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MATH > BAISC CALCULUS > Flashcards

BAISC CALCULUS Flashcards

1
Q

basic definition of limits

A

the limit of a function is ___ (y-value) as x approaches to __ (x-value) from the left and right side

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2
Q

when does a limit NOT EXIST

A

when the values of a limit from the right side is different from the left side

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3
Q

Example of an equation wherein the function value is undefined but the limit exists

A

f(x) = [(x^2 -9) / (x-3)]

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4
Q

Cite the 8 Rules on Evaluating limits

A
  1. Constant
  2. Number A
  3. Coefficient of f(x)
  4. Addition/Subtraction
  5. Multiplication
  6. Division
  7. Power
  8. Power Root (n = even; f(x) ≠ 0)
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5
Q

For equations like lim as x-> 1 =

_/x - 1
———-
x - 1

how to solve

A

rationalize
dont simplify muna

=1/2

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6
Q

if a given is

lim f(x) as x -> 1^( - ) = ?

lim f(x) as x -> 1^( + ) = ?

what are the:
1. Possible answers
2. What do you call these functions
3. Exceptions to these

A
    • or - infinity
  2. One-sided Limits
  3. Piecewise Functions
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7
Q

4 RULES for limits involving infinities + RECHECKING

A
  1. Addition/Subtraction: Highest Degree
  2. Division: Bottom Heavy = 0
  3. Division: Top Heavy = Cross out other insignificant value in the numerator then the denominator
  4. Division: Equal degree in numerator and denominator = divide coefficients

RECHECKING
divide both numerator & denominator by 1/(x^highest degree in denominator)

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8
Q

Find out whats different then solve:

lim as x->3

 1 ——— (x - 3)^2
A

THIS IS NOT A ONE SIDED FUNCTION

since = 1/0

infinity ang sagot

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9
Q

LIMITS

if k is a constant and:

k ^ ∞

what is the answer

A

∞, if k>1
0, if 0<k<1

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10
Q

3 rules for a continuous function

A
  1. f(x) exists
  2. lim exists
  3. f(x)=lim
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11
Q

CONTINUOUS FUNCTION

if piecewise eq what to do

if not piecewise

A

piecewise: for LIM, check right and left side

not piecewise: for LIM, directly substitute na

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12
Q

what to do

  1. lim ln f(x)
  2. lim sin f(x)
  3. lim e^(f(x))
A
  1. ln (lim f(x))
  2. sin (lim f(x))
  3. e^(lim f(x))
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13
Q

CONTINUITY FUNCTIONS IN

  1. open intervals
  2. closed intervals [a, b]
A
  1. all numbers in between must be continuous
  2. a. all numbers in between mus be continuous
    b. lim x-> a+ = f(a)
    c. lim x-> b- = f(b)
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14
Q

FACTORING RULES
1. a2 - b2

  1. (a+b)2
  2. (a-b)2
  3. a3 + b3
  4. a3 - b3
A
  1. (a+b) (a-b)
  2. a2 +2ab +b2
  3. a2 -2ab +b2
  4. (a + b) (a2 - 2ab + b2)
  5. (a - b) (a2 + 2ab + b2)
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15
Q

What to remember in one sided functions?

A
  1. Factor
  2. Substitute x
  3. Analyze

SUBSTITUTE X KASI approach x na sya,…

and look at if from the right/left

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16
Q

RELATIONSHIP STATEMENT between function continuity and differentiation

A

If the function f(x) is differentiable at x = c, then it is also continuous at x = c.

17
Q

If possible, give an example of the following:

a. A function which is differentiable and continuous.
b. A function which is not continuous and not differentiable.
c. A function which is continuous but not differentiable.

A

a. x^2

b. g(x) =
1 if | x < 0
0 if | x ≥ 0

c. |x|

18
Q

General equation for derivatives

A

f’(x) = lim f(x+h) - f(x)
x->h —————–
h

19
Q

11 Rules of Differentiation

A
  1. Constant
  2. Power
  3. Coefficient
  4. Sum & Difference
  5. Product [ g(x) h’(x) + g’(x) h(x) ]
  6. Quotient [ wherein g/h: {h(x) g’(x) - g(x) h’(x)} / h(x)^2}
  7. e^x
  8. ln (1/x)
  9. log (1/xln10)
  10. sin(x) (cos(x))
  11. cos(x) (-sin(x))
20
Q

What is the geometric and physical interpretation of f’(x) = a

A

geometric: slope of the tangent line to the graph of f(x) at a
physical: - instantaneous rate of change, velocity

21
Q

What are the derivatives of the ff:
1. f(x) = e^π
2. f(π) = e^x
3. f(x) = 4x^2e^x

A
  1. 0 kasi constant, π is not x
  2. e^π
  3. Product Rule: 4e^x (2x+x^2)
22
Q

MEANING OF:

dy/dx

A

y = f( )
x = x

23
Q

In finding the V’ = 4/3π r^ 3

why is the answer = to 100π?

A

because 4/3π is not a constant as naka dikit sya kay variable…

if the eq is 4/3π + r^3, then 4/3π will be = to 0

24
Q

OPTIMIZATION - Relative Extrema

f’‘(x) <0
f’‘(x) >0

A

maximum
minimum

25
Q

Unit Circle

A
26
Q

FORMULA FOR AREA AND VOLUME

  1. Circle/Sphere
  2. Cylinder (V)
  3. Rectangle/Rectangular Prism
  4. Triangle/Triangular Prism
  5. Square Pyramid
A

πr^2 & 4/3πr^3
πr^2h
LW & LWH
1/2bh & 1/2bhl
1/3b^2h

MATH (2 decks)

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