Analytical Applications of Differentiation: Curve Sketching & Optimization

Function (f)

The original function; its graph shows the output/height (position) at each x-value.

First derivative (f′)

Measures the rate of change of f; equals the slope of the tangent line to f at each x.

Second derivative (f″)

Measures how the rate of change is changing; describes the concavity of f and whether f′ is increasing or decreasing.

Monotonicity

Whether a function is increasing or decreasing on an interval, determined by the sign of its derivative.

Increasing (via f′)

f is increasing on intervals where f′(x) > 0.

Decreasing (via f′)

f is decreasing on intervals where f′(x) < 0.

Horizontal tangent

A point where the tangent line to f is flat; occurs when f′(x) = 0 (if the derivative exists).

Critical point

A point c in the domain of f where f′(c)=0 or f′(c) does not exist; local extrema can only occur at critical points.

Local maximum (First Derivative Test)

Occurs at x=c when f′ changes sign from positive to negative (increasing to decreasing).

Local minimum (First Derivative Test)

Occurs at x=c when f′ changes sign from negative to positive (decreasing to increasing).

No local extremum at a critical point

If f′(c)=0 (or undefined) but f′ does not change sign across c, then f has no local maximum/minimum there.

Concave up

f is concave up where f″(x) > 0; slopes of f are increasing (tangents get steeper).

Concave down

f is concave down where f″(x) < 0; slopes of f are decreasing (tangents get less steep).

Inflection point

A point where concavity changes (up to down or down to up); requires f″ to change sign (f″=0 alone is not sufficient).

Relationship: f″ and f′

Because f″ is the derivative of f′: if f″>0 then f′ is increasing; if f″<0 then f′ is decreasing.

Concavity from a graph of f′

f is concave up where f′ is increasing and concave down where f′ is decreasing.

Common confusion: value vs slope

f(x)>0 means f is above the x-axis; f′(x)>0 means f is increasing—these are different ideas.

Zeros of f′ (what they mean)

x-values where f′=0 correspond to horizontal tangents/critical points of f, not necessarily x-intercepts (zeros) of f.

Optimization problem

A problem that asks for a maximum or minimum value of a quantity (area, cost, distance, volume, etc.) subject to constraints.

Objective function

The quantity Q you want to maximize or minimize; you rewrite it (using constraints) as a function of one variable before differentiating.

Constraint

An equation/inequality that limits the variables in an optimization problem; used to eliminate variables via substitution.

Closed Interval Method

To find absolute extrema of a continuous function on [a,b], compare values at endpoints and at interior critical points (where Q′=0 or undefined).

First Derivative Test (optimization)

Uses sign changes of Q′ around a critical point to determine whether Q has a local maximum ( +→− ) or local minimum ( −→+ ).

Second Derivative Test

If Q′(c)=0 and Q″(c)>0 then Q has a local minimum at c; if Q″(c)<0 then Q has a local maximum at c (not applicable if Q′(c)≠0).

Minimizing squared distance

To find a closest point, minimize D² instead of D because the square root is increasing; minimizing D and D² gives the same x-location for the minimum.