IB Mathematics SL 5.7 The second derivative AA SL Paper 1- Exam Style Questions- New Syllabus
▶️ Answer/Explanation
(a)
The total area consists of 8 congruent rectangles: \(8 \times (x \times y) = 60\).
Isolating \(y\):
\(y = \frac{60}{8x} = \frac{7.5}{x}\).
(b)
In a \(4 \times 2\) grid:
– Horizontal cord lines: There are 3 lines spanning the full width (\(4x\)). Total = \(3 \times 4x = 12x\).
– Vertical cord lines: There are 5 lines spanning the full height (\(2y\)). Total = \(5 \times 2y = 10y\).
Total length \(T = 12x + 10y\).
Substituting \(y = \frac{7.5}{x}\):
\(T = 12x + 10\left(\frac{7.5}{x}\right) = 12x + \frac{75}{x}\). (Shown)
(c)
Rewrite as \(T = 12x + 75x^{-1}\).
\(\frac{dT}{dx} = 12 – 75x^{-2} = 12 – \frac{75}{x^2}\).
(d)
(i) Set \(\frac{dT}{dx} = 0\):
\(12 = \frac{75}{k^2} \implies k^2 = \frac{75}{12} = 6.25\).
\(k = \sqrt{6.25} = 2.5\) (since \(x > 0\)).
(ii) \(T = 12(2.5) + \frac{75}{2.5} = 30 + 30 = 60 \text{ m}\).
(iii) \(y = \frac{7.5}{2.5} = 3 \text{ m}\).
(e)
(i) \(\frac{d^2T}{dx^2} = \frac{d}{dx}(12 – 75x^{-2}) = 150x^{-3} = \frac{150}{x^3}\).
(ii) At \(x = 2.5\), \(\frac{d^2T}{dx^2} = \frac{150}{(2.5)^3} > 0\).
Since the second derivative is positive, the function has a local minimum at \(x = 2.5\).