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.
Heading, not arriving: 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.
Continuous or not
- 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.
Higher derivatives
First derivative
Differentiate f(x) to get f'(x) (the gradient).
Second derivative
Differentiate that to get f''(x). It tells you the rate of change of the gradient — how the slope itself is changing.
Keep going
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.