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Calculus for Dummies > Chapter 1 - 4: Calculus PreReqs > Flashcards

Chapter 1 - 4: Calculus PreReqs Flashcards

Calculus PreReqs

1
Q

solving quadratic equations: completing the square

3x² = 24x + 27

A
  1. put the variables on one side and the constant in the other: 3x² - 24x = 27
  2. divides both sides by coefficient of x²: x² - 8x = 9
  3. divide coefficient of x by 2 and square it, then add to both sides: x² - 8x + 16 = 9 + 16
  4. factor left side (factor always has same # from prior step): (x-4)(x-4) = 25 → (x-4)² = 25
  5. use square root property: x-4 = √25 → x-4 = ±5
  6. solve: x-4 = ±5 → x = 9 or -1
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2
Q
A
  • The quantity b2 - 4ac, which appears under the radical sign in the quadratic formula
  • determines the number and type of solutions
  • b2 - 4ac > 0: Two unequal real solutions
  • b2 - 4ac = 0: One solution (a repeated solution) that is a real number
  • b2 - 4ac < 0: No real solutions
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3
Q

solving quadratic equations: quadratic formula

2x² - 6x + 1 = 12

A
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4
Q

solving quadratic equations: factoring

2x² - 5x = 12

A
  • bring all terms to one side: 2x² - 5x - 12 = 0
  • factor: (2x + 3)(x -4) = 0
  • set each factor to 0 and solve: 2x + 3 = 0 → 2x = -3 → x = -3/2
  • set each factor to 0 and solve: x - 4 = 0 → x = 4
  • two solutions: -3/2, 4
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5
Q

differences of two cubes

64x³ - 125

A
  • express each term as the cube of a monomial: 64x³ + 125 = 4x³ - 5³
  • convert 4x³ - 5³ to (a³ - b³) = (a - b)(a² + ab + b²)
  • (4x - 5)(4x² - 4x5 + 5²) → (4x - 5)(4x² - 20x + 25)
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6
Q

factoring sum of cubes

x³ + 8

A
  • express each term as the cube of a monomial: x³ + 8 = x³ + 2³
  • convert x³ + 2³ to (a³ + b³) = (a + b)(a² - ab + b²)
  • (x + 2)(x² - x2 + 2²) → (x + 2)(x² - 2x + 4)
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7
Q

quadratic equation

A
  • any second degree (highest power of x is 2) polynomial equation
  • 2x² + 5x = 12
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8
Q

x² + bx + c

  • when c is positive, find 2 numbers with …
  • when c is negative, find 2 numbers with …
A
  • the same sign as the middle term: x² - 5x + 6 = (x - 3)(x - 2)
  • opposite signs whose sum is the coefficient of the middle term: x + 3x - 18 = (x + 6)(x - 3)
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9
Q

factoring trinomials steps

x² + 6x + 8

A
  1. find the first terms whose product is x²: (x, inside) (x, outside)
  2. find 2 last terms whose product is 8: 8, 1 or 4, 2 or -8, -1 or -4, -2
  3. select the terms whose outside product and inside product is 6x: 4, 2
  4. answer: (x, 4) (x, 2) (outside 2 ⋅ x + inside 4 ⋅ x = 6x)
  5. x² + 6x + 8
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10
Q

factoring by grouping

x³ + 4x² + 3x + 12

A
  • group terms w/common factor: (x³ + 4x²) + (3x + 12)
  • take out GCF: x²(x + 4) + 3(x + 4)
  • take out GCF: (x + 4)(x² + 3)
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11
Q

factoring the greatest common factor

x²(x + 3) + 5(x + 3)

A
  • if you are adding, the answer is multiplication
  • GCF is (x + 3), so pull it out
  • (x + 3)(x² + 5)
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12
Q

factoring the greatest common factor

18x³ + 27x²

A
  • 9 and x² are the greatest integer and expression that divides in 18 & 27 and x² & x³
  • 9x²(2x) + 9x²(3)
  • 9x²(2x + 3)
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13
Q

what is the degree of the following polynomial:

