(a) Sketch the graph of $y = 3 ext{cos}^{-1}(2x)$ - HSC - SSCE Mathematics Extension 1 - Question 2 - 2003 - Paper 1

To differentiate $x \tan^{-1} x$, we use the product rule:

Let $u = x$ and $v = \tan^{-1}(x)$, where:

• $\frac{du}{dx} = 1$,
• $\frac{dv}{dx} = \frac{1}{1 + x^2}$.

Using the product rule: $\frac{d}{dx}(u v) = u \frac{dv}{dx} + v \frac{du}{dx}$ We get: $\frac{d}{dx} (x \tan^{-1} x) = x \cdot \frac{1}{1 + x^2} + \tan^{-1}(x) \cdot 1.$

Thus, the derivative is: $\frac{x}{1 + x^2} + \tan^{-1}(x).$

To evaluate the integral ( \int_{0}^{1} \frac{1}{\sqrt{2 - x^2}} : dx ), we use the substitution $x = \sqrt{2} \sin \theta$. Then, we find:

• $dx = \sqrt{2} \cos \theta d\theta$.
• When $x = 0$, $\theta = 0$, and when $x = 1$, $\theta = \frac{\pi}{4}$.

The integral becomes: $\int_{0}^{\frac{\pi}{4}} \frac{\sqrt{2} \cos \theta}{\sqrt{2 - 2\sin^2 \theta}} d\theta = \int_{0}^{\frac{\pi}{4}} d\theta.$

Thus, evaluating gives: $\left[ \theta \right]_{0}^{\frac{\pi}{4}} = \frac{\pi}{4}.$

Using the binomial theorem: $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k,$ we identify $a = 2$, $b = x^2$, and $n = 5$.

To find the term containing $x^4$, we set $k = 2$ (since $x^{2k} = x^4$): $\text{Term} = \binom{5}{2} (2)^{5-2} (x^2)^2 = \binom{5}{2} (2)^3 (x^4).$

Calculating:

• $\binom{5}{2} = 10$ and $(2)^3 = 8$ gives: $10 \times 8 = 80.$ Therefore, the coefficient of $x^4$ is $80.$

To express $\cos x - \sin x$ in the form $R \cos(x + \alpha)$, we start from: $R = \sqrt{A^2 + B^2}$ where $A = 1$ and $B = -1$. Thus: $R = \sqrt{1^2 + (-1)^2} = \sqrt{2}.$

Next, to find $\alpha$, we use: $\tan \alpha = \frac{B}{A} = \frac{-1}{1} = -1 \Rightarrow \alpha = -\frac{\pi}{4}.$

Thus, we can write $\cos x - \sin x$ as: $\sqrt{2} \cos\left(x + \frac{\pi}{4}\right).$

To sketch the graph:

1. Identify the amplitude, which is $\sqrt{2}$, indicating the graph oscillates between $-\sqrt{2}$ and $\sqrt{2}$.
2. The period is $2\pi$.
3. Start at $y = \cos(0) - \sin(0) = 1 - 0 = 1$ and mark critical points within $[0, 2\pi]$.
4. Sketch the wave based on the transformations identified: the graph will reflect shifts and inverses from the standard cosine and sine functions.
5. Label key points at $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$.