Chapter 2.1 Notes: The Derivative and The Tangent Line Problem

The derivative (f'(x) ) is a fundamental concept in calculus that represents the instantaneous rate of change of a function, equivalent to the slope of the tangent line to the function's graph at a specific point. Mathematically, it is defined as the limit of the average rate of change (or the slope of secant lines) as the interval between two points on the function approaches zero. This limit is expressed as: f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} . When the derivative exists, it implies the function is differentiable at that point, leading to specific properties regarding continuity and the nature of tangent lines.

Secant lines and the derivative

  • Secant lines are lines passing through two points, (a, f(a)) and (b, f(b)), on a graph. They provide the average rate of change between these points.

    • The slope of a secant line (m_{\text{sec}}) represents the average rate of change between these two points: m_{\text{sec}} = \frac{f(b) - f(a)}{b - a}

    • As the two points get infinitesimally close to each other (i.e., when b approaches a), the secant line approaches the tangent line at that single point.

  • Alternatively, with a fixed point x and a small increment h, the secant slope (also called the difference quotient) is:
    m_{\text{sec}}(x,h) = \frac{f(x + h) - f(x)}{h}

  • The fundamental idea is that as h \to 0, the secant slope tends towards the slope of the tangent line at x.

Longer explanation

  • Secant line through two points on the graph: through (a, f(a)) and (b, f(b)).

  • Slope of the secant line (average rate of change):
    m_{\text{sec}} = \frac{f(b) - f(a)}{b - a}

  • As the two points get closer, the secant slope approaches the slope of the tangent line at the point where the two points converge.

  • The slope of the tangent line is the slope of the graph at x = c when the limit exists.

  • The slope of the secant line with a fixed x and a small increment h is:
    m_{\text{sec}}(x,h) = \frac{f(x + h) - f(x)}{h}

  • This quantity is also called the difference quotient.

  • Fundamental idea: as (h \to 0), the secant slope tends to the tangent slope at (x).

Definition of the derivative

  • If f is defined on an open interval containing c and the limit exists, the derivative at c is:
    f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}

  • The derivative is the instantaneous rate of change (the instantaneous slope of the graph).

  • Notation:

    • (f'(x))

    • (\dfrac{dy}{dx})

    • (\dfrac{d}{dx}f(x))

  • If the limit exists for every x in an interval, the function is differentiable on that interval.

When a function cannot be differentiated

  • If f is not continuous at c, then f is not differentiable at c.

    • Jumps, holes, etc. prevent differentiability.

  • Graphs with sharp turns (cusps) or corners are not differentiable at the turning point.

  • The theorem: differentiable implies continuous.

  • Continuity does not imply differentiability (example discussed in transcript: graphs can be continuous yet not differentiable at certain points).

  • Vertical tangent lines

    • A function may have a vertical tangent where the slope tends to \pm\infty so the derivative does not exist in the ordinary sense.

    • In such a case, the tangent line is vertical: (x = c).

    • Alternate form hint from transcript: the tangent can be described as ((c, f(c))) with undefined slope when the limit diverges.

    • Example noted in transcript: the function (f(x) = x^{1/3}) can have a vertical tangent at (x = 0). This is continuous there but not differentiable there due to infinite slope.

The tangent line and slope

  • When the derivative exists at c, the tangent line to the graph at ((c, f(c))) has slope (m = f'(c)).

  • Equation of the tangent line at ((c, f(c))):
    y - f(c) = f'(c)\,(x - c)
    or equivalently
    y = f(c) + f'(c)\,(x - c).

  • Horizontal tangent: if (f'(c) = 0), the tangent line is horizontal: y = f(c).

  • Vertical tangent: if the limit defining the derivative does not exist due to infinite slope, the tangent line is vertical: x = c.

Useful examples and worked steps (EX 1–EX 4 from transcript)

  • EX 1: Find the slope of the tangent to (f(x) = 2x - 3) (a line)

    • Since the function itself is a line, the slope of the tangent line is constant:
      f'(x) = 2

    • Using the difference quotient:
      f(x+h) - f(x) = (2(x+h) - 3) - (2x - 3) = 2h,
      m = \lim_{h \to 0} \frac{2h}{h} = 2.

    • Tangent line at any x is the line itself: (y = 2x - 3).

  • EX 2: Find the slope of the tangent line to (f(x) = x^2 + 1) at ((-1, 2))

    • Derivative: (f'(x) = 2x); hence (f'(-1) = -2).

    • Tangent line: use point-slope form
      y - 2 = -2\,(x + 1) \Rightarrow y = -2x.

  • EX 3: Find equation of the tangent line to a function at ((1,2)) (from transcript context, horizontal tangent at that point)

    • If (f(1) = 2) and (f'(1) = 0), then the tangent line is horizontal:
      y = 2.

  • EX 4: Tangent lines to (f(x) = x^3) that are parallel to the line (3x - y + 1 = 0)

    • The line (3x - y + 1 = 0) has slope (m = 3) (since (y = 3x + 1)).

    • Derivative: (f'(x) = 3x^2). Set to the target slope:
      3x^2 = 3 \Rightarrow x^2 = 1 \Rightarrow x = \pm 1.

    • Evaluate f at these x-values: (f(1) = 1), (f(-1) = -1).

    • Tangent lines:

    • At (x = 1): y - 1 = 3\,(x - 1) \Rightarrow y = 3x - 2.

    • At (x = -1): y + 1 = 3\,(x + 1) \Rightarrow y = 3x + 2.

    • Both lines are parallel to the given line (slope 3).

Summary of key ideas and relationships

  • The derivative at a point is the limit of the average rate of change as the interval shrinks:
    f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.

  • The derivative, when it exists, equals the slope of the tangent line to the graph at that point.

  • The tangent line at (x = c) has equation:
    y - f(c) = f'(c)\,(x - c).

  • Secant lines approximate the tangent line; their slope approaches the tangent slope as the two points coalesce.

  • Differentiability implies continuity, but continuity does not guarantee differentiability.

  • Horizontal tangents occur where (f'(c) = 0); vertical tangents occur where the derivative does not exist due to infinite slope, and the tangent line is effectively (x = c).

Quick reference: common forms to remember

  • Secant slope (two points):
    m_{\text{sec}} = \frac{f(b) - f(a)}{b - a}.

  • Secant slope with x and h:
    m_{\text{sec}}(x,h) = \frac{f(x+h) - f(x)}{h}.

  • Derivative (instantaneous rate of change):
    f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.

  • Tangent line at ((c, f(c))):
    y - f(c) = f'(c)\,(x - c).

  • Horizontal tangent if (f'(c) = 0): (y = f(c)).

  • Vertical tangent if the limit does not exist (slope infinite): tangent line is (x = c).

Real-world relevance and connections

  • The derivative concept underpins physics (instantaneous velocity), economics (instantaneous rate of change), biology (growth rates), and engineering (slope of curves and optimization).

  • Understanding when a derivative exists and when it does not is crucial for practical applications and for interpreting the behavior of functions.