Notes on Derivatives, Tangents, and Limits

Limit, Secant Lines, and Real-Life Problems

  • Calculus relies on the notion of a limit; to do calculus, we need a solid understanding of limits.
  • Real-life/problem context mentioned:
    • Minimizing time to get from one point to another (minimum time problems).
    • Maximizing volume of a box you can make from a piece of cardboard (maximum volume problems).
    • Area problems (anticipated later, after the second test, into November) when we start talking about integrals.
  • Key takeaway: limits underpin many concepts in calculus, including slopes, tangent lines, and derivatives.
  • Intro example setup: consider the function with a graph such as the parabola y = f(x) = x^2; discuss slope of secants and the idea of the tangent slope as a limit of secant slopes.

Secant Line, Slope, and Difference Quotient

  • Slope between two points is rise over run:
    • m = rac{y2 - y1}{x2 - x1}
    • or equivalently m = rac{ riangle y}{ riangle x} where
    • riangle y = y2 - y1 = f(x2) - f(x1),
    • riangle x = x2 - x1.
  • For a general function f, the slope between two points (x, f(x)) and (x+Δx, f(x+Δx)) is:
    • m_{ ext{sec}} = rac{f(x+Δx) - f(x)}{Δx}
  • Example using f(x) = x^2:
    • At x = 1 and x+Δx = 1.2, f(1) = 1, f(1.2) = 1.44, so
    • Δy = 1.44 - 1 = 0.44, Δx = 0.2, so m = 0.44 / 0.2 = 2.2.
    • At x = 1 and x+Δx = 1.1, f(1.1) = 1.21, so Δy = 0.21, Δx = 0.1, so m = 0.21 / 0.1 = 2.1.
  • As Δx gets smaller, the secant slope approaches a limiting value, which becomes the slope of the tangent line at the point x.
  • Important caution: the expression Δy is a change in the function value, not a literal "delta y" number; it is computed as f(x+Δx) - f(x).
  • For x = 1 on f(x) = x^2, the tangent slope will approach 2 as Δx → 0 (see limit definition below).

Derivative: The Slope of the Tangent Line (Limit Form)

  • Definition of the derivative at a point x is the limit of the slope of the secant line as Δx → 0:
    • f'(x) =
      \,\lim_{\Delta x \to 0} \, \frac{f(x+Δx) - f(x)}{Δx}
  • Key idea: you cannot simply plug Δx = 0 into the difference quotient because you get 0/0; you must simplify first, then take the limit.
  • Notation:
    • f'(x) is read as "f prime of x" and denotes the derivative of f at x (the slope of the tangent to y = f(x) at the point x).
    • Other common notations: dy/dx, d/dx, D_x f(x), and y' (when the context uses y = f(x)).
    • The apostrophe in f'(x) indicates a derivative; these are all equivalent ways to denote the same concept.
  • Example: If f(x) = x^2, compute f'(x) from the definition:
    • f(x+Δx) = (x+Δx)^2 = x^2 + 2xΔx + (Δx)^2
    • f(x+Δx) - f(x) = (x^2 + 2xΔx + (Δx)^2) - x^2 = 2xΔx + (Δx)^2

    • rac{f(x+Δx) - f(x)}{Δx} = rac{2xΔx + (Δx)^2}{Δx} = 2x + Δx .
    • Taking the limit as Δx → 0 gives
      f'(x) = \,\lim_{Δx \to 0} (2x + Δx) = 2x.
  • Resulting derivative function for f(x) = x^2:
    • f'(x) = 2x.
  • Consequences: the derivative at a specific x gives the slope of the tangent line there; e.g.,
    • At x = 1: f'(1) = 2.
    • At x = 3: f'(3) = 6.
    • At x = -1: f'(-1) = -2.
  • Relationship between derivative and tangent line:
    • The derivative is a function that assigns to each x the slope of the tangent line to y = f(x).
    • The tangent line at x = x0 has equation
    • y - f(x0) = f'(x0)\, (x - x_0).
    • For f(x) = x^2 at x0 = 1: slope m = f'(1) = 2, tangent line is
    • y - 1 = 2\, (x - 1) \,\Rightarrow \, y = 2x - 1.
  • The tangent line concept can be contrasted with the secant line, which uses two distinct points on the curve.

