Applications of Derivatives: every key term you need (+ practice quiz)

More AP Calculus AB guides

• Limits and Continuity
• Derivatives
• Integration
• Differential Equations

Extreme Value Theorem

Continuous function on closed interval [a,b] attains max and min; critical points found where f'(x)=0 or undefined.

Rolle's Theorem

If f continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃c in (a,b) where f'(c)=0; derivative equals zero somewhere.

Mean Value Theorem

If f continuous on [a,b], differentiable on (a,b), then ∃c in (a,b) where f'(c)=[f(b)-f(a)]/(b-a); instantaneous rate equals average rate.

First Derivative Test

Sign of f'(x): + to - means local max, - to + means local min; doesn't change sign means neither.

Critical Points

Points where f'(x)=0 or f'(x) undefined; candidates for local extrema; must be in domain of f.

Absolute vs Local Extrema

Absolute: largest/smallest on entire domain; local: largest/smallest in neighborhood; absolute extrema occur at critical points or endpoints.

Monotonicity

f increasing where f'(x)>0, decreasing where f'(x)<0; analyzing sign of derivative determines intervals of increase/decrease.

Second Derivative Test

If f'(c)=0: f''(c)>0 means local min, f''(c)<0 means local max; inconclusive if f''(c)=0.

Inflection Points

Points where concavity changes; f''(x)=0 or undefined (and changes sign); graph changes from concave up to down or vice versa.

Concavity Analysis

f concave up where f''(x)>0, concave down where f''(x)<0; second derivative test for concavity.

Curve Sketching

Find domain, intercepts, asymptotes, critical points, extrema, inflection points, intervals of increase/decrease/concavity; sketch graph.

Optimization Problems

Maximize/minimize quantity subject to constraint; write objective function, use constraint to simplify, find critical points.

Closed Interval Extrema

Evaluate f at critical points and endpoints; absolute max/min occurs at one of these points.

Implicit Differentiation for Extrema

When curve defined implicitly, find dy/dx via implicit differentiation; find critical points where dy/dx=0 or undefined.

Absolute Extrema Location

Global max/min on closed interval: evaluate at critical points and endpoints; on open interval: approach limits at boundaries.

Business Optimization

Revenue, cost, profit functions; marginal cost = derivative of cost; optimize production levels, pricing; real-world applications.

Related Rates Revisited

Optimization in motion: volumes, areas changing with time; differentiate constraint equation, substitute known rates, solve.

Antiderivatives Preview

Function F where F'(x) = f(x); integration reverses differentiation; fundamental connection to integrals.

Slope Fields (Direction Fields)

Visual representation of dy/dx at points (x,y); shows direction of solutions to differential equations.

Optimization with Constraints

Lagrange multipliers concept; use constraint to eliminate variable or solve system of equations from optimization conditions.

Mean Value Theorem Conclusion

If f is continuous on [a,b] and differentiable on (a,b), some c in (a,b) satisfies f'(c) = [f(b)-f(a)]/(b-a): an instantaneous rate equals the average rate.

When MVT Fails

|x| on [-1,1] has no c with f'(c) = 0 because differentiability fails at the interior point x = 0, so the hypotheses are not met.

First Derivative Test

At a critical point, a sign change of f' from positive to negative gives a local maximum; negative to positive gives a local minimum; no change gives neither.

Repeated Root in f'

A factor like (x-1)² in f' touches zero without changing sign, so x = 1 is a critical point that is not an extremum.

Second Derivative Test Limits

If f'(c) = 0 and f''(c) = 0, the test is inconclusive and the first derivative test or higher analysis must be used.

Inflection Points

Occur where f'' changes sign, not merely where f'' = 0; concavity must actually reverse for the point to qualify.

Candidates Test

On a closed interval, evaluate f at every critical point and at both endpoints; the largest and smallest values are the absolute extrema.

Optimization Workflow

Write the objective, use a constraint to reduce to one variable, state the domain, differentiate, and justify the extremum before answering in context.

Open-Top Box Result

Minimizing x² + 4xh subject to x²h = 32 gives surface area x² + 128/x, minimized at x = 4 with height 2.

Related Rates Setup

Write an equation relating the quantities first, differentiate with respect to time, and only then substitute the instantaneous values.

Ladder Problem

For x² + y² = L², differentiating gives x(dx/dt) + y(dy/dt) = 0, so the top slides down at rate -(x/y)(dx/dt).

Shadow and Similar Triangles

Proportions from similar triangles relate a walker's distance to the shadow-tip distance; differentiating that proportion gives the tip's speed.

Linear Approximation

L(x) = f(a) + f'(a)(x-a) estimates nearby values; concave-down curves make the estimate an overestimate, concave-up an underestimate.

L'Hôpital on 0/0

lim(x→0)(e^x - 1 - x)/x² = 1/2 after applying the rule twice; check the indeterminate form again before each application.

Speeding Up vs Slowing Down

An object speeds up when velocity and acceleration share a sign and slows down when their signs differ, regardless of direction of travel.

Reading a Graph of f'

Where the graph of f' is above the axis f increases; where f' increases f is concave up; x-intercepts of f' with sign change locate extrema of f.

Monotonicity Proof

To show a function is increasing on an interval, demonstrate f'(x) > 0 throughout it, not merely at sample points.

Endpoint Extrema

Absolute extrema can occur at endpoints even where f' is nonzero, which is why endpoints must be tested separately.

Global vs Local

A local extremum compares nearby values only; a global extremum compares across the whole domain and may not exist on an open interval.

Rate of Change of a Rate

When a problem asks how fast a rate itself is changing, the answer involves the second derivative, so differentiate the rate expression once more.