Comprehensive University Calculus: Limits, Derivatives, and Integrals

Chapter 1: Limits and Sequences

• Definition of Sequences ($a_n$): A sequence is an ordered list of numbers assigned to the natural numbers $n \in \mathbb{N}$. The transcript illustrates sequences using notation like $(a_1, a_2, ..., a_n)$.
• Specific Example (Page 10): Consider the sequence where the terms are defined relative to a value $r$, such as $r, 2r, 4r$.
• Examples of Sequences and Convergence (Page 11):
  • Sequence $a_n = \frac{n-1}{n}$: As $n$ increases, the values follow: $0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, ...$. This sequence approaches the limit of $1$.
  • Sequence $b_n = \frac{2n+1}{n+1}$: Values include $\frac{3}{2}, \frac{5}{3}, \frac{7}{4}, \frac{9}{5}, ...$. This sequence approaches the value $2$.
  • Sequence $c_n = \frac{1}{n}$: Values include $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ...$. This sequence approaches $0$.
• The Concept of Limit (Page 12): The limit represents the value that a function or sequence approaches as the input (or index) travels toward a specific point.
  • Notation: $\lim_{x \to a} f(x) = l$.
  • Numerical Proximity Example: To find the limit of $x$ as it approaches $9$, we look at values from both sides:
    • Left side ($x \to 9^{-}$): $8.9, 8.99, 8.999, 8.9999, ...$
    • Right side ($x \to 9^{+}$): $9.1, 9.01, 9.001, 9.0001, ...$
• Formal (Epsilon-Delta) Definition: A function $f(x)$ has a limit $l$ as $x$ approaches $a$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - l| < \epsilon$.

Properties and Calculation of Limits

• Basic Rules (Page 16):
  • Sum/Difference Rule: $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$.
  • Product Rule: $\lim_{x \to a} [f(x) \times g(x)] = (\lim_{x \to a} f(x)) \times (\lim_{x \to a} g(x))$.
  • Constant Multiple Rule: $\lim_{x \to a} [C \times f(x)] = C \times \lim_{x \to a} f(x)$.
  • Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \neq 0$.
• Evaluating Limits of Polynomials (Page 19):
  • Example: $\lim_{x \to 2} (2x + 1) = 2(2) + 1 = 5$.
  • Example: $\lim_{x \to 4} (x^2 - 16) = 4^2 - 16 = 0$.
• Squeeze Theorem (Sandwich Theorem) (Page 19): If $f(x) \leq g(x) \leq h(x)$ for all $x$ near $a$, and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$.

Indeterminate Forms and Advanced Limits

• The $0/0$ Indeterminate Form (Page 21):
  • When direct substitution results in $0/0$, factorization is required.
  • Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x + 2) = 4$.
  • Example: $\lim_{x \to 4} \frac{x - 4}{\sqrt{x} - 2}$. Multiplying by the conjugate: $\lim_{x \to 4} \frac{(x-4)(\sqrt{x}+2)}{x-4} = 4$.
• Infinite Limits and $\infty/\infty$ (Page 24):
  • For rational functions $f(x) = \frac{a_n x^n + ...}{b_m x^m + ...}$ as $x \to \infty$:
    1. If $n < m$, the limit is $0$.
    2. If $n = m$, the limit is the ratio of leading coefficients $\frac{a_n}{b_m}$.
    3. If $n > m$, the limit is $\pm \infty$.
• Euler’s Number ($e$):
  • Defined as $\lim_{n \to \infty} (1 + \frac{1}{n})^n = e \approx 2.718281828$.
  • Alternative form: $\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$.
• Trigonometric Limits (Page 32-34):
  • Fundamental limit: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$.
  • Application: $\lim_{x \to 0} \frac{\tan(x)}{x} = 1$.
  • Example: $\lim_{x \to 0} \frac{\sin(5x)}{\sin(3x)} = \frac{5}{3}$.

Continuity of Functions

• Definition (Page 36): A function $f(x)$ is continuous at a point $a$ if:
  1. $f(a)$ is defined (exists).
  2. $\lim_{x \to a} f(x)$ exists.
  3. $\lim_{x \to a} f(x) = f(a)$.
• Types of Discontinuity:
  • Removable: The limit exists, but does not equal the function value.
  • Jump/Essential: The left-hand limit and right-hand limit are not equal.
• Properties: If $f$ and $g$ are continuous, then $f \pm g, f \times g$, and $f/g$ (where $g \neq 0$) are also continuous.