Derivative of Simple Functions (Intuition and Checks)

  • Constant function: f(x) = c
    • Slope/derivative is zero: f'(x) = 0 for all x.
    • Reason: f(x+Δx) - f(x) = c - c = 0, so the quotient is 0 for all Δx ≠ 0, and the limit is 0.
  • Linear function: f(x) = m x + b
    • Derivative is the constant slope: f'(x) = m for all x.
    • Example: f(x) = 3x + 7 → f'(x) = 3.
  • The derivative concept is not limited to easy cases; for many other functions, you use the limit (and later learn shortcut rules) to compute f'(x).

Tangent Line Formulations and Notation Details

  • Equations of tangent lines given a point and slope can be written in two common forms:
    • Point-slope form: y - y1 = m \,(x - x1) where (x1, y1) lies on the tangent line (indeed on the curve at that x).
    • If you know the slope and want a y-intercept form, you can rearrange to slope-intercept form: y = m x + b (with b determined from the point).
  • In the presented example, starting from the derivative, one builds the tangent line using the point on the curve and the slope given by the derivative at that point.

The Difference Quotient and the Secant Line Slopes

  • Before taking the limit, the slope between two points is the slope of a secant line:
    • m_{ ext{sec}} = \frac{f(x+Δx) - f(x)}{Δx}
  • The limit of this slope as Δx → 0 is the derivative f'(x):
    • This is how the tangent slope is defined.
  • Important nuance: as you reduce Δx, the secant line better approximates the tangent line at x; the tangent line is the limiting position of the secant line as the two points collide.

Differentiability and Special Cases

  • Differentiable at a point means the derivative exists there (the limit exists).
  • Points where the slope differs from either side (sharp corners) are not differentiable at that point.
  • A vertical tangent may occur when the slope tends to ±∞; in such cases the limit does not exist in the finite sense, so f'(x) is undefined there.
  • The formal definition of the derivative explicitly includes the caveat: the limit must exist for f'(x) to be defined. This is sometimes emphasized as "provided that the limit exists."
  • Practical implications for exams/homework: some problems use the limit definition directly, while others will leverage shortcut rules learned later.

Practical Techniques and Notation Summary

  • Notation recap:
    • f'(x): derivative of f with respect to x.
    • dy/dx: derivative of y with respect to x (if y = f(x)).
    • D_x f(x): another common operator notation for taking the derivative with respect to x.
    • f'(x) is itself a function of x, often denoted as the derivative function.
  • Quick mental rules (to be covered next) include shortcut differentiation rules for common families of functions (polynomials, exponentials, etc.).

Quick Reference: Key Formulas to Remember

  • Secant slope (difference quotient):
    • m_{ ext{sec}} = \frac{f(x+Δx) - f(x)}{Δx}
  • Derivative (definition):
    • f'(x) = \lim_{Δx \to 0} \frac{f(x+Δx) - f(x)}{Δx}
  • For the specific case f(x) = x^2:
    • f'(x) = 2x
    • At x = 1: f'(1) = 2
    • At x = 3: f'(3) = 6
    • At x = -1: f'(-1) = -2
  • Tangent line at x0:
    • y - f(x0) = f'(x0) (x - x_0)
  • Examples:
    • If f(x) = 3x + 7, then f'(x) = 3 (constant slope).
    • If f(x) = 7 (constant), then f'(x) = 0.
  • Important caveat:
    • The derivative exists only where the limit exists; some problems may have no derivative at certain points (non-differentiable points).

Forward Look: What’s Next in the Course

  • Friday: introduction to derivative shortcut rules that simplify computing derivatives for common functions.
  • Subsequent work will connect derivatives to integrals and other calculus topics.
  • Revisit and practice differentiability, limits, and tangent-line problems, including more complex functions and their slopes at various x-values.