Calculus I — Derivatives, Notation, and Course Logistics (Lecture Notes)

Schedule and Announcements

• Quizzes: Quizzes were graded over the weekend; instructor had a travel mishap (ran out of red pen ink) but finished grading after returning.
• Upcoming quiz: Second quiz this Friday on continuity and the intermediate value theorem; topics likely sections 4 and 6; a quick review planned.
• Trig review: Beginning this Friday, some trig content will appear in class; students should refresh beforehand; there will be a smaller review in the first half of Friday's class.
• Midterm: A half-semester course; midterm scheduled about a week after this coming Friday (week from Friday). A full day review will be offered the day before the midterm; students can ask questions in class.
• Online posting of midterm: Not posted online; grades and progress are better tracked by students themselves via online assessments, skill tests, and homework grades (student self-awareness). The instructor notes potential quirks with online posting and alphabetization of rosters.
• Registration for calculus program (Dr. Kazmarek): Important weekend email with a registration link; two to three week registration window; must submit first attempt in the first week but can register any time within the window; slots may fill up, and the instructor has no control over registration—advises contacting registration if full.
• Drop deadline: Today is the drop deadline; students should ensure they’re comfortable with their status.
• Homeworks schedule for this week:
  • Homework due tonight (originally due yesterday).
  • Warm-up due tomorrow.
  • One assignment due Friday.
  • Homework due Friday night.
  • Next week will have slightly less homework due to the exam.
• Special note about the Friday warm-up: It’s two problems and contributes only a small portion to the grade; students shouldn’t stress too much about it.
• Skills test: The course includes a skills test starting Monday next week. A practice test is available on WebAssign; the first one has 13 questions drawn randomly from a pool; students have three attempts; the score rounds up toward 10 points (assuming no tailoring). More details to come after the midterm.
• Topics coverage for the skills test: Should cover up to today’s material; instructor will confirm tomorrow (or students can check tonight).
• Registration reminders: Expect an email this coming weekend about registration with a link; you may have a 2–3 week window to complete registration; first attempt should be in the first week to ensure a slot; if not available, instructor cannot guarantee a fix; contact registration if needed.
• Questions: Open floor for questions about today’s material and announcements.

Derivatives: Core Concepts and Foundations

• Derivative defined as the slope of the tangent line; historically from the slope between two points and taking the limit as the points come together.
• Two common limit definitions of the derivative:
  • At a fixed point a: f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}
  • As a function of x: f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
• A constant function has derivative 0 (horizontal line slope).
• Treating the derivative as its own function (f′) is helpful: it allows taking higher derivatives and simpler expressions in many cases. Replacing the constant a with a variable x makes formulas neater:
  • f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
• Differentiability versus continuity:
  • If f is differentiable at a point a, then f is continuous at a (proof sketch using the limit definition and algebra):
  • Start from \lim_{x\to a} \frac{f(x)-f(a)}{x-a} exists because the derivative exists.
  • Multiply by 1 = \frac{x-a}{x-a} to get f'(a)=\lim_{x\to a} \frac{f(x)-f(a)}{x-a} and then use the product rule to show that
    • (\lim_{x\to a} (f(x)-f(a)) = 0), hence
    • (\lim_{x\to a} f(x) = f(a)) and f is continuous at a.
  • Continuous does not imply differentiable: example f(x) = |x| is continuous everywhere but not differentiable at x = 0 (sharp corner).
• Notation for derivatives (wide variety):
  • f′(x) (prime notation)
  • y = f(x); y′ or f′(x)
  • dy/dx or df/dx (Weibnitz notation)
  • Df or D/dx (operator notation)
  • Different notations are interchangeable in practice; choice depends on context.
• Differentiability criteria and examples:
  • Absolute value: f(x) = |x| is differentiable for x ≠ 0 but not at x = 0 due to a cusp (left and right derivatives differ: −1 vs 1).
  • Cube root: f(x) = \sqrt[3]{x} = x^{1/3} is continuous everywhere but its derivative has a vertical tangent at x = 0; f′(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \sqrt[3]{x^2}}; domain of f′ excludes x = 0.
  • Three main ways a function can fail to be differentiable at a point:
  • Discontinuity at that point
  • Sharp corner or cusp at that point
  • Vertical tangent line at that point (derivative tends to ±∞ or does not exist)
• Practical takeaway: differentiability implies continuity; the converse is false in general.

