Video Notes: Limits, Continuity, Differentiation, and Chain Rule

Unit 1 – Limits & Continuity

• What is a limit?
  • Informal idea: as x approaches a, f(x) approaches some value L.
  • The limit describes the value that f(x) gets arbitrarily close to near x = a, even if f(a) is undefined or different.
• Finding limits from tables
  • Look at values of f(x) for x values near a from both sides.
  • Observe the trend of the values as x gets closer to a to identify L.
• Finding limits from graphs
  • Read off the y-values as x approaches a from the left and from the right.
  • If both sides approach the same y-value, that is the limit; if they differ, the limit does not exist.
• One-Sided limits
  • Left-hand limit: \lim_{x \to a^-} f(x)
  • Right-hand limit: \lim_{x \to a^+} f(x)
  • The (two-sided) limit exists only if both one-sided limits exist and are equal to L.
• Properties of Limits (including composition)
  • Sum and difference: \lim{x\to a} [f(x) \pm g(x)] = \lim{x\to a} f(x) \pm \lim_{x\to a} g(x) (when both limits exist)
  • Product: \lim{x\to a} [f(x) \cdot g(x)] = \left(\lim{x\to a} f(x)\right) \cdot \left(\lim_{x\to a} g(x)\right)
  • Quotient (when denominator limit ≠ 0): \lim{x\to a} \frac{f(x)}{g(x)} = \frac{\lim{x\to a} f(x)}{\lim_{x\to a} g(x)}
  • Constant multiple: \lim{x\to a} [c \cdot f(x)] = c \cdot \lim{x\to a} f(x)
  • Composition (continuity): if \lim{x\to a} g(x) = L and f is continuous at L, then \lim{x\to a} f(g(x)) = f(L).
• Finding Limits using algebra (0/0 indeterminate form)
  • When direct substitution yields 0/0, simplify algebraically:
  • Factor and cancel common factors.
  • Rationalize (where applicable).
  • Substitute after simplification.
  • Example:
  • Evaluate \lim_{x\to 1} \frac{x^2 - 1}{x - 1}
    • Factor: \frac{(x-1)(x+1)}{x-1}, cancel (x−1) (for x ≠ 1), obtain \lim_{x\to 1} (x+1) = 2.
• Types of Discontinuities
  • Removable discontinuity: a hole where the limit exists but is not equal to f(a).
  • Jump discontinuity: left and right limits exist but are unequal.
  • Infinite discontinuity: function grows without bound as x approaches a.
  • Oscillatory discontinuity: limits do not settle to a single value due to oscillation.
• Defining/Showing a function is continuous at a point
  • A function f is continuous at a point a if \lim_{x\to a} f(x) = f(a).
  • Continuity on an interval means the function is continuous at every point in that interval.
• Vertical and Horizontal Asymptotes
  • Vertical asymptote at x = a if \lim_{x\to a} f(x) = \pm \infty.
  • Horizontal asymptote as x → ±∞ if \lim_{x\to \pm\infty} f(x) = L (y = L).
• Limits to Infinity
  • Describes end behavior of f as x grows without bound or decreases without bound.
• The Intermediate Value Theorem (IVT)
  • If f is continuous on [a, b] and k is any value between f(a) and f(b), then there exists c in (a, b) such that f(c) = k.