Chapter 2: The Derivative

• Historical context (Page 49): Development attributed to Pierre de Fermat, Isaac Newton, and Gottfried Wilhelm Leibniz.
• Definition of the Derivative: The derivative measures the instantaneous rate of change or the slope of the tangent line to a curve.
• Limit Definition (Newton’s Difference Quotient):     $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Physical Interpretation: If $s = f(t)$ is a position function, then $v(t) = f'(t)$ is the velocity at time $t$.

Rules of Differentiation

• Constant Rule: If $f(x) = C$, then $f'(x) = 0$.
• Power Rule: If $f(x) = x^n$, then $f'(x) = n x^{n-1}$.
• Sum and Difference Rule: $(u \pm v)' = u' \pm v'$
• Product Rule: $(u \times v)' = u'v + v'u$
• Quotient Rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$
• Chain Rule: If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$.

Derivatives of Transcendental and Trigonometric Functions

• Trigonometric Derivatives (Page 74-78):
  • $\frac{d}{dx}(\sin(x)) = \cos(x)$
  • $\frac{d}{dx}(\cos(x)) = -\sin(x)$
  • $\frac{d}{dx}(\tan(x)) = \sec^2(x)$
  • $\frac{d}{dx}(\cot(x)) = -\csc^2(x)$
  • $\frac{d}{dx}(\sec(x)) = \sec(x) \tan(x)$
  • $\frac{d}{dx}(\csc(x)) = -\csc(x) \cot(x)$
• Implicit Differentiation (Page 80): Used for equations where $y$ is not explicitly isolated (e.g., $x^2 + y^2 = 25$). Both sides are differentiated with respect to $x$, treating $y$ as a function of $x$ ($y'$ notation is used).
• Higher-Order Derivatives (Page 84): The second derivative $y''$ or $f''(x)$ is the derivative of the first derivative. The $n$-th derivative is denoted as $f^{(n)}(x)$.

Chapter 3: Applications of the Derivative

• Critical Points: Points where $f'(x) = 0$ or $f'(x)$ is undefined.
• First Derivative Test (Monotonicity):
  • If $f'(x) > 0$ on an interval, the function is increasing.
  • If $f'(x) < 0$ on an interval, the function is decreasing.
• Extrema (Maximum and Minimum):
  • Relative/Local Max: Occurs at critical point $a$ if the sign of $f'(x)$ changes from positive to negative.
  • Relative/Local Min: Occurs at critical point $a$ if the sign of $f'(x)$ changes from negative to positive.
• Concavity and Inflection Points (Page 97):
  • If $f''(x) > 0$, the curve is concave up (U-shape).
  • If $f''(x) < 0$, the curve is concave down (inverted U-shape).
  • An Inflection Point occurs where the concavity changes (and $f''(x) = 0$).
• L'Hôpital's Rule (Page 130): If $\lim_{x \to a} \frac{f(x)}{g(x)}$ results in $0/0$ or $\infty/\infty$, then:     $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$
• Mean Value Theorem (MVT): If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c \in (a,b)$ such that:     $f'(c) = \frac{f(b) - f(a)}{b - a}$
• Rolle's Theorem: A special case of MVT where if $f(a) = f(b)$, there exists $c \in (a,b)$ such that $f'(c) = 0$.

Chapter 4: Integrals

• Indefinite Integral (Antiderivative) (Page 148): The inverse operation of differentiation.
  • $\int f(x) dx = F(x) + C$, where $F'(x) = f(x)$.
• Fundamental Rules of Integration:
  • $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$).
  • $\int k f(x) dx = k \int f(x) dx$.
  • $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$.
• Definite Integral (Page 154): Represents the net signed area under a curve between bounds $a$ and $b$.
  • $\int_{a}^{b} f(x) dx = F(b) - F(a)$.
• Riemann Sums: The area is approximated by dividing the interval $[a,b]$ into $n$ sub-intervals of width $\Delta x = \frac{b-a}{n}$ and summing the areas of rectangles.
• Mean Value Theorem for Integrals (Page 161): If $f$ is continuous on $[a,b]$, there exists a point $c \in [a,b]$ such that:     $\int_{a}^{b} f(x) dx = f(c)(b - a)$     The value $f(c)$ is called the average value of the function over the interval.