IB Mathematics AA SL The second derivative Study Notes - New Syllabus
IB Mathematics AA SL The second derivative Study Notes
LEARNING OBJECTIVE
- The second derivative.
Key Concepts:
- The second derivative.
- Graphical behaviour of functions.
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The Second Derivative and Notation
The Second Derivative and Notation
The second derivative tells us how the rate of change of a function is itself changing. It provides information about the concavity of a function and the nature of turning points.
Definition: If \( f'(x) \) is the first derivative of a function \( f(x) \), then the second derivative is:
- \( f”(x) = \dfrac{d}{dx}\left[f'(x)\right] = \dfrac{d^2y}{dx^2} \)
- It measures the curvature or “bending” of the graph of \( f(x) \).
Notation:
- Lagrange: \( f”(x) \)
- Leibniz: \( \dfrac{d^2y}{dx^2} \)
Interpretation:
- \( f”(x) > 0 \): graph is concave up (∪ shape)
- \( f”(x) < 0 \): graph is concave down (∩ shape)
- \( f”(x) = 0 \): possible inflection point (concavity may change)
Example:
For the function \( f(x) = x^3 – 3x \), determine the second derivative and identify the concavity of the graph.
▶️ Answer/Explanation
\( f”(x) = 6x \)
• If \( x > 0 \), \( f”(x) > 0 \) → concave up
• If \( x < 0 \), \( f”(x) < 0 \) → concave down
Relationship Between Graphs of \( f \), \( f' \), and \( f'' \) And Technology Use
Relationship Between Graphs of \( f \), \( f’ \), and \( f” \) And Technology Use
Graphical Behavior:
- \( f'(x) \) gives the gradient/slope of \( f(x) \)
- \( f”(x) \) tells whether the slope is increasing or decreasing
- Inflection points occur when \( f”(x) = 0 \) and the sign of \( f” \) changes
| Interval | \( f'(x) \) | \( f”(x) \) | Shape |
|---|---|---|---|
| Increasing, steepening | \( > 0 \) | \( > 0 \) | Concave up |
| Increasing, flattening | \( > 0 \) | \( < 0 \) | Concave down |
Technology Integration (GDC/Desmos):
- Graph \( f(x) \), \( f'(x) \), and \( f”(x) \) simultaneously to observe relationships
- Use GDC’s
nDeriv()orgraph derivativetools - Zoom in near inflection points to confirm concavity changes
Example:
Using technology, investigate the concavity of the function \( f(x) = \cos(x) \) by analyzing its second derivative.
▶️ Answer/Explanation
\( f(x) = \cos(x) \)
\( f'(x) = -\sin(x) \)
\( f”(x) = -\cos(x) \)
Interpret concavity
• Where \( f”(x) < 0 \), the graph of \( f(x) \) is concave down.
Application: Simple Harmonic Motion (Physics)
Application: Simple Harmonic Motion (Physics)
In simple harmonic motion (SHM), the second derivative of displacement gives acceleration. SHM is defined by the condition that acceleration is proportional and opposite to displacement.
Let: \( x(t) = A \cos(\omega t) \)
- Velocity: \( v(t) = \dfrac{dx}{dt} = -A\omega \sin(\omega t) \)
- Acceleration: \( a(t) = \dfrac{d^2x}{dt^2} = -A\omega^2 \cos(\omega t) = -\omega^2 x(t) \)
Conclusion: Acceleration is proportional to the negative of displacement: \( a(t) = -\omega^2 x(t) \). This confirms SHM behavior.
Example:
Show that \( x(t) = 4\sin(3t) \) satisfies the simple harmonic motion equation using the second derivative.
▶️ Answer/Explanation
Solution:
\( v(t) = \dfrac{dx}{dt} = 12\cos(3t) \)
\( a(t) = \dfrac{d^2x}{dt^2} = -36\sin(3t) \)
Since \( a(t) = -36\sin(3t) = -9 \cdot 4\sin(3t) = -\omega^2 x(t) \),
where \( \omega^2 = 9 \), this satisfies the SHM condition:
\( a(t) = -\omega^2 x(t) \)
