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
logarithms rewrite log₂8 with base b
* log₂8 = logb8 / logb8
26
logarithms convert log134 to natural logs (base e)
* log134 = loge4 / loge13 * loge4 / loge13 = ln4 / ln13
27
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 \_\_\_\_\_.
* absolute * positive * bars
28
rewrite
29
(x2)3=
x6
30
x-4
31
x2 • x3 =
x5
32
rewrite as a root
33
rewrite x-3 =
1/x3
34
x0 =
* 1 * if x = 0, then 0⁰ = undefined
35
is cancelling allowed? a²b³(xy - pq)⁴(c + d) / ab⁴z(xy - pq)³ + 1
* no * + 1 breaks the chain of multiplication
36
is cancelling allowed? a²b³(xy - pq)⁴(c + d) / ab⁴z(xy - pq)³
* yes * a(xy - pq)(c + d) / bz
37
multiplication rule for cancelling
can cancel in a fraction only when it has an unbroken chain of multiplication through the entire numerator and denominator
38
algebraic expression
* anything w/out an equal sign * if it has an equal sign its an equation * behave like variables
39
dividing fractions
* flip the second fraction * multiply the numerators * multiply the denominators
40
multiplying fractions
* multiply the numerators * multiply the denominators
41
limits
* the microscope that zooms in on a curve * zooms in on a curve until its straight * curves represent changing quantities
42
convergent series
* 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
divergent series
* series adds up to infinity * there are some exceptions
44
slope =
45
when you zoom in on curves they become \_\_\_\_\_
straight
46
integration
adding up small parts of something to get the total
47
differentiation
finding a derivative
48
derivative
* slope of a curve or steepness * simple rate (i.e. miles per hour)
49
**exponential** For any base where b \> 0 (bx)y =
bxy
50
**exponential** For any base where b \> 0 1 / by =
b-y
51
**exponential** For any base where b \> 0 bx / by =
bx / by = bx-y
52
**exponential** For any base where b \> 0 bxby =
bxby = bx+y
53
**logarithmic** For any base where b \> 0, b ≠ 1 logb 1 =
logb 1 = 0
54
**logarithmic** For any base where b \> 0, b ≠ 1 logb x =
logb x = loge x = ln x
55
**logarithmic** For any base where b \> 0, b ≠ 1 logb b =
logb b = 1
56
**logarithmic** For any base where b \> 0, b ≠ 1 logb xz =
logb xz = z logb x
57
**logarithmic** For any base where b \> 0, b ≠ 1 logb 1/y =
logb 1/y = - logby
58
**logarithmic** For any base where b \> 0, b ≠ 1 logb x/y =
logb x/y = logbx - logby
59
**logarithmic** For any base where b \> 0, b ≠ 1 logbxy =
logbxy = logbx + logby
60
**logarithmic** For any base where b \> 0, b ≠ 1 logbbx =
logbbx = x
61
**logarithmic** For any base where b \> 0, b ≠ 1 blogbx =
blogbx = x
62
Natural exponential functions
* 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
Natural logarithm functions
* 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
formula for area of triangle
(base • height) / 2
65
formula for area of square
width • height
66
**Natural logarithm** ln xy =
ln x + ln y
67
**Natural logarithm** ln (x/y) =
ln x - ln y
68
**Natural logarithm** ln |x| =
1 / x
69
**The number *e*** ln *e*p =
70
**The number *e*** *e*x + y =
*e*x*e*y
71
**The number *e*** *e*x - y =
*e*x / *e*y
72
**The number *e*** *(e*x)p =
*ex*y
73
**The number *e*** ln *(e*x) =
x
74
**The number *e*** *b* x =
75
**The number *e*** *b*x =
76
**simplifying √ of exponents** when the exponent is even √x6
* divide exponent by 2 * write result outside of **√** * leave no variable inside √ √x6 * exponent = 6 * 6 / 2 = 3 * x3√ = x3 * x3 = x6
77
**simplifying √ of exponents** when the exponent is odd √x3
* 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
**simplifying √ of exponents** √80x3y2
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
exponential growth & decay f(t) = A0ekt
* 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
Equation for doubling time time required for a quantity (A0ekt) to double
81
Equation for half life