Podcast
Questions and Answers
What is the primary focus when using the 'method of exhaustion' to find the area of a shape?
What is the primary focus when using the 'method of exhaustion' to find the area of a shape?
- Measuring the weight of the shape to estimate the area.
- Dividing the shape into equal squares and counting them.
- Calculating the precise dimensions using complex formulas.
- Inscribing and circumscribing polygons to approximate the area. (correct)
The tangent line to a curve at a given point $P$ is best described as:
The tangent line to a curve at a given point $P$ is best described as:
- A line that is perpendicular to the curve at point $P$.
- A line that touches the curve at $P$ and represents the limiting position of secant lines through $P$. (correct)
- A line that intersects the curve at two distinct points close to $P$.
- A line that is parallel to the x-axis and passes through point $P$.
What condition must be met for $\lim_{x \to a} f(x)$ to exist?
What condition must be met for $\lim_{x \to a} f(x)$ to exist?
- The one-sided limits, $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$, must both exist and be equal. (correct)
- The value of $f(a)$ must be equal to zero.
- The function $f(x)$ must be defined at $x = a$.
- The function $f(x)$ must be continuous at $x = a$.
Given $f(x) = \frac{x^2 - 4}{x - 2}$, how can the limit as $x$ approaches 2 be determined algebraically?
Given $f(x) = \frac{x^2 - 4}{x - 2}$, how can the limit as $x$ approaches 2 be determined algebraically?
If $\lim_{x \to a^-} g(x)$ and $\lim_{x \to a^+} g(x)$ both do not exist, what can be concluded about $\lim_{x \to a} g(x)$?
If $\lim_{x \to a^-} g(x)$ and $\lim_{x \to a^+} g(x)$ both do not exist, what can be concluded about $\lim_{x \to a} g(x)$?
According to the definition of a limit, what does the statement $\lim_{x \to a} f(x) = L$ imply?
According to the definition of a limit, what does the statement $\lim_{x \to a} f(x) = L$ imply?
What is the significance of evaluating one-sided limits?
What is the significance of evaluating one-sided limits?
Given the function $H(t) = \begin{cases} 0 & \text{if } t < 0 \ 1 & \text{if } t \geq 0 \end{cases}$, what can be said about $\lim_{t \to 0} H(t)$?
Given the function $H(t) = \begin{cases} 0 & \text{if } t < 0 \ 1 & \text{if } t \geq 0 \end{cases}$, what can be said about $\lim_{t \to 0} H(t)$?
Which of the following is true regarding the fundamental limit laws?
Which of the following is true regarding the fundamental limit laws?
What must be true for a function $f$ to be continuous at a number $a$?
What must be true for a function $f$ to be continuous at a number $a$?
When is a discontinuity considered 'removable'?
When is a discontinuity considered 'removable'?
How is continuity defined over a closed interval $[a, b]$?
How is continuity defined over a closed interval $[a, b]$?
What can be said about the continuity of a polynomial function?
What can be said about the continuity of a polynomial function?
If function $g$ is continuous at $a$ and function $f$ is continuous at $g(a)$, what can be said about the composite function $f(g(x))$?
If function $g$ is continuous at $a$ and function $f$ is continuous at $g(a)$, what can be said about the composite function $f(g(x))$?
Which statement best describes the Squeeze Theorem?
Which statement best describes the Squeeze Theorem?
What does it mean for $\lim_{x \to a} f(x) = \infty$?
What does it mean for $\lim_{x \to a} f(x) = \infty$?
What is an 'indeterminate form' when evaluating limits?
What is an 'indeterminate form' when evaluating limits?
What general strategy can be used to evaluate limits of rational functions at infinity when an indeterminate form is encountered?
What general strategy can be used to evaluate limits of rational functions at infinity when an indeterminate form is encountered?
Given that $f(x) = \frac{\sin(x)}{x}$, what is the limit as $x$ approaches 0?
Given that $f(x) = \frac{\sin(x)}{x}$, what is the limit as $x$ approaches 0?
Which of the following functions is continuous at every point in its domain?
Which of the following functions is continuous at every point in its domain?
A rational function $f(x) = \frac{P(x)}{Q(x)}$ is continuous everywhere except where?
A rational function $f(x) = \frac{P(x)}{Q(x)}$ is continuous everywhere except where?
To show that $\lim_{x \to 0} x^2 \sin(\frac{1}{x}) = 0$, what theorem is most applicable?
To show that $\lim_{x \to 0} x^2 \sin(\frac{1}{x}) = 0$, what theorem is most applicable?
What is implied by the statement $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$?
What is implied by the statement $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$?
In the context of limits, what does the term 'vertical asymptote' indicate?
In the context of limits, what does the term 'vertical asymptote' indicate?
How are one-sided limits related to vertical asymptotes?
