Calculus I Unit 1-3 Notations Flashcards

1
Q

Change (y)

A

Δy = y₂ - y₁
or ƒ(x)₂ - ƒ(x)₁

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2
Q

Rate of Change (x)

A

Δx = x₂ - x₁

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3
Q

Average Rate of Change

A

Δy/Δx

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4
Q

Secant Line of a Function

A

Avg rate of change between 2 points/positions

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5
Q

Tangent Line of a Function

A

Slope at a curve of a particular point

(Instantaneous ROC at particular position)

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6
Q

Difference Equation

A

ƒ(x + h) - ƒ(x) / h

where h = Δx

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7
Q

Directional Limit Properties

A

If:
limit as x → a⁻ ƒ(x) = limit as x → a⁺ ƒ(x)
then:
limit as x → a ƒ(x) exists

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8
Q

Definition of
limit as x → a ƒ(x) = ∞

A

ƒ(x) grows arbitrarily large for all x sufficiently close to a

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9
Q

Definition of
limit as x → a ƒ(x) = -∞

A

ƒ(x) is negative and grows arbitrarily large for all x sufficiently close to a

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10
Q

Explain Solution for:

limit as x → 0⁺ (k + x) / x

where k is constant

A

Since k is constant and in the numerator, as the value of x decreases: the numerator approaches k while the denominator gets infinitely closer to 0.

limit as x → 0⁺ (k + x) / x = ∞

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11
Q

Explain Solution for:

limit as x → 0⁻ (k + x) / x

where k is constant

A

Since k is constant and in the numerator, as the value of x increases: the numerator approaches k while the denominator becomes infinitely smaller in magnitude through negative values

limit as x → 0⁻ (k + x) / x = -∞

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12
Q

Explain Solution for:

limit as x → 3⁺ (2 - 5x) / (x - 3)

A

Since as x approaches 3, the numerator becomes closer to -13, and the dominator becomes infinitely smaller from a positive directions
( -/+) = -

limit as x → 3⁺ (2 - 5x) / (x - 3) = -∞

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13
Q

Explain Solution for:

limit as x → 3⁻ (2 - 5x) / (x - 3)

A

Since as x approaches 3, the numerator becomes closer to -13, and the dominator becomes infinitely smaller from a negative directions ( -/-) = +

limit as x → 3⁺ (2 - 5x) / (x - 3) = ∞

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14
Q

How do you find the Vertical Asymptote of ƒ(x) ?

A

The Vertical Asymptote is the value of limit value of x where ƒ(x) = ∞

This is noted by x = a

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15
Q

Sum of Cubes Factorization:

A

(a³ + b³) = (a + b)(a² - ab + b²)

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16
Q

Difference of Cubes Factorization

A

(a³ - b³) = (a - b)(a² + ab + b²)

17
Q

How do you find the horizontal asymptote of ƒ(x)

A

If a given Limit of ƒ(x) has an x → ± ∞ , the value of ƒ(x) = Horizontal Asymptote

18
Q

Explain limit as x → ∞ of
(2 + 7x + 2x²)/(x²)

A

limit as x → ∞ = 2

Since there is a constant of 2 in the numerator while the highest degree (x²) is in both the numerator and denominator.

Both the numerator and denominator are divided by x² leaving us with 2

19
Q

Explain solution to
limit as x → -∞ of:

√ (16x² + x) / (x)

A

Divide each part of the numerator and denominator by their respective highest degree

since the numerator is powered to an ‘odd’ degree and x → -∞ it will be divided by -x.

Since the denominator is powered by an even degree it will be divided by a x². Solve with new constant.

20
Q

How do we know that a function doesn’t have an Horizontal Asymptote ?

A

The degree of X in the numerator is higher than the highest degree in the denominator

21
Q

Explain the solution to:

limit as x → 0⁺ of (5eˣ) / (1 - eˣ)

A

limit as x → 0⁺ of (5eˣ) / (1 - eˣ): Since eⁿ when n > 0 is greater than 1, the denominator (1 - x) will always be negative. Since eˣ can never be negative, the numerator (5eˣ) must always be positive. As the value of x approaches 0 from the positive direction:

limit as x → 0⁺ = -∞

22
Q

Explain the solution to:

limit as x → 0⁻ of (5eˣ) / (1 - eˣ)

A

limit as x → 0⁻ of (5eˣ) / (1 - eˣ): Since eⁿ when n < 0 is less than 1, the denominator will always be positive. Since eˣ can never be negative, the numerator (5eˣ) must always be positive. As the value of x approaches 0 from the positive direction:

limit as x → 0⁻ = ∞

23
Q

Explain the Interval of:

limit as x → 0 of (5eˣ) / (1 - eˣ)

A

(-∞, 0) , (0,∞), Since the value of ƒ(x) exists at all numbers except when eˣ = 1 and
(e⁰ = 1)

24
Q

What creates a Slanted Asymptote ?

A

A Slanted Asymptote is created when the degree of the independent variable in the numerator is EXACTLY ONE degree higher than the degree of the denominator.

25
How do you solve for a equation of the line of the **Slanted Asymptote** ?
Perform Polynomial Long Division , the slope and y intercept which is found is the equation that describes the line of our limit.
26
3 Conditions of Continuity:
#1: ƒ(a) is defined #2 the Limit as x → a of ƒ(x) exists #3: Limit as x → a of ƒ(x) = ƒ(a) (All conditions must be met)
27
3 Types of Discontinuity:
Removeable Discontinuité Jump Discontinuité Infinite Discontinuité
28
Explain the Continuity of Rational Functions 𝑝(x)/𝑔(x) :
Rational functions are continuous everywhere except when 𝑔(x) = 0 / or the denominator = 0
29
Explain Limits of Composite Functions ƒ(𝑔(x))
If 𝑔(x) is continuous at **a** and ƒ is continuous at 𝑔(x) then: Limit as x → a of ƒ(𝑔(x)) = ƒ(Limit as x → a of 𝑔(x)) Take the limit inside the function then evaluate at the composite.
30
Equation for Instantaneous Rate of Change (The Derivative) ƒ'(x)
ƒ'(x) = limit as h → 0 of: ƒ(x + h) - ƒ(x) / h
31
How to find the equation of the Tangent Line of a point ?
Solve for Tangent Line (mₜₐₙ) using the Instantaneous Rate of Change formula, then solve for 𝑦 using: 𝑦 - 𝑦₁ = mₜₐₙ(x - x₁)
32
What is Intermediate Value Theorem ?
In mathematical analysis, the intermediate value theorem states that if ƒ is a continuous function whose domain contains the interval [a, b], then it takes on any given value between ƒ(a) and ƒ(b)
33
Equation for Instantaneous Rate of Change at a particular given point ƒ'(a)
limit as x → a of: ƒ'(a) = ƒ(x) - ƒ(a) / x -a where **a** is a given and ƒ(a) is solved for
34
Identify the Horizontal Asymptote of: limit as x → -∞ of: √ (9x⁴ + 25x) + x² / x² - 9
Divide all values by the highest degree in the denominator (x²), then solve. since ƒ(x) = 4, our horizontal asymptote will also = 4.