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9 multiple-choice questions and 22 flashcards on Analytical Applications of Differentiation, about 6% of the High School Calculus bank. Every one carries a written rationale.
Analytical 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 derivative: d/dx[x⁵]
Answer: A — 5x⁴ (power rule: d/dx[xⁿ] = n·x^(n-1))
Power rule: d/dx[xⁿ] = n·x^(n-1). Bring exponent down, decrement exponent. x⁵ → 5x⁴. Most-used differentiation rule. Verify: tangent slope of x⁵ at x=1 should be 5(1)=5.
Chain rule: d/dx[f(g(x))] = ?
Answer: E — f'(g(x))·g'(x) (chain rule)
Chain rule: outer × inner. d/dx[f(g(x))] = f'(g(x))·g'(x). Example: d/dx[sin(x²)] = cos(x²)·2x. Most important rule for composite functions.
A spherical balloon has radius growing at 3 cm/s. At r = 4, dV/dt = ? (V = (4/3)πr³)
Answer: C — 192π cm³/s (dV/dt = 4πr²·dr/dt = 4π(16)(3))
C) Differentiate V = (4/3)πr³ → dV/dt = 4πr²·dr/dt. At r=4, dr/dt=3: 4π·16·3 = 192π. A) Used r·dr/dt only — forgets squaring. B) Forgot the dr/dt factor. D) Used r=4 but only 4π·4·3 — dropped r².
Product rule: d/dx[u(x)·v(x)] = ?
Answer: B — u'·v + u·v' (product rule)
Product rule: derivative of u·v is u'v + uv'. Order doesn't matter due to commutativity. Common error: just multiplying u'·v'. Practice with x·sin(x), x²·eˣ, etc.
Derivative of f(x) = 3x² - 4x + 7:
Answer: A — 6x - 4 (sum/constant rule: 3·2x - 4·1 + 0 = 6x - 4)
Apply power rule + sum/constant rules: d/dx[3x²] = 6x; d/dx[-4x] = -4; d/dx[7] = 0. Sum: 6x - 4. Derivative of polynomial reduces degree by 1.
4 cards from the 22 in this chapter.
A rectangle has perimeter 40. Maximize its area.
P = 2L + 2W = 40 → L + W = 20. A = LW = L(20 − L). A' = 20 − 2L = 0 → L = 10. Max area = 100 (a square).
What is the derivative of ln(x²)?
2/x (using chain rule on ln(u): u'/u).
If f' > 0 and f'' < 0, what does the graph look like?
Increasing and concave down (think of leveling off).
A balloon's radius grows at 2 cm/s. How fast is the volume changing when r = 5? (V = (4/3)πr³)
dV/dt = 4πr² · dr/dt = 4π(25)(2) = 200π cm³/s.
These are a sample. The full Analytical Applications of Differentiation chapter runs 31 items with per-chapter progress tracking, on the web and in the iOS app.
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