How are one-sided limits related to vertical asymptotes?
Assuming a polynomial function $p(x)$ of degree $n > 0$ and a real number $a_n > 0$, what is the $\lim_{x \to \infty} p(x)$?
Assuming a polynomial function $p(x)$ of degree $n > 0$ and a real number $a_n > 0$, what is the $\lim_{x \to \infty} p(x)$?
If a function has a removable discontinuity at $x=a$, what can be done to make the new function continuous?
If a function has a removable discontinuity at $x=a$, what can be done to make the new function continuous?
The velocity of a skydiver is modeled by $v(t) = \sqrt{\frac{32}{k}} \frac{1-e^{-2t\sqrt{32k}}}{1+e^{-2t\sqrt{32k}}}$. What does $\lim_{t \to \infty} v(t)$ represent?
The velocity of a skydiver is modeled by $v(t) = \sqrt{\frac{32}{k}} \frac{1-e^{-2t\sqrt{32k}}}{1+e^{-2t\sqrt{32k}}}$. What does $\lim_{t \to \infty} v(t)$ represent?
What type of function is $f(x) = x^{100} - 2x^{37} + 75$, and where is it continuous?
What type of function is $f(x) = x^{100} - 2x^{37} + 75$, and where is it continuous?
What is the defining characteristic of the 'greatest integer function', denoted as $[x]$?
What is the defining characteristic of the 'greatest integer function', denoted as $[x]$?
$f(x) = \begin{cases} cx^2 + 2x & \text{if } x < 2 \ x^3 - cx & \text{if } x \geq 2 \end{cases}$. What value of $c$ makes this function continuous?
$f(x) = \begin{cases} cx^2 + 2x & \text{if } x < 2 \ x^3 - cx & \text{if } x \geq 2 \end{cases}$. What value of $c$ makes this function continuous?
Given the function $f(x)=\frac{x+1}{x-2};$,$;\lim_{x \to 2^+}f(x)$ equals
Given the function $f(x)=\frac{x+1}{x-2};$,$;\lim_{x \to 2^+}f(x)$ equals
Given the graphed function find $\lim_{x \to 5^-} g(x)$
Given the graphed function find $\lim_{x \to 5^-} g(x)$
Given the graphed function what number is $\lim_{x \to 5} g(x)$ equal to?
Given the graphed function what number is $\lim_{x \to 5} g(x)$ equal to?
When an expression equals an Indeterminate form can use which of the following rule or therom?
When an expression equals an Indeterminate form can use which of the following rule or therom?
Flashcards
Area via Exhaustion Method
Area via Exhaustion Method
Calculus approach using inscribed polygons. Area of circle is limit of inscribed polygon areas.
Area Problem
Area Problem
The central problem in the branch of calculus.
Tangent Line
Tangent Line
A line that touches a curve at a point.
Pierre Fermat
Pierre Fermat
French mathematician (1601-1665); ideas behind differential calculus.
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Limit
Limit
Condition where function values get indefinitely close to a value as input approaches some point.
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Left-Hand Limit
Left-Hand Limit
The limit of f(x) as x approaches 'a' from the left.
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Right-Hand Limit
Right-Hand Limit
The limit of f(x) as x approaches 'a' from the right.
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One-Sided Limits
One-Sided Limits
Limits evaluated from the left and right.
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Existence of a Limit
Existence of a Limit
The limit exists if and only if both one-sided limits exist and are equal.
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Notion of Limit
Notion of Limit
Analyzing function behavior nearby, not exactly at, a point of interest.
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Definition of a Limit
Definition of a Limit
lim f(x) = L if f(x) approaches L as x approaches a.
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Formal Definition of Limit
Formal Definition of Limit
Suppose f(x) is defined when x is near the number a. We write lim x→a f(x) = L
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Limit Does Not Exist
Limit Does Not Exist
If the left-hand limit does not equal the right-hand limit, the limit does not exist.
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Factor then find limit
Factor then find limit
Evaluate both sides for same value
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Continuous Function
Continuous Function
A function f is continuous at a number a if lim x→a f(x) = f(a).
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Continuity Conditions
Continuity Conditions
Function f is continuous at x = a if:
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Which functions are continous?
Which functions are continous?
Types of functions that are continuos at every number in their domains.
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Infinite Limit
Infinite Limit
lim h(x) = ∞ means h(x) increases without bound as x gets close.
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Horizontal Asymptote
Horizontal Asymptote
y = f(x) approaches the line y = b as x increases or decreases without bound.
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Highest Power of x in Denominator
Highest Power of x in Denominator
Rule to solve indeterminate form by numerator times 1/x, and denominator times 1/x
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ENGM1180 – Mathematics for Engineers II: Lecture 1 – Limits & Continuity
- Course Code: ENGM1280
- Course Title: Mathematics for Engineers II
- Ramone R. Jackson is the lecturer.
