Bertopia

Calculus Workout

Limits, derivatives, integrals and series, worked out

Calculus Workout is a Bertopia flashcard pack of 61 cards: limits, derivatives, integrals and series, worked out. It covers limits and continuity; derivatives; applications of derivatives; integrals; sequences and series; differential equations.

Sample cards

Evaluate lim⁡x→3(x2−2x)\lim_{x \to 3} (x^2 - 2x).
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Answer

3
Polynomials are continuous, so substitute: 9 − 6 = 3.
If f(x) = x³ − 4x, what is f′(2)?
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Answer

8
f′(x) = 3x² − 4, so f′(2) = 12 − 4 = 8.
What are the critical points of f(x) = x³ − 3x?
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Answer

x = −1 and x = 1
f′(x) = 3x² − 3 = 0 gives x = ±1. ±√3 are where f itself is 0.
Evaluate ∫023x2 dx\int_0^2 3x^2\,dx.
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Answer

8
Antiderivative x³; 2³ − 0 = 8.
Evaluate ∑n=0∞(23)n\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n.
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Answer

3
Geometric with a = 1, r = 2/3: 1/(1 − 2/3) = 3.
Solve dy/dt = 0.5y with y(0) = 4.
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Answer

y = 4e0.5t
Growth proportional to size is exponential: y = y₀ekt with k = 0.5 and y₀ = 4.

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