Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Calculus — Topic 5.10
📋 Learning Intentions
- Find the second derivative \( f''(x) \) by differentiating \( f'(x) \) a second time.
- Determine the concavity of a function at a point using the sign of \( f''(x) \).
- Identify points of inflexion by locating where \( f''(x) = 0 \) and verifying a change of concavity.
- Classify stationary points as local maxima, local minima, or points of inflexion using the second derivative test.
- Interpret the second derivative in context — for example, as acceleration in kinematics or the rate of change of a rate of change.
- Use the GDC to graph \( f''(x) \) and confirm concavity and inflexion points numerically.
Real-World Introduction
When a country's GDP is growing, economists distinguish between two very different situations: growth that is accelerating (the rate of increase is itself increasing) and growth that is decelerating (the rate of increase is slowing down). Both situations show positive growth, yet they feel completely different to policymakers. This distinction — between the rate of change and the rate of change of the rate of change — is precisely what the second derivative captures.
In engineering, the second derivative of a beam's deflection curve determines whether the beam is bending concave upward (like a bowl) or concave downward (like an arch). In medicine, the second derivative of a drug concentration curve identifies the inflexion point — the moment when the drug's concentration is rising most rapidly before it begins to level off. These are not abstract ideas; they are the mathematical tools that make real modelling precise.
🌍 Why This Matters In kinematics, the second derivative of displacement with respect to time is acceleration. A car's speedometer measures velocity (first derivative); the feeling of being pushed back into your seat measures acceleration (second derivative). The second derivative test is the mathematical language of "speeding up" and "slowing down" applied to any function.
Core Concepts & Definitions
The first derivative \( f'(x) \) measures the gradient of \( f(x) \) — how steeply the function is rising or falling at each point. But the gradient itself is a function, and it too can be increasing or decreasing. Differentiating \( f'(x) \) a second time gives a new function that measures how the gradient is changing. This is the second derivative.
📖 Key Definition — The Second Derivative If \( f(x) \) is a differentiable function, the second derivative is defined as:
\[ f''(x) = \frac{d^2y}{dx^2} = \frac{d}{dx}\left[f'(x)\right] \] It is obtained by differentiating \( f'(x) \) with respect to \( x \). Alternative notations include \( y'' \), \( \frac{d^2y}{dx^2} \), and \( f''(x) \).
To find \(