Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
Solution to Test 1
SEMESTER A

Write down your name and student number on answer booklet

Only approved calculators are allowed

2015/2016
Please answer all the SIX questions, and justify your answers
1. [10+10 marks] Evaluate the following limi
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
PS5 extras
5b
5c
5a
1. Find the determinant of A 2a a 2b a c
3a 2a 3b 2a c
.
x1 x2 2 x3 5
2. Solve the system 2 x1 3x2 x3 2 by Gaussian Elimination.
5 x x 3x 2
3
1 2
x1 5 x2 2 x3 8
3. Solve the system 2 x1 10 x2 5 x3 19 by Gaussian Elimina
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA2176a1
1.
Functions, Limits and Continuity 0910B
Find the largest possible domain of each of the following functions:
2x
(a) y = f ( x) = 2
(b) y = f ( x) = 25 x 2
x 4x 5
(c)
( x) =
x2 1
x 1
Solution:
(a)
x 2 4 x 5 = (x 5)( x + 1) = 0 x = 1 or x = 5
Th
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
ma2176a3
Applications of Derivatives
1.
An airplane, flying horizontally at an altitude of 1 km, passes directly over an observe. If the constant
speed of the plane is 240 km per hour, how fast is its distance from the observer increasing 30 seconds
later
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA2176a2 Derivatives 0910B
1. If y = x sin x , prove that x 2 y ' 2 xy '+ ( 2 + x 2 ) y = 0 .
Proof:
y = x sin x y ' = sin x + x cos x y " = cos x + cos x x sin x = 2 cos x x sin x .
So
x 2 y ' '2 xy '+ (2 + x 2 ) y = x 2 (2 cos x x sin x ) 2 x(sin x + x
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
PS3 extras solutions
1. For the following two functions:
(a) Find the intervals on which f is increasing or decreasing.
(b) Find the local maximum and minimum values of f .
(c) Find the intervals of concavity and the inection points.
f (x) = x4 2x2
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
1. Find the domain and sketch the graph of the function.
G(x) =
Solution:
3x + x
x
When x = 0, G(x) has no denition; when x = 0, G(x) is well dened. So the domain of G is
(, 0) (0, ). The sketch of G is
y
4
y = G(x)
2
x
0
2. Find the domain and sketch t
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA2176a7 Matrices, Determinants and Systems of Linear Equations 0910B
1. Evaluate
b
7
2
1 1 1
1 + a
3
(a) det 2
(b) det a b c (c) det
a 1+b
4 3
1 3
2
a
b 1+
7
a b2 c2
Solution:
(a)
3 7 2
0 2 19
2 4 3 = det 0 10 17 = det 2 19 = 34 + 190 = 224
d
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA2176a4
Integrations and Its Applications 0910B
x
3 x if 0 x < 1
1. Let f ( x) =
. Find F ( x) = f (t ) dt , 0 x 2 .
2 + x if 1 x 2
0
Solution:
x
x
t2 x
x2
For 0 x < 1 , F ( x) = f ( t ) dt = ( 3 t ) dt = 3t 0 = 3 x .
2
2
0
0
1 x 2
x
1
x
x
1
5
t2
F (
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA2176
Basic Calculus and Linear Algebra
Chapter 6
Matrices, Determinants and Systems of Linear Equations
Additional Examples
Determinant
6.1.
5b
5c
5a
Find the determinant of A = 2a
a + 2b a + c .
3a 2a 3b 2a c
Solution
5a
5b
5c
a
b
c
a
b
c
A = 2a
a
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA2176 Semester A 20082009 Basic Calculus and Linear Algebra
Chapter 3 Integration and its Applications Additional Examples
Indenite Integrals
1. Evaluate the following integrals.
dx
7 5x
(a)
(b)
x sin x2 dx
(c)
(d)
(e)
dx
x loge x
cos3 x sin x dx
dx
1 +
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
Problem Set 2 , Solutions
1. Find an equation of the tangent line to the curve at the given point.
y=
x, (1, 1).
Solution:
dy
dx
x=1
1
=
2 x
x=1
1
= ,
2
1
equation: y 1 = (x 1).
2
2. Find an equation of the tangent line to the graph of y = g(x) at
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Spring 2016
MA1006 Solutions of Problem Set 2A
MA2176a2 Derivatives 0910B
1. If y = x sin x , prove that x 2 y ' 2 xy '+ ( 2 + x 2 ) y = 0 .
Proof:
y = x sin x y ' = sin x + x cos x y " = cos x + cos x x sin x = 2 cos x x sin x .
So
x 2 y ' '2 xy '+ (2 + x 2 ) y = x
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Spring 2016
MA1006
Problem Set 2B
1. Find an equation of the tangent line to the curve at the given point.
y=
x, (1, 1).
Solution:
1
dy
1
= ,
=
dx x=1
2 x x=1 2
1
equation: y 1 = (x 1).
2
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
