A limit is the value a function heads towards: Writing limx→a f(x) = L means: as x gets closer and closer to a (from either side), the output f(x) gets closer and closer to L.
Key point: this is about where f(x) is heading, not necessarily its value at a. The function might not even be defined exactly at a.
Continuous = you can draw it without lifting your pen: A function is continuous at a point if there is no jump, hole or break there — the graph flows through in one unbroken stroke. Informally, f is continuous at a when limx→a f(x) = f(a): where the curve is heading is exactly where it actually is.
Polynomials (like x², x³ + 2x) are continuous everywhere. A graph with a sudden jump, or a gap (a 'hole'), is not continuous there.
IB-style question — evaluating a simple limit
A function is given by f(x) = x² + 1.
Write down limx→2 f(x), and explain why it equals f(2).
Step by step
- f(x) = x² + 1 is a polynomial, so it is continuous everywhere.
- For a continuous function the limit equals the value, so just substitute x = 2.
- Evaluate.
Final answer
limx→2 f(x) = 5, and it equals f(2) because the graph is unbroken (continuous) at x = 2.
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Differentiate, then differentiate again: The second derivative is just the derivative of the derivative: differentiate f(x) to get f'(x) (the gradient), then differentiate that to get f''(x).
f''(x) tells you the rate of change of the gradient — how the slope itself is changing. Keep differentiating for f'''(x), f⁽⁴⁾(x), and so on.
Two notations, same thing: Prime notation: f'(x), f''(x), f'''(x), then f⁽⁴⁾(x) for the 4th and beyond.
Leibniz notation: dy/dx, d²y/dx², d³y/dx³.
Read 'd²y/dx²' as 'd-two-y by d-x-squared' — it does not mean anything is squared; it's the symbol for differentiating twice.
IB-style question — find the first and second derivatives
A curve is defined by y = x⁴ − 2x³ + 5x.
Find dy/dx and d²y/dx².
Step by step
- Differentiate once (power rule on each term).
- Differentiate the result again to get the second derivative.
Final answer
dy/dx = 4x³ − 6x² + 5 and d²y/dx² = 12x² − 12x.
IB-style question — evaluate a second derivative
For f(x) = x³ − 4x², find f''(2).
Step by step
- First derivative.
- Second derivative (differentiate again).
- Substitute x = 2.
Final answer
f''(2) = 4.