Key Concepts: Rate of Change, Secants, Tangents, Extrema, and Concavity

Average Rate of Change (Secant)

  • Defined on an x-interval [a, b].

  • It equals the slope of the secant line through (a, f(a)) and (b, f(b)).

  • Formula: m_s = \frac{f(b) - f(a)}{b - a}.

Tangent Line and Instantaneous Rate of Change

  • The tangent line at a point approximates the function in a neighborhood of that point.

  • As the second point on the secant moves toward the fixed point, the secant line becomes indistinguishable from the tangent line.

  • The slope of the tangent line at x = a is the instantaneous rate of change: m_t = f'(a).

  • If you have two points on the tangent line, say (a, f(a)) and (b, f(b)), the line’s slope is m_t = \frac{f(b) - f(a)}{b - a} as b \to a in the limit.

What the Slopes Mean: Increasing/Decreasing

  • Positive slope => function is increasing in that region.

  • Negative slope => function is decreasing in that region.

  • Sign of the tangent slope indicates local behavior of the y-values with respect to x.

Local vs Absolute Extrema

  • Local maximum: a point where the y-value is greater than nearby y-values (a peak).

  • Local minimum: a point where the y-value is smaller than nearby y-values (a valley).

  • Absolute (global) maximum/minimum: the largest/smallest y-value on the entire domain.

  • Horizontal tangent (slope 0) often occurs at local extrema.

  • Domain matters: on all real numbers, a cubic may have no absolute max/min; on a closed, bounded domain, a continuous function attains both absolute max and min.

Peaks, Valleys, and Sketching with Domain

  • If a function goes from increasing to decreasing, you get a local maximum (peak).

  • If it goes from decreasing to increasing, you get a local minimum (valley).

  • Endpoints and domain bounds influence absolute extrema on a given interval.

Concavity and Inflection Points

  • Concave up: looks like a smile; tangent lines tend to get steeper as x increases; second derivative > 0.

  • Concave down: looks like a frown; tangent lines tend to get less steep; second derivative < 0.

  • Inflection point: a point where concavity changes.

  • Exponential growth/decay are concave up; square root graph is concave down. Calculus helps identify inflection points precisely.

Quick Takeaways for Sketches

  • Domain controls possible extrema; bounded domains allow absolute extrema.

  • Increasing/decreasing intervals identify local extrema and overall shape.

  • Tangent lines give instantaneous rate of change; secants give average rate of change.

  • Use concavity to understand the curve’s bending and to locate inflection points.