Calculus I Unit 1-3 Notations

Question 1

Change (y)

Answer 1

Δy = y₂ - y₁
or ƒ(x)₂ - ƒ(x)₁

Question 2

Rate of Change (x)

Answer 2

Δx = x₂ - x₁

Question 3

Average Rate of Change

Answer 3

Δy/Δx

Question 4

Secant Line of a Function

Answer 4

Avg rate of change between 2 points/positions

Question 5

Tangent Line of a Function

Answer 5

Slope at a curve of a particular point

(Instantaneous ROC at particular position)

Question 6

Difference Equation

Answer 6

ƒ(x + h) - ƒ(x) / h

where h = Δx

Question 7

Directional Limit Properties

Answer 7

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

Question 8

Definition of
limit as x → a ƒ(x) = ∞

Answer 8

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

Question 9

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

Answer 9

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

Question 10

Explain Solution for:

limit as x → 0⁺ (k + x) / x

where k is constant

Answer 10

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 = ∞

Question 11

Explain Solution for:

limit as x → 0⁻ (k + x) / x

where k is constant

Answer 11

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 = -∞

Question 12

Explain Solution for:

limit as x → 3⁺ (2 - 5x) / (x - 3)

Answer 12

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) = -∞

Question 13

Explain Solution for:

limit as x → 3⁻ (2 - 5x) / (x - 3)

Answer 13

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) = ∞

Question 14

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

Answer 14

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

This is noted by x = a

Question 15

Sum of Cubes Factorization:

Answer 15

(a³ + b³) = (a + b)(a² - ab + b²)

Question 16

Difference of Cubes Factorization

Answer 16

(a³ - b³) = (a - b)(a² + ab + b²)

Question 17

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

Answer 17

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

Question 18

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

Answer 18

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

Question 19

Explain solution to
limit as x → -∞ of:

√ (16x² + x) / (x)

Answer 19

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.

Question 20

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

Answer 20

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

Question 21

Explain the solution to:

limit as x → 0⁺ of (5eˣ) / (1 - eˣ)

Answer 21

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⁺ = -∞

Question 22

Explain the solution to:

limit as x → 0⁻ of (5eˣ) / (1 - eˣ)

Answer 22

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⁻ = ∞

Question 23

Explain the Interval of:

limit as x → 0 of (5eˣ) / (1 - eˣ)

Answer 23

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

Question 24

What creates a Slanted Asymptote ?

Answer 24

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.

Question 25

How do you solve for a equation of the line of the Slanted Asymptote ?

Answer 25

Perform Polynomial Long Division , the slope and y intercept which is found is the equation that describes the line of our limit.

Question 26

3 Conditions of Continuity:

Answer 26

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)

Question 27

3 Types of Discontinuity:

Answer 27

Removeable Discontinuité

Jump Discontinuité

Infinite Discontinuité

Question 28

Explain the Continuity of Rational Functions 𝑝(x)/𝑔(x) :

Answer 28

Rational functions are continuous everywhere except when 𝑔(x) = 0 / or the denominator = 0

Question 29

Explain Limits of Composite Functions ƒ(𝑔(x))

Answer 29

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.

Question 30

Equation for Instantaneous Rate of Change (The Derivative) ƒ’(x)

Answer 30

ƒ’(x) = limit as h → 0 of:
ƒ(x + h) - ƒ(x) / h

Question 31

How to find the equation of the Tangent Line of a point ?

Answer 31

Solve for Tangent Line (mₜₐₙ) using the Instantaneous Rate of Change formula, then solve for 𝑦 using: 𝑦 - 𝑦₁ = mₜₐₙ(x - x₁)

Question 32

What is Intermediate Value Theorem ?

Answer 32

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)

Question 33

Equation for Instantaneous Rate of Change at a particular given point ƒ’(a)

Answer 33

limit as x → a of: ƒ’(a) = ƒ(x) - ƒ(a) / x -a

where a is a given and ƒ(a) is solved for

Question 34

Identify the Horizontal Asymptote of:

limit as x → -∞ of:

√ (9x⁴ + 25x) + x² / x² - 9

Answer 34

Divide all values by the highest degree in the denominator (x²), then solve.

since ƒ(x) = 4, our horizontal asymptote will also = 4.