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9 multiple-choice questions and 15 flashcards on Contextual Applications of Differentiation, about 6% of the High School Calculus bank. Every one carries a written rationale.
Contextual Applications of Differentiation is one of 8 chapters in CoStudy's High School Calculus bank, and it holds 9 of the bank's 150 multiple-choice questions — roughly 6% 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.
5 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.
Find the maximum of f(x) = -x² + 4x + 1.
Answer: A — Maximum value 5 at x = 2 (f'(x) = -2x + 4 = 0 → x = 2; f(2) = -4 + 8 + 1 = 5; concave down)
Find critical point: f'(x) = -2x + 4 = 0 → x = 2. Second derivative test: f''(x) = -2 < 0, concave down, MAX at x = 2. f(2) = 5. Optimization: classic parabola example.
On [0, 4], f(x) = x² − 4x + 1 attains its minimum value of:
Answer: A — −3
A) f'(x) = 2x − 4 = 0 at x = 2; f(2) = 4 − 8 + 1 = −3. Endpoint values f(0) = 1, f(4) = 1 are both larger. B) Used an endpoint value. C) Off-by-one slip. D) Common arithmetic error.
Find the antiderivative of f(x) = 1/x.
Answer: C — ln|x| + C (key exception to power rule since n = -1)
∫1/x dx = ln|x| + C. Absolute value matters because ln only defined for positive arguments; |x| handles both sides. Power rule fails for n = -1 (would give x⁰/0).
Use substitution to evaluate ∫(2x)(x² + 1)³ dx.
Answer: B — (x² + 1)⁴/4 + C (let u = x²+1, du = 2x dx; then ∫u³ du = u⁴/4)
u-substitution: let u = x²+1, du = 2x dx. Integral becomes ∫u³ du = u⁴/4 + C = (x²+1)⁴/4 + C. Reverses chain rule. Look for inner function whose derivative matches part of integrand.
If f(x) = x³ - 6x² + 9x, find the local maximum value.
Answer: D — 4 (critical points: f'=3x²-12x+9=0 → x=1, 3; f(1)=1-6+9=4 is local max, f(3)=0 is local min)
D) f'(x) = 3x² - 12x + 9 = 3(x-1)(x-3). Critical points: x=1, x=3. f''(x) = 6x - 12. f''(1) = -6 < 0 → local max; f(1) = 1 - 6 + 9 = 4. B) f(3), the local min. C) Sign error. A) f(0) = 0 or boundary confusion.
4 cards from the 15 in this chapter.
What is a related rates problem?
Finding the rate of change of one quantity given the rate of change of another (use chain rule and given relationship).
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 concave down (cap-shaped).
What is the Second Derivative Test?
At critical point c: if f''(c) > 0, local min; if f''(c) < 0, local max; if f''(c) = 0, inconclusive.
These are a sample. The full Contextual Applications of Differentiation chapter runs 24 items with per-chapter progress tracking, on the web and in the iOS app.
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