Unit 2 – Differentiation

• What is a Derivative?
  • The derivative at a point a is the instantaneous rate of change of f, i.e., the slope of the tangent line to y = f(x) at x = a.
  • Formal limit definition: f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}.
• How is the derivative its own function?
  • The derivative is a function, denoted f'(x), that maps x to the slope of the tangent to the curve y = f(x) at x.
  • Notation: f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
• Average rates of change turning into Instantaneous Rates of Change
  • Average rate of change on [a, b]: \frac{f(b) - f(a)}{b - a}
  • Instantaneous rate: limit of the average rate as h → 0: f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
• Equations of Tangent Lines
  • Point-slope form using derivative at a: if f is differentiable at a, the tangent line at (a, f(a)) is:
  • y = f(a) + f'(a)\,(x - a).
• Derivative Rules
  • Power Rule: for n ≠ 0, \frac{d}{dx} x^n = n x^{n-1}.
  • Product Rule: for functions u(x) and v(x): \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x).
  • Quotient Rule: for u(x) and v(x) with v(x) ≠ 0: \frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{[v(x)]^2}.
• Derivatives of six trig Functions
  • \frac{d}{dx}[\sin x] = \cos x
  • \frac{d}{dx}[\cos x] = -\sin x
  • \frac{d}{dx}[\tan x] = \sec^2 x
  • \frac{d}{dx}[\csc x] = -\csc x \cot x
  • \frac{d}{dx}[\sec x] = \sec x \tan x
  • \frac{d}{dx}[\cot x] = -\csc^2 x
• Derivatives of e^x and ln(x)
  • \frac{d}{dx}[e^x] = e^x
  • \frac{d}{dx}[\ln x] = \frac{1}{x},\quad x>0.
• Finding derivatives using multiple representations (tables and graphs as well as analytical)
  • You can differentiate from:
  • Analytical form directly using rules.
  • Tables of derivatives for common functions.
  • Graphical/intuition-based approaches (e.g., slope of secant approximations) as a check.
• What does it mean to be differentiable?
  • Differentiable at a point implies continuity at that point.
  • A function can be continuous at a point but not differentiable there (e.g., cusp or corner).
  • Differentiability requires a well-defined tangent; breaks at sharp corners or vertical tangents.

Unit 3 – More Differentiation

• The Chain Rule
  • If a function is a composition h(x) = f(g(x)), then the derivative is:
  • h'(x) = f'(g(x)) \cdot g'(x).
  • Concept: differentiate the outer function evaluated at the inner function, times the derivative of the inner function.
  • Example: If h(x) = \sin(3x), then h'(x) = \cos(3x) \cdot 3 = 3\cos(3x).
• Notes on using the Chain Rule
  • Apply the chain rule repeatedly for nested compositions.
  • When combining chain rule with product and quotient rules, treat inner/outer components carefully and apply the correct rule to each part.
• Summary of key ideas from Unit 1–3
  • Limits establish the rigorous notion of approaching a value.
  • Continuity ties limits and function values together at a point.
  • Derivatives measure instantaneous rate of change and are the slopes of tangents.
  • Rules (Power, Product, Quotient, Chain) provide shortcuts to compute derivatives efficiently.
  • Trigonometric, exponential, and logarithmic derivatives form the backbone of differentiation in many applications.
  • Real-world relevance: rates of change, optimization, physics (velocity/acceleration), engineering, economics, etc.

Quick recap of useful formulas

• Limit definitions and basic algebra:
  • \lim_{x\to a} f(x) = L (when it exists)
  • Example: \lim_{x\to 1} \frac{x^2 - 1}{x - 1} = 2 via factoring.
• Derivative definitions and rules:
  • f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
  • Power Rule: \frac{d}{dx} x^n = n x^{n-1}
  • Product Rule: \frac{d}{dx}[u v] = u' v + u v'
  • Quotient Rule: \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u' v - u v'}{v^2}
  • Chain Rule: \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
• Derivatives of common functions:
  • \frac{d}{dx}[\sin x] = \cos x,\quad \frac{d}{dx}[\cos x] = -\sin x,\quad \frac{d}{dx}[\tan x] = \sec^2 x
  • \frac{d}{dx}[\csc x] = -\csc x \cot x,\quad \frac{d}{dx}[\sec x] = \sec x \tan x,\quad \frac{d}{dx}[\cot x] = -\csc^2 x
  • \frac{d}{dx}[e^x] = e^x,\quad \frac{d}{dx}[\ln x] = \frac{1}{x}.