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Calculus Revision Flashcards
Free A Level Maths Cards
Cover differentiation from first principles, integration techniques, and the practical applications of calculus in A Level Maths with these free revision flashcards.
Question
What is differentiation?
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Answer
Finding the derivative — the rate of change of a function with respect to a variable. Geometrically, it gives the gradient of the tangent to a curve at any point.
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Question
What is the power rule for differentiation?
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Answer
If y = axⁿ, then dy/dx = naxⁿ⁻¹. Multiply by the power and reduce the power by 1.
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Question
What is the derivative of a constant?
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Answer
Zero. Constants have no rate of change.
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Question
What does dy/dx = 0 indicate?
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Answer
A stationary point (turning point) — where the gradient of the curve is zero. Could be a local maximum, local minimum, or point of inflection.
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Question
How do you distinguish between a maximum and minimum turning point?
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Answer
Find d²y/dx² (second derivative). If d²y/dx² < 0 → maximum. If d²y/dx² > 0 → minimum. If d²y/dx² = 0 → may be a point of inflection.
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Question
What is integration?
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Answer
The reverse process of differentiation — finding the area under a curve. Indefinite integration gives a family of functions (+ C); definite integration gives a numerical value.
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Question
What is the power rule for integration?
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Answer
∫axⁿ dx = axⁿ⁺¹/(n+1) + C, for n ≠ −1. Add 1 to the power and divide by the new power.
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Question
What is the fundamental theorem of calculus?
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Answer
Differentiation and integration are inverse processes. A definite integral ∫ₐᵇ f(x)dx = F(b) − F(a), where F is the antiderivative of f.
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Question
What does a definite integral represent geometrically?
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Answer
The signed area between the curve y = f(x) and the x-axis between x = a and x = b. Area below the x-axis is counted as negative.
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Question
What is the chain rule?
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Answer
For composite functions: dy/dx = (dy/du) × (du/dx). Example: if y = (3x+1)⁵, let u = 3x+1, then dy/dx = 5(3x+1)⁴ × 3 = 15(3x+1)⁴.
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