The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 (Calculus and Linear Algebra)
Assignment 02
Students should submit their solutions to the General Office of the Department of
Applied Mathematics (TU732) by 5:00pm on the due d
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 11
Infinite Series
1. Let I n = x cos xdx and J n = x n sin xdx , where n is a non-negative integer. Using
n
0
0
integration by parts, expr
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 10
Improper Integrals
1. Which of the following integrals is/are improper? Why?
2
(a)
1
2 x 1dx ;
1
1
(b)
1
2 x 1 dx ;
0
(c)
sin x
1+ x
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 8
Definite Integrals and Fundamental Theorem of Calculus
1. Find the values of the following definite integrals using the integration table
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 9
Application of Integration
1. Find the area of the region enclosed by the given curves.
1
(a) y = x , y = x ,and x = 0 , x = 9 ;
2
(b) y
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 7
Mean Value Theorem and Indefinite Integrals
1. Apply Mean-value theorem to prove that if 0 < a < b ,
ba
ba
,
< tan 1 b tan 1 a <
2
1+ b
1
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 2
Limits and Continuity
1. Evaluate the limit, if it exists.
1
(a) lim 1 + x 3 ( 2 6 x 2 + x3 ) ;
x 8
2
(e) lim x 4 cos ;
x 0
x
1 1
(f) li
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 6
Differentials and LHopitals Rule
1. Let y = f ( x ) be a differentiable function. Show that
y
=1
dy
where y is the increment in y and dy
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 4
Differentiation techniques
1. Differentiate from (a) to (c) with respect to x .
5 x + 3 2 x
tan 3 x
(a) y =
; (c) y = 5 x3 3 x5 + 5ecos x
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 1
Elementary function and Partial fractions
1. Consider the functions f and g defined by f ( x)= 2 x 2 and g ( =
x)
x+2 .
(a) Find domains
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 5
Higher derivatives and Application of Differentiation
1. (a) Find y " if y x3 ( x + 1) ;
=
2
(b) Find y "' if y = x 3 cos 2 x ;
(c) Show
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 3
Differentiability
1. Prove, from the first principle, that if f ( x ) =
2
1
, then f ' ( a ) = 3 , for a 0 .
2
a
x
2. Prove the following
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 (Calculus and Linear Algebra)
Assignment 04
Students should submit their solutions to the General Office of the Department of
Applied Mathematics (TU732) by 5:00pm on the due d
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 (Calculus and Linear Algebra)
Assignment 03
Students should submit their solutions to the General Office of the Department of
Applied Mathematics (TU732) by 5:00pm on the due d
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 (Calculus and Linear Algebra)
Assignment 01
Students should submit their solutions to the General Office of the Department of
Applied Mathematics (TU732) by 5:00pm on the due d
The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA1007 Calculus and Linear Algebra
Tutorial 13
Matrices
1. Which of the following are elementary matrices
1 0
(a)
;
5 1
5 1
(b)
;
1 0
1 0
(c)
;
0
3
1 1 0
(e) 0 0 1 ;
0 0