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10 multiple-choice questions and 32 flashcards on Integration and Accumulation of Change, about 7% of the High School Calculus bank. Every one carries a written rationale.
Integration and Accumulation of Change 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.
Derivative of e^x:
Answer: E — e^x (exponential function is its own derivative — fundamental property)
d/dx[eˣ] = eˣ. Unique self-derivative property. This is why e is special. Also d/dx[ln x] = 1/x. d/dx[a^x] = a^x · ln(a).
Evaluate ∫₁³ (2x + 1) dx using the FTC.
Answer: C — 10 (antiderivative is x² + x; (9+3) - (1+1) = 12 - 2 = 10)
C) Antiderivative of 2x+1 is x² + x. Evaluate at 3: 9+3=12. At 1: 1+1=2. Difference: 12-2 = 10. A) Forgot to subtract the lower bound. B) Stopped at the upper bound value. D) Used wrong antiderivative for the linear term.
Fundamental Theorem of Calculus (Part 2): if F is an antiderivative of f, then ∫ from a to b of f(x) dx = ?
Answer: B — F(b) - F(a) (evaluation theorem)
FTC Part 2: definite integral = antiderivative evaluated at upper minus lower limit. F(b) - F(a). Connects derivatives and integrals. Foundation of integration.
∫ x² dx =
Answer: C — x³/3 + C
C) Reverse the power rule: increase exponent by 1, divide by new exponent → x³/3 + C. A) Differentiated instead of integrated. B) Used the antiderivative formula for x. D) Multiplied instead of dividing by the new exponent.
Mean Value Theorem states: if f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) such that:
Answer: E — f'(c) = (f(b) - f(a))/(b - a) (some point's instantaneous rate equals average rate)
MVT: average rate of change on [a,b] equals instantaneous rate at some c. Geometric: secant line slope equals tangent at some point. Rolle's special case: if f(a)=f(b), then f'(c)=0 somewhere.
1 cards from the 32 in this chapter.
What is an antiderivative?
A function F such that F'(x) = f(x). Sometimes called an indefinite integral, ∫f(x)dx = F(x) + C.
These are a sample. The full Integration and Accumulation of Change chapter runs 42 items with per-chapter progress tracking, on the web and in the iOS app.
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