The Derivative as a Function and Basic Sign Analysis

• The derivative provides information about monotonicity and shape:
  • If f′(x) > 0 for x in an interval, f is increasing on that interval.
  • If f′(x) < 0 for x in an interval, f is decreasing on that interval.
  • If f′(x) = 0 at a point, there is a horizontal tangent there; not necessarily a local max/min, but often important for extrema after the midterm.
• Example: f(x) = x^2
  • Derivative via limit definition (illustrative):
  • f(x+h) = (x+h)^2 = x^2 + 2xh + h^2
  • (f(x+h) - f(x))/h = (2xh + h^2)/h = 2x + h
  • As h → 0, f′(x) = 2x
  • Sign analysis: for x > 0, f′(x) > 0 => f is increasing; for x < 0, f′(x) < 0 => f is decreasing; at x = 0, f′(0) = 0 => horizontal tangent.
• The derivative as a function helps with later topics (e.g., maxima/minima, curve sketching) and will be used for sign analysis and level of detail in graph matching problems.

The Power Rule (and a brief justification)

• Power rule (for monomial f(x) = x^n):
  • \frac{d}{dx} x^n = n x^{n-1}
• The rule holds for positive integers (proof sketch provided via expansion of f(x+h) and collecting the terms with h):
  • Expand f(x+h) = (x+h)^n = \sum_{k=0}^n {n\choose k} x^{n-k} h^k
  • The quotient (f(x+h) - f(x))/h equals a sum with a term n x^{n-1} + terms involving h; as h → 0, all terms with h vanish, leaving n x^{n-1}.
• The rule is often accepted for all real n in calculus, though a rigorous justification for non-integer exponents and logs exists but is beyond this quick sketch.
• Practical note: you will typically be asked to compute a derivative using the limit definition on exams, but the power rule is a standard tool for simpler problems.
• A quick application (re-stated): for f(x) = x^2, f′(x) = 2x, consistent with the limit-based calculation above.

Differentiability, Notation, and Higher Derivatives

• Differentiability at a point a means the derivative exists there:
  • f′(a) exists if the limit (\lim_{h\to0} (f(a+h) - f(a))/h) exists.
• Higher derivatives: derivative of the derivative creates f′′(x) = d^2f/dx^2, and so on:
  • f″(x) = \frac{d}{dx} (f′(x)),
  • f^{(n)}(x) for higher orders, using appropriate notation (e.g., Df, d f/dx, d^n f/dx^n).
• Second derivative sign and curvature: f″(x) gives information about concavity and inflection points (not deeply covered in this session, but introduced for context).
• Domains and derivatives: the domain of a function is the set of x-values for which the function is defined; the domain of f′ can be smaller than the domain of f (e.g., sqrt(6 - x) has domain x < 6, but f′(x) exists only for x < 6 as well in that example).

Worked Example: Derivative of f(x) = x^2 via Limit Definition

• Setup: f(x) = x^2; compute f′ via the limit definition:
  • f(x+h) = (x+h)^2 = x^2 + 2xh + h^2
  • (f(x+h) - f(x))/h = (2xh + h^2)/h = 2x + h
  • Let h → 0: f′(x) = 2x
• Conceptual takeaway: the derivative at any x-value is obtained by doubling that x-value; the slope of the tangent line at x is 2x.

Visual and Conceptual Connection: Derivative Sign and Graphs

• Sign of derivative and monotonicity (illustrated with x^2):
  • For x > 0, f′(x) > 0 ⇒ f is increasing on (0, ∞)
  • For x < 0, f′(x) < 0 ⇒ f is decreasing on (-∞, 0)
  • For x = 0, f′(0) = 0 ⇒ horizontal tangent at the bottom of the parabola
• Interpreting derivative signs on graphs helps with multiple-choice questions that ask to match a graph with its derivative or to identify where the derivative is positive, negative, or zero.
• It’s important to be able to reason about intervals of increase/decrease from derivative signs rather than only evaluating function values.

