Download Review for Midterm

Review Material (I)
1. The Laws of Exponents:
i. If a, b are reals and r , s are intgers then
  a r   a rs
s
 ar a s  ar s
  ab   a b
r
2
.23.310 
23
 3.2 
1
r
5
ar
 s  a r s
a
r
2

Ans
.

27 
5
Method: Simplify numerator and denominator
separately. [Patience is required for simplification]

2x 1
6x
3
x5 
 2

Ans. 

x  4x  4 x  4 x  2
( x  2) 2 

Method: Factorize the denominators and find the
LCM.
a.
2
2  4 x 2  9   (2 x  3)  12   4 x 2  9  (8 x)
1
b.
1
2
 4x  9 
1
2
2
2
2


 Ans. 6(3  2 x)3 

 4 x2  9  2 

Method: Take out common factors in the numerator.
Remember:
iv. Rationalize the Denominators
 When is n a defined?
81x 2  16 y 2

3 x 2 y
If a  0 then n must be ODD.
 When x is a variable,
Method: Use of formula:
x2  x .
2.


a b


a  b  a b
Simplifications
i. Simplify complex expression, e.g.,
5

x 1
x

x 1

 Ans.  9 x  4 y  3 x  2 y 


2x
x3
7
x3
3 2

u  3 uv  3 v 2 
 Ans.

uv


Method: Use of formula:
1
 3
u3v

2 x 2  7 x  15 
Ans
.


x 2  10 x  7 


Method: Simplify numerator and denominator
separately. [Patience is required for simplification]
3
a3b

3

a 2  3 ab  3 b 2  a  b
ii. Simplify the following:
9 x 2  4 9 x 4  6 x3  4 x 2 
x 
.
Ans.
2
4

3x  5 x  2
27 x  8 x
x  1 

5 x 2  12 x  4 25 x 2  20 x  4
b.

x 4  16
x2  2x


x
 Ans. 2


 x  4   5 x  2  
3. Review Trigonometric Identities
a.
Method: Factorize numerators and denominators and
use the formulas:
 a3  b3  (a  b)(a 2  ab  b2 )
 a3  b3  (a  b)(a 2  ab  b2 )
iii. Simplify the following:
a.
1  tan 2   sec2 
b.
cos 2  cos 2   sin 2 
c.
sin
d.


csc  x     sec x
2

e.
cos(   )   cos 

2

1  cos 
2
tan(a  b) 
f.
tan a  tan b
1  tan a tan b
7.
[See similar other Identities in your Textbook]
Vertical & Horizontal
Translation of Simple Graphs
e.g. Sketch the graphs of
y  ( x  1)3  2; y  x  2  1; y  sin( x  1)  2
y  e x 1  2; y  ln( x  1)  2;
4. Use of Right Angled Triangle to
find Trigonometric Ratios
e.g. (i) If sin x  h in the diagram, then
h
x
a.
sec x 
b.
sin(2 x)  2h 1  h2
c.
cos
x

2
1
8. Writing Quadratic Function in
Standard Form
Find a, b and c when y = 5  4 x  2 x 2 is written in
the form y = a  b( x  c)2 ? Then sketch the graph
of the function
(Check: Vertex, x & y intercepts, parabola opens
upwards or downwards) See page:
1 h2
9.
Long Division
Divide 5 x  3 x 2  4 x  5 by x 2  2 x  4 and
find Divisor; Quotient; Remainder.
4
1 1 h 2
2
(ii) Use Right Angled Triangle to check:
10. Area of Geometrical Figures
Triangle, Trapezoid, Circle; Sector.
If x  3sec , then
x2  9  3tan x .
11.
Read Page 44-48 & 64-74 of the Text for
the following Topics
5.
Sketch the Graph of Basic
Functions
e.g. i. y  x ; y  x 2 ; x  y 2 ; y  x3 ; y  x
ii. y  e x ; y  ln x;
x
1
iii. y  2 ; y    ;
2
iv. y  log 2 x; y  log 1 x
x
2
v. All trigonometric Functions
6. Symmetry about Axes and the
Origin
Recall the Tests for Symmetries.
Factorize
x3  64 ; x 6  5 x3  6 ; 3x3  x 2  3x  1
12. Chain Rule for differentiation
Rule: Differentiate Extreme outer function, the
next outer function, and then next….
e.g. find the derivative of

 y  sin 3 ( x 2  3)  ecos x


3

4
6
 y  ln sin 5 x3  5 .
[Identify the outer functions in order.]