Chapter 1 - 4: Calculus PreReqs

Question 1

solving quadratic equations: completing the square

3x² = 24x + 27

Answer 1

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

Question 2

Answer 2

• 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

Question 3

solving quadratic equations: quadratic formula

2x² - 6x + 1 = 12

Answer 3

Question 4

solving quadratic equations: factoring

2x² - 5x = 12

Answer 4

• 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

Question 5

differences of two cubes

64x³ - 125

Answer 5

• 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)

Question 6

factoring sum of cubes

x³ + 8

Answer 6

• 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)

Question 7

quadratic equation

Answer 7

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

Question 8

x² + bx + c

• when c is positive, find 2 numbers with …
• when c is negative, find 2 numbers with …

Answer 8

• 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)

Question 9

factoring trinomials steps

x² + 6x + 8

Answer 9

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

Question 10

factoring by grouping

x³ + 4x² + 3x + 12

Answer 10

• 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)

Question 11

factoring the greatest common factor

x²(x + 3) + 5(x + 3)

Answer 11

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

Question 12

factoring the greatest common factor

18x³ + 27x²

Answer 12

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

Question 13

what is the degree of the following polynomial:

4x⁵ - 6x³ + x² - 5x + 2

Answer 13

5

Question 14

degree of a polynomial

Answer 14

the polynomial’s highest power of x

Question 15

polynomial

Answer 15

• 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

Question 16

difference of cubes

Answer 16

(a³ - b³) = (a - b)(a² + ab + b²)

Question 17

sum of cubes

Answer 17

(a³ + b³) = (a + b)(a² - ab + b²)

Question 18

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

Answer 18

• can
• can’t

Question 19

factor the following

(3x²)² - (5)²

Answer 19

(3x² - 5)(3x² + 5)

Question 20

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

9x⁴ - 25

Answer 20

(3x²)² - (5)²

Question 21

LogcC =

Answer 21

1

Question 22

Logc1 =

Answer 22

0

Question 23

rewrite log₃81 = x

Answer 23

3ˣ = 81

Question 24

logarithms

2³=8 as log is …

Answer 24

• log₂8 = 3
• log base 2 of 8 = 3

Question 25

logarithms rewrite log₂8 with base b

Answer 25

* log₂8 = log_b8 / log_b8

Question 26

logarithms convert log₁₃4 to natural logs (base e)

Answer 26

* log₁₃4 = log_e4 / log_e13 * log_e4 / log_e13 = ln4 / ln13

Question 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 \_\_\_\_\_.

Answer 27

* absolute * positive * bars

Question 28

rewrite

Answer 28

Question 29

(x²)³=

Answer 29

x⁶

Question 30

Answer 30

x^-4

Question 31

x² • x³ =

Answer 31

x⁵

Question 32

rewrite as a root

Answer 32

Question 33

rewrite x^-3 =

Answer 33

1/x³

Question 34

x⁰ =

Answer 34

* 1 * if x = 0, then 0⁰ = undefined

Question 35

is cancelling allowed? a²b³(xy - pq)⁴(c + d) / ab⁴z(xy - pq)³ + 1

Answer 35

* no * + 1 breaks the chain of multiplication

Question 36

is cancelling allowed? a²b³(xy - pq)⁴(c + d) / ab⁴z(xy - pq)³

Answer 36

* yes * a(xy - pq)(c + d) / bz

Question 37

multiplication rule for cancelling

Answer 37

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

Question 38

algebraic expression

Answer 38

* anything w/out an equal sign * if it has an equal sign its an equation * behave like variables

Question 39

dividing fractions

Answer 39

* flip the second fraction * multiply the numerators * multiply the denominators

Question 40

multiplying fractions

Answer 40

* multiply the numerators * multiply the denominators

Question 41

limits

Answer 41

* the microscope that zooms in on a curve * zooms in on a curve until its straight * curves represent changing quantities

Question 42

convergent series

Answer 42

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

Question 43

divergent series

Answer 43

* series adds up to infinity * there are some exceptions

Question 44

slope =

Answer 44

Question 45

when you zoom in on curves they become \_\_\_\_\_

Answer 45

straight

Question 46

integration

Answer 46

adding up small parts of something to get the total

Question 47

differentiation

Answer 47

finding a derivative

Question 48

derivative

Answer 48

* slope of a curve or steepness * simple rate (i.e. miles per hour)

