AP Calculus AB Unit 5 Practice Test: Analytical Applications of Differentiation

Using Derivatives to Analyze Functions

Unit 5 is the culmination of differential calculus. You now use derivatives not just to compute rates of change, but to understand the complete behavior of a function — where it increases or decreases, where it reaches extreme values, how it curves, and how to find the optimal value of a quantity. These are the analytical skills that define AP Calculus AB FRQs focused on graph analysis and optimization.

Topics in This Unit Practice Test

• The Mean Value Theorem (MVT) — statement, conditions, and AP justification language
• The Extreme Value Theorem (EVT) — guaranteeing absolute extrema on closed intervals
• Critical points and their classification
• The first derivative test for local maxima and minima
• The second derivative test for local extrema
• Concavity and points of inflection
• Relationships between f, f′, and f″ graphs
• Curve sketching using calculus-derived information
• Optimization problems — finding absolute maxima and minima in applied contexts

The Mean Value Theorem on the AP Exam

The MVT states that if a function is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) where f′(c) equals the average rate of change over the interval. On the AP exam, MVT questions often ask you to either justify the existence of such a c or to find its value. Justification requires explicitly stating both conditions — continuity and differentiability — before invoking the theorem.

Reading f′ and f″ to Sketch f

A high-frequency AP skill is interpreting the graph of a derivative to draw conclusions about the original function. Where f′ > 0, f is increasing. Where f′ < 0, f is decreasing. A sign change in f′ indicates a local extremum. Where f″ > 0, f is concave up; where f″ < 0, f is concave down. A sign change in f″ indicates a point of inflection. The ability to move fluently between the graphs of f, f′, and f″ is essential for both MCQ and FRQ performance.

Optimization FRQ Strategy

1. Define the quantity to be optimized and write its formula as a function of one variable
2. Use a constraint equation to eliminate extra variables
3. Differentiate and set the derivative equal to zero to find critical points
4. Use the first or second derivative test to confirm whether each critical point is a maximum or minimum
5. Check endpoints if the problem specifies a closed interval (EVT applies)
6. State your answer clearly with appropriate units and context