- Office hours operate under an open door policy and are subject to availability.
- Location: Room #4, Ground floor, Faculty of Engineering
- Email: ramerjackson01@gmail.com
- Class Times & Venue: Tuesdays – Thursdays 8:00am - 9:00am SLT 2
- Assessments:
- Assignments: 20%
- Project: 20%
- In-Course: 20%
- Final exam: 40%
Formulas and Theorems
- The slides include various algebraic, arithmetic, and geometric formulas.
- It also include factoring, binomial theorem, trigonometric identities and functions.
Area Problem
- Ancient Greeks used the "method of exhaustion" to find areas around 2500 years ago.
- They could find the area A of any polygon by dividing it into triangles.
- The Greek method of exhaustion was to inscribe polygons in a figure and circumscribe polygons around the figure.
- They would then increase the number of sides of the polygons.
- Eudoxus (fifth century BC) used exhaustion to prove the formula for the area of a circle: A = πr².
- The area problem is the central problem in integral calculus.
Tangent Problem
- Differential calculus came about due to the tangent problem.
- It was not invented until 2000 years after integral calculus.
- Key figures: Pierre Fermat (1601-1665), John Wallis (1616-1703), Isaac Barrow (1630-1677), Isaac Newton (1642-1727), and Gottfried Leibniz (1646-1716).
Limits
- Functions can be undefined at certain points like x = 2.
- Tables and graphs can give clues about a function's behavior.
One-Sided Limits
- Notation x → 2⁻ indicates x approaches 2 from the left side.
- Notation x → 2⁺ indicates x approaches 2 from the right side.
- lim x→2⁻ f(x) and lim x→2⁺ f(x) are one-sided limits.
- If the two one-sided limits of f(x) are the same as x approaches a, then the limit of f(x) as x approaches a exists.
Limit Definition
- If f(x) is defined when x is near a number a, then lim x→a f(x) = L if f(x) can be made arbitrarily close to L by taking x sufficiently close to a, but not equal to a.
- lim x→a f(x) = L if and only if lim x→a⁻ f(x) = L and lim x→a⁺ f(x) = L.
Limit Laws
- Suppose that c is a constant and the limits lim x→a f(x) and lim x→a g(x) exist. Then:
- lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
- lim x→a [f(x) - g(x)] = lim x→a f(x) - lim x→a g(x)
- lim x→a [cf(x)] = c lim x→a f(x)
- lim x→a [f(x)g(x)] = lim x→a f(x) ⋅ lim x→a g(x)
- lim x→a [f(x) / g(x)] = lim x→a f(x) / lim x→a g(x) if lim x→a g(x) ≠ 0
- lim x→a [f(x)]ⁿ = [lim x→a f(x)]ⁿ where n is a positive integer.
- lim x→a c = c
- lim x→a x = a
- lim x→a xⁿ = aⁿ where n is a positive integer.
- lim x→a ⁿ√x = ⁿ√a where n is a positive integer. (If n is even, we assume that a > 0.)
- lim x→a ⁿ√f(x) = ⁿ√lim x→a f(x) where n is a positive integer. (If n is even, we assume that lim x→a f(x) > 0.)
- If f is a polynomial or a rational function and a is in the domain of f, then lim x→a f(x) = f(a)
Theorems
- If lim x→a f(x) = L and lim x→a g(x) = L, then lim x→a f(x) = L
- Suppose that f(x) ≤ g(x) ≤ h(x) for all x in some interval (c, d), except possibly at the point a ∈ (c, d) and that lim x→a f(x) = lim x→a h(x) = L. Then, it follows that lim x→a g(x) = L.
- Any polynomial is continuous everywhere on R = (-∞, ∞).
- Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.
- If f is continuous at b and lim x→a g(x) = b, then lim x→a f(g(x)) = f(b).
- If g is continuous at a and f is continuous at g(a), then the composite function f ∘ g given by (f ∘ g)(x) = f(g(x)) is continuous at a.
Function Types
- The following types of functions are continuous at every number in their domains
- polynomials
- rational functions
- root functions
- trigonometric functions
Definition of Continuity
- A function f is continuous at a number a if lim x→a f(x) = f(a).
- The definition implicitly requires: f(a) is defined, lim x→a f(x) exists, and lim x→a f(x) = f(a).
- If a function f is continuous at every point on an open interval (a, b), then it is continuous on (a, b).
- If it is continuous on the open interval (a, b) and lim x→a+ f(x) = f(a) and lim x→b- f(x) = f(b), then the function f is continuous on the closed interval [a, b].
- A function f is continuous on an interval if it is continuous at every number in the interval.
- If f is defined only on one side of an endpoint of the interval, continuous at the endpoint is understood to mean continuous from the right or continuous from the left.
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