Question 49

**exponential** For any base where b \> 0 (b^x)^y =

Answer 49

b^xy

Question 50

**exponential** For any base where b \> 0 1 / b^y =

Answer 50

b^-y

Question 51

**exponential** For any base where b \> 0 b^x / b^y =

Answer 51

b^x / b^y = b^x-y

Question 52

**exponential** For any base where b \> 0 b^xb^y =

Answer 52

b^xb^y = b^x+y

Question 53

**logarithmic** For any base where b \> 0, b ≠ 1 log_b 1 =

Answer 53

log_b 1 = 0

Question 54

**logarithmic** For any base where b \> 0, b ≠ 1 log_b x =

Answer 54

log_b x = log_e x = ln x

Question 55

**logarithmic** For any base where b \> 0, b ≠ 1 log_b b =

Answer 55

log_b b = 1

Question 56

**logarithmic** For any base where b \> 0, b ≠ 1 log_b x^z =

Answer 56

log_b x^z = z log_b x

Question 57

**logarithmic** For any base where b \> 0, b ≠ 1 log_b 1/y =

Answer 57

log_b 1/y = - log_by

Question 58

**logarithmic** For any base where b \> 0, b ≠ 1 log_b x/y =

Answer 58

log_b x/y = log_bx - log_by

Question 59

**logarithmic** For any base where b \> 0, b ≠ 1 log_bxy =

Answer 59

log_bxy = log_bx + log_by

Question 60

**logarithmic** For any base where b \> 0, b ≠ 1 log_bb^x =

Answer 60

log_bb^x = x

Question 61

**logarithmic** For any base where b \> 0, b ≠ 1 b^log_bx =

Answer 61

b^log_bx = x

Question 62

Natural exponential functions

Answer 62

* 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 e^x equals the value of e^x * when x = 0, e⁰ = 1 → value of e^x = 1, and slope = 1 * when x = 1, e¹ = e → the value of e^x = 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 e^x has both value and slope equal to 1 at x = 0 * Simplifies calculations * Inverse of natural logarithm function: ln(x) * ln(e^x) = x * They are the same curve with x-axis and y-axis flipped. * e(ln x) = x

Question 63

Natural logarithm functions

Answer 63

* with base b = e * log_e(x) which is more commonly written ln(x) * Inverse of natural exponential function: e^x * 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)

Question 64

formula for area of triangle

Answer 64

(base • height) / 2

Question 65

formula for area of square

Answer 65

width • height

Question 66

**Natural logarithm** ln xy =

Answer 66

ln x + ln y

Question 67

**Natural logarithm** ln (x/y) =

Answer 67

ln x - ln y

Question 68

**Natural logarithm** ln |x| =

Answer 68

1 / x

Question 69

**The number *e*** ln *e*^p =

Answer 69

Question 70

**The number *e*** *e*^{x + y} =

Answer 70

*e*^x*e*^y

Question 71

**The number *e*** *e*^{x - y} =

Answer 71

*e*^x / *e*^y

Question 72

**The number *e*** *(e*^x)^p =

Answer 72

*ex*^y

Question 73

**The number *e*** ln *(e*^x) =

Answer 73

x

Question 74

**The number *e*** *b* ^x =

Answer 74

Question 75

**The number *e*** *b*^x =

Answer 75

Question 76

**simplifying √ of exponents** when the exponent is even √x⁶

Answer 76

* divide exponent by 2 * write result outside of **√** * leave no variable inside √ √x⁶ * exponent = 6 * 6 / 2 = 3 * x³√ = x³ * x³ = x⁶

Question 77

**simplifying √ of exponents** when the exponent is odd √x³

Answer 77

* 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

Question 78

**simplifying √ of exponents** √80x³y²

Answer 78

80x³y² * exponent = 3 * 3 - 1 = 2 * 2 / 2 = 1 x√80xy²​ * exponent = 2 * 2 / 2 = 1 xy√80x * 4 \* 4 \* 5 = 80 4xy√5x √80x³y^{2 =} 4xy√5x

Question 79

exponential growth & decay f(t) = A₀e^kt

Answer 79

* 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

Question 80

Equation for doubling time time required for a quantity (A₀e^kt) to double

Answer 80

Question 81

Equation for half life

Answer 81