4x⁵ - 6x³ + x² - 5x + 2

A

5

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14
Q

degree of a polynomial

A

the polynomial’s highest power of x

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15
Q

polynomial

A
  • all terms have a variable raised to a positive integral power (no fraction or negative powers)
  • just terms with a coefficient
  • no logs, sines or cosines allowed
  • ex: 4x⁵ - 6x³ + x² - 5x + 2
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16
Q

difference of cubes

A

(a³ - b³) = (a - b)(a² + ab + b²)

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17
Q

sum of cubes

A

(a³ + b³) = (a + b)(a² - ab + b²)

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18
Q

a difference of squares (a² - b²) _____ be factored, but a sum of squares (a² + b²) _____

A
  • can
  • can’t
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19
Q

factor the following

(3x²)² - (5)²

A

(3x² - 5)(3x² + 5)

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20
Q

rewrite the following to show a difference of squares (a² - b²)

9x⁴ - 25

A

(3x²)² - (5)²

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21
Q

LogcC =

A

1

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22
Q

Logc1 =

A

0

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23
Q

rewrite log₃81 = x

A

3ˣ = 81

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24
Q

logarithms

2³=8 as log is …

A
  • log₂8 = 3
  • log base 2 of 8 = 3
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25
Q

logarithms

rewrite log₂8 with base b

A
  • log₂8 = logb8 / logb8
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26
Q

logarithms

convert log134 to natural logs (base e)

A
  • log134 = loge4 / loge13
  • loge4 / loge13 = ln4 / ln13
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27
Q

if you have an even number root/index, you need the _____ value on the answer, because whether the root is positive or negative, the answer is _____. if it’s an odd number root, you don’t need the _____.

A
  • absolute
  • positive
  • bars
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28
Q

rewrite

A
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29
Q

(x2)3=

A

x6

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30
Q
A

x-4

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31
Q

x2 • x3 =

A

x5

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32
Q

rewrite as a root

A
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33
Q

rewrite

x-3 =

A

1/x3

34
Q

x0 =

A
  • 1
  • if x = 0, then 0⁰ = undefined
35
Q

is cancelling allowed?

a²b³(xy - pq)⁴(c + d) / ab⁴z(xy - pq)³ + 1

A
  • no
    • 1 breaks the chain of multiplication
36
Q

is cancelling allowed?

a²b³(xy - pq)⁴(c + d) / ab⁴z(xy - pq)³

A
  • yes
  • a(xy - pq)(c + d) / bz
37
Q

multiplication rule for cancelling

A

can cancel in a fraction only when it has an unbroken chain of multiplication through the entire numerator and denominator

38
Q

algebraic expression

A
  • anything w/out an equal sign
  • if it has an equal sign its an equation
  • behave like variables
39
Q

dividing fractions

A
  • flip the second fraction
  • multiply the numerators
  • multiply the denominators
40
Q

multiplying fractions

A
  • multiply the numerators
  • multiply the denominators
41
Q

limits

A
  • the microscope that zooms in on a curve
  • zooms in on a curve until its straight
  • curves represent changing quantities
42
Q

convergent series

A
  • you keep adding numbers
  • sum keeps growing
  • even though you add # forever and sum grows forever, sum of all infinitely is a finite number
43
Q