Special Cases: Non-Differentiability and Visual Checks

• Absolute value example (non-differentiable at 0):
  • f(x) = |x| = \begin{cases} x, & x \ge 0 \ -x, & x < 0 \end{cases}
  • f′(x) = \begin{cases} 1, & x>0 \ -1, & x<0 \end{cases}
  • f′(0) does not exist due to a cusp (left and right derivatives differ).
• Cube root example (vertical tangent):
  • f(x) = \sqrt[3]{x} = x^{1/3}; f′(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \sqrt[3]{x^2}}.
  • At x = 0, f′(x) blows up (vertical tangent); derivative domain excludes x = 0, though f is defined at x = 0.
• Quick checklist for not differentiable points on a graph:
  • Jump/discontinuity → not differentiable
  • Sharp corner/cusp → not differentiable
  • Vertical tangent → derivative does not exist (infinite slope)
• Curve sketching and derivative analysis are connected: horizontal tangents indicate potential local extrema, and derivative information guides sketching and end behavior.

Rate of Change in Physics: Velocity, Speed, Acceleration, and Jerk

• Distance function: s(t) denotes distance as a function of time t.
• Velocity: v(t) = s′(t) = \frac{ds}{dt}; the rate of change of distance with respect to time.
• Acceleration: a(t) = v′(t) = \frac{d^2 s}{dt^2} = \frac{dv}{dt}; the rate of change of velocity.
• Jerk: j(t) = a′(t) = \frac{d^3 s}{dt^3}; the rate of change of acceleration.
• Speed vs velocity:
  • Velocity includes direction; speed is the magnitude of velocity: speed = |v(t)| when motion is along a line.
  • This distinction matters in problems about speeding up or slowing down:
  • Speeding up when velocity and acceleration have the same sign (same direction of motion).
  • Slowing down when velocity and acceleration have opposite signs (opposite directions of motion).
• These physics connections illustrate how derivatives describe motion and how higher-order derivatives describe changes in those changes (acceleration, jerk).

Practice and Exam Preparation Implications

• Expect some exam problems to require the limit definition of the derivative; not all problems will be limit-based, but a representative problem type will appear.
• Expect problems involving polynomials, quotients, or radicals where the derivative is computed via the limit definition or the power rule.
• After the midterm, emphasis shifts toward practical use of derivatives for curve sketching and identifying maxima/minima via horizontal tangents.
• You will see both theoretical questions (e.g., differentiability- continuity relationships) and practical graph-based questions (matching a function to its derivative or vice versa).
• Domain considerations are important: the domain of f and the domain of f′ may differ; watch for division by zero or radicals that restrict the domain.

Quick Reference: Key Formulas and Notations (to memorize)

• Derivative definitions:
  • At a fixed point a: f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}
  • As a function: f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
• Power rule (monomial): \frac{d}{dx} x^n = n x^{n-1}
• Common notations for the derivative:
  • f'(x), \frac{dy}{dx}, \frac{df}{dx}, D f, etc.
• Example derivatives:
  • For f(x) = x^2: f'(x) = 2x
  • For f(x) = \sqrt{6 - x}: f'(x) = -\frac{1}{2\sqrt{6 - x}}\quad\text{with domain } x < 6
  • For f(x) = |x|: f'(x)=\begin{cases} 1 & x>0 \ -1 & x<0 \ \text{undefined} & x=0 \end{cases}
  • For f(x) = \sqrt[3]{x}: f'(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3\sqrt[3]{x^2}}
• Distance/velocity/acceleration notation:
  • v(t) = s'(t) = \frac{ds}{dt}
  • a(t) = v'(t) = \frac{d^2 s}{dt^2}
  • j(t) = a'(t) = \frac{d^3 s}{dt^3}
• Speed vs velocity:
  • speed = |v(t)|
• Conceptual relationships:
  • Differentiable at a point ⇒ continuous at that point
  • Non-differentiability can arise from discontinuity, a cusp/corner, or a vertical tangent
  • Horizontal tangents often signal local extrema or flat spots on the graph

End-of-Session Reminders

• Review upcoming quizzes and the midterm schedule; prepare for a mix of limit-based and rule-based derivative problems.
• Bring questions to class for the trig and other review segments.
• Keep in mind the difference between derivative concepts and their geometric interpretations to strengthen curve-sketching skills.