f with domain D is one-to-one if for any x1 , x2 D,
x1 6= x2 implies f (x1 ) 6= f (x2 ).
Exa
8.3
Arc Lengths of Plane Curves
(Reference: TC, 6.3)
Goal: To use definite integrals to find lengths of plane curves.
Definition
A function f is said to be continuously differentiable or smooth if f is
continuous.
Lengths of Curves
1. Let f be continuousl
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2009/2010 Sem 1
MA1102R Calculus
Tutorial 4
sin
= 1, find the following limits.
0
1. Using lim
sin(x 1)
,
x1 x2 + x 2
(a) lim
sin ax
, (a 6= 0, b 6= 0).
x0 sin bx
(b) lim
2. Find the derivatives
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1102R Calculus (2009/2010 Sem 1)
Challenge of the Week 2
1. Find the limit lim ( x2 + x x2 x).
x
2. (a) Prove using the , -definition that for a > 0,
lim 3 x = 3 a.
xa
[Hint:
Recall that b3 c3 =
CHAPTER 5: INTEGRALS
5.1
The Definite Integral
(Reference: TC, 5.1, 5.2, 5.3)
Approximation of Area Under a Graph
Consider the area under the graph of y =
1 x2 , x [0, 1], in the first quadrant.
Divide the interval [0, 1] into n equal subintervals.
Use re
CHAPTER 8: APPLICATIONS OF
INTEGRATION
8.1
Area Between Curves
(Reference: TC, 5.6)
Goal: To use definite integrals to find areas of regions between curves.
Recall (see 5.1) that for f (x) 0 for all x, the area between y = f (x) and the x-axis
Z b
from x
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1102R Calculus (2009/2010 Sem 1)
Homework Assignment 3
Solution:
1. Let f (x) = x3 + 9x2 + 33x 8. Then f is continuous and differentiable on R.
(i) Since f (0) = 8 < 0 and f (1) = 35 > 0, applyi
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1102R Calculus (2009/2010 Sem 1)
Challenge of the Week 4
1.
(i) Using the definition of derivative, show that for any n Z+ ,
d 1
1 1
(x n ) = x n 1 ,
dx
n
x > 0.
(ii) Deduce the Power Rule for r
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1102R Calculus (2009/2010 Sem 1)
Challenge of the Week 4
Solution:
1.
(i) By the definition of derivative,
1
1
1
1
yn xn
y n xn
d 1
(x n ) = lim
= lim 1
n1
n2
n1
1
1
yx y x
yx (y n x n )(y n + y
9.3
tions
Applications of First-Order Differential Equa-
(Reference: TC, 7.5, 9.1, 9.2, 9.5)
Goal: To use differential equations to model and solve real-life problems in chemistry and
physics (radioactive decay), life science (bacteria growth), physics (h
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1102R Calculus (2009/2010 Sem 1)
Homework Assignment 5
IMPORTANT:
Please write your name, matriculation card number and tu-
torial group number on the answer script, and submit during either Gro
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1102R Calculus (2009/2010 Sem 1)
Extra Problems Set 2
Topics: Chapter 3 (Derivatives), Part I of Chapter 4 (Applications of Differentiation)
1. Using the definition of derivative, find the deriv
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1102R Calculus (2009/2010 Sem 1)
Challenge of the Week 9
Solution:
1.
(i) Let f (x) = ex (1 + x). Then f (x) = ex 1. Recall that ex is increasing.
If x > 0, then f (x) > e0 1 = 0; so f is increa
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2009/2010 Sem 1
MA1102R Calculus
Tutorial 9
1. Use logarithmic differentiation to find the derivative of y with respect to x.
x2 2x
x
(a) y =
,
(b) y = x(x ) , x > 0.
3
sin 3x
2. Find the followi
4.3
Behaviors of Functions from First and Second
Derivatives
(Reference: TC, 4.3, 4.4)
Increasing and Decreasing Tests
Definition
Let f be a function defined on an interval I.
1. f is said to be increasing on I if for any x1 , x2 I, where x1 < x2 ,
f (x1
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2009/2010 Sem 1
MA1102R Calculus
Tutorial 10
Partial Solution:
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
x=2
Z
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2009/2010 Sem 1
MA1102R Calculus
Tutorial 9
Partial Solution:
1
ln | sin 3x|.
3
1 dy
2
1 3 cos 3x
Differentiate with respect to x:
= + ln 2
.
y dx
x
3 sin 3x
dy
2
x2 2x
2
Then
=y
+ ln 2 cot 3x =
The Substitution Rule
5.4
(Reference: TC, 5.5, 5.6)
Goal: Use a change of variable to convert an unfamiliar integral into one that we know
how to evaluate.
Evaluating Indefinite Integrals
Theorem (Substitution in Indefinite Integrals): Let u = g(x) be a d
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2009/2010 Sem 1
MA1102R Calculus
Tutorial 8
Partial Solution:
du
1. (a) Let u = . Then
= 2 . So
x
dx Z x
Z
1
1
1
cos(/x)
dx
=
cos
u
du
=
sin
u
+
C
=
sin
+ C.
x2
x
Z
Z
2
(b) (2 + tan ) d = (1 + s
MA1102R CALCULUS
2016/2017 SEMESTER 1
FORMULA SHEET
Definitions
1. (a) The limit of f (x) equals L as x approaches a, denoted by lim f (x) = L, if for any > 0 there is a
xa
> 0 such that 0 < |x a| < |f (x) L| < .
(b) lim f (x) = L: for any > 0 there is a
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
MA1102R Calculus
Solution to Tutorial 7
Tutorial Part I (Partial)
(t3 4t + 15)0
3(3)2 4
23
t3 4t + 15
3t2 4
=
lim
=
=
.
=
lim
t3 (t2 t 12)0
t3 t2 t 12
t3 2t 1
2(3) 1
7
1. (a) lim
8x2
(8x2 )0
x
16x
= l
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
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)