Basic Cal PDF
Basic Cal PDF
Basic Cal PDF
CALCULUS
LESSON 1: INTRODUCTION TO CALCULUS
Prepared by:
KELVIN ROY L. SAN MATEO
Faculty, Mathematics Department DHLA ONLINE SUMMER CLASS 2020
REVIEW OF THE
FUNDAMENTALS OF
ADVANCED ALGEBRA
(Factoring Polynomials)
COMMON MONOMIAL FACTOR ax ay a x y
ILLUSTRATION
2 2 3
1. xyz wxy 3. pq r p qr
Solution: Solution:
ILLUSTRATION
2
5. 3x y 9 xy 12 x y 2 2 6. x y
Solution: Solution:
3x xy 3 3 xy 3 4 x xy y x y 1 x 1 y
3x xy 3 3xy 3 4 x xy y 1 x 1 y
3xy x 3 4 x2 y 1 x y
y x
2 2
DIFFERENCE OF TWO SQUARES x y x y x y
ILLUSTRATION
2 2 2 2
1. a b 3. 9a 25b
Solution: Solution:
2 2 2 2 2 2
a b a b a b 9a 25b 3a 5b
3a 5b 3a 5b
2 2
2. 4x y 4. 2x 2 y2
Solution: Solution:
2 2 2 2 2 2 2 2
4x y 2x y 2x y 2x y
2x y 2x y
2x y 2x y
2 2
DIFFERENCE OF TWO SQUARES x y x y x y
ILLUSTRATION
2 2
5. x 1 y
Solution:
2 2
x 1 y x 1 y x 1 y
x 1 y x 1 y
2 2 2 2
6. a b
2
4a 2 a b 4a a b 2a
Solution:
a b 2a a b 2a
a b 2a a b 2a
b a 3a b
SUM AND DIFFERENCE OF TWO CUBES
3 3 2 2
x y x y x xy y
x3 y3 x y x2 xy y2
ILLUSTRATION
3 3 3 3
1. a b 3. 8x y
Solution: Solution:
3 3 2 2 3
a b a b a ab b 2x y3 2x y 4x2 2 xy y2
2. x 3
1 4. 27x3 8 y3
Solution: Solution:
3 3 2 3 3 2 2
x 1 x 1 x x 1 3x 2y 3x 2 y 9x 6 xy 4y
SUM AND DIFFERENCE OF TWO CUBES
3 3 2 2
x y x y x xy y
x3 y3 x y x2 xy y2
ILLUSTRATION
3 3 3
5. 2a 1 Note: ax3 by 3 3
ax 3
by a2 x2 3
abxy b2 y 2
Solution: If a and b are not perfect cube numbers
3 3
3
2a 1 3
2a 1 3
4a 2 3
2a 12
3 3 3 3
6. 3x 2y 3
3x 3
2y 3
3x 3
2y 3
32 x 2 3
3x 3 2 y 3
22 y 2
Solution:
3
3x 3
2y 3
9 x2 3
6 xy 3
4 y2