AP Calculus BC Unit 5 practice questions: Analytical Applications of Differentiation
8 original multiple-choice questions on AP Calc BC Unit 5, analytical applications of differentiation. Answer each one in your browser and read the explanation, then review the key ideas below.
Question 1 of 8
Where does f(x) = x³ − 3x have a relative maximum?
Question 2 of 8
For f(x) = x² on [1, 3], what value of c satisfies the Mean Value Theorem?
Question 3 of 8
f″(x) = 6x − 12. At what x-value does the graph of f have a point of inflection?
Question 4 of 8
f′(x) = (x − 1)(x + 2). On which intervals is f increasing?
Question 5 of 8
What is the absolute maximum of f(x) = x² − 4x on [0, 5]?
Question 6 of 8
Where is f(x) = x³ concave up?
Question 7 of 8
The Extreme Value Theorem guarantees an absolute maximum and minimum when f is
Question 8 of 8
Two positive numbers add up to 20. What is the largest possible value of their product?
Key ideas in Unit 5
- The Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) equal to the average rate of change.
- Critical points are where f′ = 0 or f′ is undefined. Use the first or second derivative test to classify them.
- The Extreme Value Theorem: a continuous function on a closed interval has an absolute max and min, found among critical points and endpoints (the candidates test).
- f is concave up where f″ > 0; points of inflection are where f″ changes sign.
- Optimization: write the quantity as a function of one variable, find its critical points, and check the endpoints.
More AP Calc BC practice
← Unit 4: Contextual Applications of Differentiation
Unit 6: Integration and Accumulation of Change →
The full AP Calculus BC practice test, all units
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