Solved: Consider a function $f$, such that $f(x) = 5.8\sin\left(\frac{\pi}{6}(x +... [IB Mathematics Analysis and Approaches (AA)]

Question

SLPaper 2

Consider a function $f$ , such that $f(x) = 5.8\sin\left(\frac{\pi}{6}(x + 1)\right) + b$ , $0 \leq x \leq 10$ , $b \in \mathbb{R}$ .

The function $f$ has a local maximum at the point (2, 21.8), and a local minimum at (8, 10.2).

A second function $g$ is given by $g(x) = p\sin\left(\frac{2\pi}{9}(x - 3.75)\right) + q$ , $0 \leq x \leq 10$ ; $p$ , $q \in \mathbb{R}$ .

The function $g$ passes through the points (3, 2.5) and (6, 15.1).

Find the period of ${f}$ .

[2]

Verified

Solution

correct approach A1 eg ${\frac{\pi }{6} = \frac{{2\pi }}{{period}}}$ (or equivalent) period = 12 A1 [2 marks]

Note: It seems like the LaTeX equation in the original HTML was invalid.

Find the value of $b$ .

[2]

Verified

Solution

validapproach (M1) eg $\frac{{\text{max}} + \text{min}}{2}\,\,b = \text{max} - \text{amplitude}$ $\frac{{21.8 + 10.2}}{2}$ , or equivalent $b$ = 16 A1 [2 marks]

Hence, find the value of $f(6)$ .

[2]

Verified

Solution

attempt to substitute into their function (M1) $5.8\,\sin\left( {\frac{\pi }{6}\left( {6 + 1} \right)} \right) + 16$ $f(6) = 13.1$ (A1) [2 marks]

Find the value of ${p}$ and the value of ${q}$ .

[5]

Verified

Solution

valid attempt to set up a system of equations (M1) two correct equations A1 $p\cdot \sin\left( \frac{2\pi }{9}\left( 3 - 3.75 \right) \right) + q = 2.5$ , $p\cdot \sin\left( \frac{2\pi }{9}\left( 6 - 3.75 \right) \right) + q = 15.1$ valid attempt to solve system (M1) $p = 8.4;\ q = 6.7$ A1A1 [5 marks]

Find the value of ${x}$ for which the functions have the greatest difference.

[2]

Verified

Solution

attempt to use $|\left( f(x) - g(x) \right)|$ to find maximum difference (M1) $x$ = 1.64 A1

[2 marks]

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