IB Mathematics SL 2.4 Determine key features of graphs AI HL Paper 1- Exam Style Questions- New Syllabus
▶️ Answer/Explanation
(a)
A horizontal translation of \(0.3\) units left is represented by \((x+0.3)\) and a vertical translation of \(0.2\) units down by \(-0.2\):
\(g(x) = 4\cos\left(\frac{\pi}{8}(x+0.3)\right) – 0.2\).
(b)
To find the intercept \(a\), solve \(g(a) = 0\):
\(4\cos\left(\frac{\pi}{8}(a+0.3)\right) = 0.2\).
Using a GDC: \(a \approx 3.57\).
(c)
(i) Since the rotation is only through \(\pi\) radians (a half-revolution), the volume is \(\frac{1}{2}\) of the standard volume of revolution (\(V = \frac{1}{2} \cdot \pi \int y^2 dx\)).
Volume = \(\frac{\pi}{2} \int_0^4 (f(x))^2 dx – \frac{\pi}{2} \int_0^a (g(x))^2 dx\).
(ii) Evaluating the integrals:
\(\frac{\pi}{2} \int_0^4 \left(4\cos(\frac{\pi}{8}x)\right)^2 dx \approx 50.2655\dots\).
\(\frac{\pi}{2} \int_0^{3.57} \left(4\cos(\frac{\pi}{8}(x+0.3)) – 0.2\right)^2 dx \approx 37.3414\dots\).
Difference: \(50.2655 – 37.3414 \approx 12.9241\dots\).
Volume \(\approx 12.9\) cm\(^3\).