Second derivative
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What is the second derivative?
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5.7.19 cards
What is the second derivative?
The derivative of the first derivative — differentiate f(x) twice.
How is the second derivative written?
f''(x) or d²y/dx².
What does f''(x) > 0 mean for the curve?
It is concave up (∪).
What does f''(x) < 0 mean for the curve?
It is concave down (∩).
State the second-derivative test.
At a stationary point: f'' > 0 → minimum, f'' < 0 → maximum.
What if f''(x) = 0 at a stationary point?
The test is inconclusive; check the sign of f' on each side instead.
How do you find where a curve is concave up?
Solve f''(x) > 0.
If f(x) = x³ − 4x², what is f''(x)?
f'(x) = 3x² − 8x, so f''(x) = 6x − 8.
What does the second derivative measure?
How the gradient (first derivative) is changing — the curve's bending.
Topic 5.7 study notes
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Math AA SL exam skills
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