Limits, continuity & higher derivatives
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Flip to reveal answersWhat does limx→a f(x) = L mean informally?
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All 8 Flashcards — Limits, continuity & higher derivatives
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Question
What does lim_{x→a} f(x) = L mean informally?
Answer
As x gets close to a (from either side), f(x) gets close to L — the value f is heading towards.
Question
What does it mean for a function to be continuous at a point?
Answer
No jump, hole or break there — you can draw through it without lifting your pen. Formally lim_{x→a} f(x) = f(a).
Question
How do you find the second derivative f''(x)?
Answer
Differentiate f(x) to get f'(x), then differentiate f'(x) again.
Question
What does the second derivative tell you?
Answer
The rate of change of the gradient — how the slope itself is changing.
Question
Write the second derivative in Leibniz notation.
Answer
d²y/dx² (read 'd-two-y by d-x-squared'); the same as f''(x).
Question
Does d²y/dx² mean (dy/dx)²?
Answer
No — the 2's are notation for differentiating twice; nothing is being squared.
Question
Find f''(x) if f(x) = x³ − 4x².
Answer
f'(x) = 3x² − 8x, then f''(x) = 6x − 8.
Question
What is d³y/dx³ for y = 2x³ + x²?
Answer
dy/dx = 6x² + 2x, d²y/dx² = 12x + 2, d³y/dx³ = 12.
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Full study notes for Limits, continuity & higher derivatives
Topic 5.12 hub
First principles (HL only)
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