Home / Courses / Calculus I (Differential & intro Integral) / Unit 5

Calculus I (Differential & intro Integral) Unit 5: Analytical applications of derivatives

The first and second derivatives tell you the shape of a graph: where it rises and falls, where it bends, and where its highest and lowest points are.

Covers: Mean Value Theorem, EVT, increasing/decreasing, concavity, curve sketching, optimization

Study guide

Derivatives reveal the shape of a graph. Critical points are x-values in the domain where f′(x) = 0 or f′ is undefined; they're the only places a local max or min can occur. f is increasing where f′ > 0 and decreasing where f′ < 0. The first derivative test classifies a critical point: f′ changing from positive to negative gives a local max, and negative to positive gives a local min.

The second derivative describes concavity: f″ > 0 means concave up, f″ 0 there's a local min, and if f″(c) < 0 a local max.

The Mean Value Theorem says that if f is continuous on [a, b] and differentiable on (a, b), then at some point c the instantaneous rate equals the average rate: f′(c) = [f(b) − f(a)]/(b − a). The Extreme Value Theorem guarantees a continuous function on a closed interval has an absolute max and min. To find them, evaluate f at every critical point and both endpoints, then compare (the closed interval method).

Curve sketching combines all of this: intercepts, asymptotes, intervals of increase and decrease, extrema, concavity and inflection points. Optimization applies it: write a formula for the quantity to maximize or minimize, use a constraint to reduce it to one variable, find critical points, and justify that you found the max or min.

From a graph of f′: f increases where the graph of f′ is above the axis and decreases where it is below. f has extrema where f′ crosses the axis, f is concave up where f′ is increasing, and f has inflection points where f′ has a local max or min. To justify a local max or min, cite a sign change of f′ (or use the second derivative test). Saying f′(c) = 0 is not enough, because a critical point where f′ doesn't change sign is neither a max nor a min.

Before using the MVT, state that f is continuous on [a, b] and differentiable on (a, b). Rolle's theorem is the special case f(a) = f(b), which guarantees some c in (a, b) with f′(c) = 0. If only f′ and one value of f are known, find the values needed for the candidates test with f(x) = f(a) + ∫ₐˣ f′(t) dt, then compare the values at the endpoints and critical points.

Key ideas

• Critical points are where f′ = 0 or f′ doesn't exist.
• First derivative test: f′ changes + to − → local max; − to + → local min.
• Second derivative test: f′(c) = 0 and f″(c) 0 → local min.
• Concave up where f″ > 0, concave down where f″ < 0. An inflection point is where concavity changes.
• Absolute extrema on a closed interval: check critical points and both endpoints (Candidates Test).
• Optimization: write the quantity to optimize as a function of one variable, find critical points, justify max or min.

Formulas and rules

Common mistakes

• Calling a point an inflection point just because f″ = 0. The sign has to change.
• Forgetting to check endpoints for absolute extrema.

Words to know

Critical point

An x in the domain where f′(x) = 0 or f′ is undefined.

First derivative test (local max)

f′ changes from positive to negative.

Concave up

f″ > 0: the graph bends upward and f′ is increasing.

Inflection point

A point where concavity changes (f″ changes sign).

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = [f(b) − f(a)]/(b − a).

Closed interval method

Compare f at critical points and both endpoints to find absolute extrema.

Extreme Value Theorem

A continuous function on [a, b] has an absolute max and min.

Increasing / decreasing test

f′ > 0 → increasing; f′ < 0 → decreasing.

Absolute (global) extremum

The largest or smallest value on the whole domain or interval.

Local (relative) extremum

A max or min compared to nearby points.

Second derivative test

f′(c) = 0 and f″(c) > 0 → local min; f″(c) < 0 → local max.

Optimization

Write the quantity, use a constraint to get one variable, find critical points, justify.

Rolle's theorem

If f(a) = f(b), then f′(c) = 0 for some c in (a, b).

Concave down

f″ < 0: the graph bends downward.

Curve sketching

Use intercepts, asymptotes, f′ (increase/decrease, extrema) and f″ (concavity, inflection points).

Sample practice questions

35 more questions like these, from level 1 to 5, are in the practice set.