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Analytical Applications of Differentiation — High School Calculus practice questions

24 multiple-choice questions and 20 flashcards on Analytical Applications of Differentiation, about 16% of the High School Calculus bank. Every one carries a written rationale.

Written and maintained by the CoStudy team · last updated 2026-10-02 · how we write and review questions

What this chapter covers

Analytical Applications of Differentiation is one of 8 chapters in CoStudy's High School Calculus bank, and it holds 24 of the bank's 150 multiple-choice questions — roughly 16% of the total. That proportion is not arbitrary: chapters follow the certifying body's published exam outline, and the number of questions in each is set by that domain's published weight, so the share of your practice time this chapter takes matches the share of the real exam it accounts for.

Studying by chapter is worth doing once you have a diagnostic score. A single overall percentage tells you whether you are close; it does not tell you which domain is dragging. Working a weak chapter in isolation, and re-testing it in isolation, is the fastest way to move a score that has stalled — and it is why the mock exams in CoStudy report by domain rather than as one number.

Free Analytical Applications of Differentiation practice questions

3 questions drawn from this chapter, with the full rationale shown — the controlling principle behind the right answer, and why each wrong option tempts and fails.

f is continuous on [a, b] and differentiable on (a, b). The Mean Value Theorem guarantees some c in (a, b) with:

  1. f(c) = 0
  2. f'(c) = 0
  3. f(c) = [f(a) + f(b)]/2
  4. f'(c) = [f(b) − f(a)]/(b − a)

Answer: D — f'(c) = [f(b) − f(a)]/(b − a)

D) Correct — Some instantaneous rate equals the average rate over the interval. A) Is about zeros (the IVT), not about slopes. B) Is Rolle's Theorem, which also needs f(a) = f(b). C) Averages the function values instead of matching slopes.

Two positive numbers add to 20. Which pair has the largest product?

  1. 1 and 19
  2. 10 and 10
  3. 20 and 0
  4. 5 and 15

Answer: B — 10 and 10

B) Correct — P = x(20 − x) is maximized at x = 10. A) Has product 19. C) Has product 0, and 0 is not positive. D) Has product 75, less than 100.

f'(c) = 0 and f''(c) = 0. What does the second derivative test say?

  1. c is an inflection point
  2. c is a local maximum
  3. Nothing; the test is inconclusive
  4. c is a local minimum

Answer: C — Nothing; the test is inconclusive

C) Correct — For example, x⁴, −x⁴, and x³ all have f' = f'' = 0 at 0 but behave differently. A) Not always: x⁴ has f''(0) = 0 but a minimum at 0. B) Not always: x⁴ has a minimum at 0. D) Not always: −x⁴ has a maximum at 0.

Analytical Applications of Differentiation flashcards

4 cards from the 20 in this chapter.

What does f''(x) > 0 tell you?

f is concave up (cup-shaped).

Find local extrema of f(x) = x³ − 3x.

f' = 3x² − 3 = 0 → x = ±1. f''(1) = 6 > 0 (min, value −2). f''(−1) = −6 < 0 (max, value 2).

What does f'(x) > 0 tell you?

f is increasing on that interval.

What is the Mean Value Theorem (MVT)?

If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) such that f'(c) = (f(b) − f(a))/(b − a).

Practise the full chapter

These are a sample. The full Analytical Applications of Differentiation chapter runs 44 items with per-chapter progress tracking, on the web and in the iOS app.

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