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10 multiple-choice questions and 16 flashcards on Differential Equations, about 7% of the High School Calculus bank. Every one carries a written rationale.
Differential Equations is one of 8 chapters in CoStudy's High School Calculus bank, and it holds 10 of the bank's 150 multiple-choice questions — roughly 7% 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.
∫₀² x dx =
Answer: B — 2
B) Antiderivative is x²/2; evaluate: 2²/2 − 0 = 2. A) Used the antiderivative of a constant. C) Forgot to divide by 2. D) Used x²/2 incorrectly as x³/3 type evaluation.
By the Fundamental Theorem of Calculus, d/dx [∫₀^x sin(t²) dt] =
Answer: A — sin(x²)
A) FTC Part 1: d/dx [∫_a^x f(t) dt] = f(x). Substitute upper limit x into the integrand → sin(x²). B) Differentiated the integrand. C) Applied chain rule unnecessarily; upper limit is x, not g(x). D) Random.
Position s(t), velocity v(t) = s'(t), acceleration a(t) = ?
Answer: A — v'(t) = s''(t) (second derivative of position)
Kinematic chain: position s(t), velocity v = s', acceleration a = v' = s''. Each derivative measures instantaneous rate of change of previous. Foundation of physics calculus.
Disk method: volume of solid of revolution of y = f(x) about x-axis from a to b:
Answer: A — π∫(f(x))²dx from a to b (each cross-section is a disk with radius f(x))
Disk method: V = π∫[f(x)]² dx. Each cross-section is circle of radius f(x), area π·r². Variation: washer (with inner radius), shells. Standard application of integration.
Area under curve y = x from 0 to 4:
Answer: D — 8 (triangle area: (1/2)·4·4 = 8, also ∫₀⁴ x dx = x²/2 |₀⁴ = 8)
Geometric: triangle with base 4 and height 4 has area 8. Or integral: ∫₀⁴ x dx = x²/2 from 0 to 4 = 8. Geometric meaning of integral confirms.
1 cards from the 16 in this chapter.
What does ∫_a^b f(x) dx represent geometrically?
The signed area between the curve and the x-axis from a to b.
These are a sample. The full Differential Equations chapter runs 26 items with per-chapter progress tracking, on the web and in the iOS app.
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