divergent series

A
  • series adds up to infinity
  • there are some exceptions
44
Q

slope =

A
45
Q

when you zoom in on curves they become _____

A

straight

46
Q

integration

A

adding up small parts of something to get the total

47
Q

differentiation

A

finding a derivative

48
Q

derivative

A
  • slope of a curve or steepness
  • simple rate (i.e. miles per hour)
49
Q

exponential

For any base where b > 0

(bx)y =

A

bxy

50
Q

exponential

For any base where b > 0

1 / by =

A

b-y

51
Q

exponential

For any base where b > 0

bx / by =

A

bx / by = bx-y

52
Q

exponential

For any base where b > 0

bxby =

A

bxby = bx+y

53
Q

logarithmic

For any base where b > 0, b ≠ 1

logb 1 =

A

logb 1 = 0

54
Q

logarithmic

For any base where b > 0, b ≠ 1

logb x =

A

logb x = loge x = ln x

55
Q

logarithmic

For any base where b > 0, b ≠ 1

logb b =

A

logb b = 1

56
Q

logarithmic

For any base where b > 0, b ≠ 1

logb xz =

A

logb xz = z logb x

57
Q

logarithmic

For any base where b > 0, b ≠ 1

logb 1/y =

A

logb 1/y = - logby

58
Q

logarithmic

For any base where b > 0, b ≠ 1

logb x/y =

A

logb x/y = logbx - logby

59
Q

logarithmic

For any base where b > 0, b ≠ 1

logbxy =

A

logbxy = logbx + logby

60
Q

logarithmic

For any base where b > 0, b ≠ 1

logbbx =

A

logbbx = x

61
Q

logarithmic

For any base where b > 0, b ≠ 1

blogbx =

A

blogbx = x

62
Q

Natural exponential functions

A
  • b is a positive number not equal to 1
  • Inverse of exponential functions
  • In the example, function passes through (0,1) and (1,e)
  • slope of ex equals the value of ex
  • when x = 0, e0 = 1 → value of ex = 1, and slope = 1
  • when x = 1, e1 = e → the value of ex = e, and slope = e
  • The area up to any x-value is also equal to ex
  • only exponential function with the property that the slope of the tangent line at x = 0 is 1, so ex has both value and slope equal to 1 at x = 0
    • Simplifies calculations
  • Inverse of natural logarithm function: ln(x)
    • ln(ex) = x
    • They are the same curve with x-axis and y-axis flipped.
  • e(ln x) = x
63
Q

Natural logarithm functions

A
  • with base b = e
  • loge(x) which is more commonly written ln(x)
  • Inverse of natural exponential function: ex
  • e(ln x) = x
  • They are the same curve with x-axis and y-axis flipped.
  • In the example, function passes through (0,1) and (e,1)
64
Q

formula for area of triangle

A

(base • height) / 2

65
Q

formula for area of square

A

width • height

66
Q

Natural logarithm

ln xy =

A

ln x + ln y

67
Q

Natural logarithm

ln (x/y) =

A

ln x - ln y

68
Q

Natural logarithm

ln |x| =

A

1 / x

69
Q

The number e

ln ep =

A
70
Q

The number e

ex + y =

A

exey

71
Q

The number e

ex - y =

A

ex / ey

72
Q

The number e

(ex)p =

A

exy

73
Q

The number e

ln (ex) =

A

x

74
Q

The number e

b x =

A
75
Q

The number e

bx =

A
76
Q

simplifying √ of exponents

when the exponent is even

√x6

A
  • divide exponent by 2
  • write result outside of √
  • leave no variable inside √

√x6

  • exponent = 6
  • 6 / 2 = 3
  • x3√ = x3
  • x3 = x6
77
Q

simplifying √ of exponents

when the exponent is odd

√x3

A
  • subtract one from the exponent
  • divide exponent by 2
  • write result outside of √
  • leave variable inside √ w/no exponent

√x3

  • exponent = 3
  • 3 - 1 = 2
  • 2 / 2 = 1
  • 1x√
  • 1x√x = x√x
78
Q

simplifying √ of exponents

√80x3y2

A

80x3y2

  • exponent = 3
  • 3 - 1 = 2
  • 2 / 2 = 1

x√80xy2​

  • exponent = 2
  • 2 / 2 = 1

xy√80x

  • 4 * 4 * 5 = 80

4xy√5x

√80x3y2 = 4xy√5x

79
Q

exponential growth & decay

f(t) = A0ekt

A
  • A0 = original amount/size at time t = 0
  • t = time elapsed
  • A = amount at time t
  • k
    • rate constant
    • represents growth rate
    • > 0 = growth
    • < 0 = decay
80
Q

Equation for doubling time

time required for a quantity (A0ekt) to double

A
81
Q

Equation for half life

A

Calculus for Dummies (6 decks)

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