BASIC CALCULUS PT 2

Question 1

COMMON equation of integral-differentiation

Answer 1

∫f(x) dx = f(x) + C

OR

f’(x) = f(x) + C

wherein ∫f(x) dx = derivative of f(x)

Question 2

Basic Definition of integrals?

Answer 2

Integral is the DIFFERENTIATED FORM of a function

Question 3

INDEFINITE vs DEFINITE INTEGRAL

Answer 3

Indefinite - an f(x) answer and no numbers sa integral

Definite - whole number ang answer and may upper limit and lower limit

Question 4

8 RULES in indefinite integrals

Answer 4

1. constant
2. power
3. e^x
4. ln (x)
5. sin (x)
6. cos (x)
7. constant multiple
8. sum and difference rule

Question 5

ANSWER:

∫ 5dx

∫ 7dy

∫ 8dr

∫ [6z+4] dz

∫ πdn

∫ 7/x^3 dx

Answer 5

1. 5x + C
2. 7y + C
3. 8r + C
4. 3 z^2 + 4z + C
5. πn + C
6. 7 (-1 / 2x^2) + C

Question 6

MOST important thing in INTEGRALS

Answer 6

+ C

Question 7

When to use integration by substitution?

Answer 7

If MUKHANG chain rule

Question 8

INTEGRATION BY SUBSTITUTION STEPS

Answer 8

1. Find u
2. Find du (derivative of u)
3. du = ? dx always
4. find value of dx and substitute

EXAMPLE
u = 2x
du = 2 dx

Question 9

ANSWER:

∫ cos (2x) sin (2x) dx

∫ [ (ln (2x)) / x ]dx

∫ (x^3) / [(2+x^4)^2] dx

Answer 9

u = sin (2x)

u = ln (2x)

u = 2+x^4

Question 10

Palatandaan sa pag-aassign ng u (3)

Answer 10

1. derivative of u can be seen in function and can be cancelled
2. more complicated
3. higher exponential value

Question 11

When to use integration by parts? (2)

Answer 11

1. if product of 2 functions denoted by the same variable (x e^x)
2. integrals of ln or log in simples form (purely lnx or logx)

Question 12

general equation in INTEGRAL BY PARTS

Answer 12

∫ u dv = u v - ∫ v du

Question 13

4 main basis in INTEGRAL BY PARTS

Answer 13

u and dv

then
du and v

Question 14

DEFINITE INTEGRALS

in [a,b]

how do u write the eq?

and how to read this?

Answer 14

(b upper limit) ∫ (a lower limit) f(x) dx = F (b) - F (a)

The (definite) integral of f(x) with respect to x from a to b

Question 15

in AREA UNDER PLANE:

Arx
- direction of rectangles
- upper limit and lower limit of integral
- direction of equation : (a-b)

Ary
- direction of rectangles
- upper limit and lower limit of integral
- direction of equation : (a-b)

Answer 15

Arx (y = x variable) find y
- vertical (patayo)
- x-axis so left (lower limit) to right (upper limit) value
- taas - baba

Ary ( x = y variable) find x
- horizontal (pahiga)
- y-axis so baba (lower limit) to taas (upper limit)
- right - left

Question 16

(x,y,z)

• what do you call this?
• what symbol to denote this?
• how many divisions?

Answer 16

3 dimensional space

R^3

8 - octants

Question 17

limits for:

1. roots
2. denominators
3. logarithms

Answer 17

1. even roots CANNOT contain negative numbers ( bawal yung √-27 or what)
   * pero pwede sa odd roots
2. bawal 0
3. bawal 0 and negative numbers

Question 18

1 possible shape for R^2

3 possible shapes for R^3

Answer 18

R^2 = plane

R^3 = plane, cylinder, surface

Question 19

Lagrange Extrema step by step

Answer 19

1. Write function and constraint
2. Write Lagrange EQ ( fx = λgx ; fy = λgy ; constraint)
3. Solve for x and y
4. Substitute x and y in constraint
5. Find value of lambda
6. Substitute to x and y eq in #3
7. Find max and min values+