1. Evaluate the limit
t2 9
t3 2t2 + 7t + 3
2. Evaluate the limit
PS1 extras
x+2
x2 x3 + 8
3. Evaluate the limit
lim
lim
lim
t0
1
1
2
t
t +t
4. Use the Squeeze Theorem to show that
lim
x0
x3 + x2 sin
= 0.
x
For Questions 5 7, nd the limit, if it ex
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
PS4 extras
Indefinite Integrals
1. Evaluate the following integrals.
dx
7 5x
(a)
(b)
x sin( x ) dx
(c)
x log
2
dx
e
x
(d)
cos x
(e)
1 cos x
3
sin x dx
dx
2. Evaluate the following integrals.
(a)
x
(b)
a
(c)
2
2
5
dx
2x 2
dx
x2
dx
a x2
2
(a 0)
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
Solution to Test 2
SEMESTER A 2015/2016
 Write down your name and student number on answer booklet
 Only approved calculators are allowed
 Please answer all the FIVE questions, and justify your answers
Question 1 [10x2 marks]:
Ship A is 15 meter
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
Chapter 2
Differentiability
1
Differentiability of functions
Consider the graph of the function y f x as shown below:
The points P and Q have coordinates
c, f c
secant line PQ is given by
slope of PQ
and
c h, f c h
respectively. The slope of t
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
1
Chapter 3
Applications of Derivatives
Derivative as Rate of Change
dy
is interpreted as the rate of change of y with
dx
respect to x. For example the concept of derivative can be applied to solve some dynamics problems.
Consider an object which i
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006 Chapter 5 Matrix Algebra and System of Linear Algebraic Equations
1. Introduction
A matrix of order m n or an m n matrix is a rectangular array of numbers having m rows and n
a11 a12 . a1n
a
21 a22 . a2 n
. aij .
columns. It can be written A .
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
Calculus and Linear Algebra for Business
Review on Standard Functions
Section I Polynomials and Rational Functions
A function f x an x n an1 x n1 a0 (where the a' i s are real numbers and n is a nonnegative
integer) is called a polynomial function
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
Chapter 4
Integration and its Applications
1
Indefinite Integrals
1.1 Introduction
The process of finding the integral of a function is called integration. Integration is the inverse operation
to differentiation, also known as antidifferentiation.
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA1006
Chapter 1
Functions, Limits and Continuity
1
Functions of a single real variable
1.1 Sets
A set is a collection of distinct objects called elements or members of that set. For example,
A 1, 2,3, 4,5 is a set and a list of all its elements is given.
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA2176a1 Functions, Limits and Continuity 0910B
1.
Find the largest possible domain of each of the following functions:
2x
x2 1
(a) y = f ( x) = 2
(b) y = f ( x) = 25 x 2
(c) ( x) =
x 1
x 4x 5
2.
Let f ( x ) = x 3 + 2, g ( x ) =
2
.
x 1
g
Find formulas fo
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
1. Find the domain and sketch the graph of the function
G(x) =
3x + x
.
x
2. Find the domain and sketch the graph of the function
f (x) =
x + 2 if x 1
x2
if x > 1.
3. In a certain country, income tax is assessed as follows. There is no tax on income up
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA2176a2
Derivatives 0910B
1.
If y = x sin x , prove that x 2 y ' 2 xy '+ ( 2 + x 2 ) y = 0 .
2.
If u = ax 2 + 2bx + c , prove that
3.
Differentiate the following functions with respect to x:
(a)
2 3000
(1 + 2 x 5x )
d
2ax 2 + 3bx + c
( xu ) =
.
dx
u
(
)
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA2176a3
Applications of Derivatives 0910B
1.
An airplane, flying horizontally at an altitude of 1 km, passes directly over an observer. If the constant
speed of the plane is 240 km per hour, how fast is its distance from the observer increasing 30 second
Calculus and Linear Algebra for Business Review on Standard Functions
MATHEMATIC 1006

Fall 2015
MA2176a4
1.
2.
3.
Integration and Its Applications 0910B
x
3 x if 0 x < 1
Let f ( x) =
. Find F ( x) = f (t )dt , 0 x 2
2 + x if 1 x 2
0
Evaluate the following integrals:
x +1
x2
dx
(a)
(b)
dx
1+ x2
x
(c)
e3x + 1
e x + 1 dx .
Evaluate the following in