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Evaluate $$\lim_{x \to 0} \frac{\sin 3x}{x}$$ - HSC - SSCE Mathematics Extension 1 - Question 1 - 2002 - Paper 1

Question 1

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Evaluate $$\lim_{x \to 0} \frac{\sin 3x}{x}$$. Find $$\frac{d}{dx}\left(3x^2 \ln x\right)$$ for $$x > 0$$. Use the table of standard integrals to evaluate $$\int_{... show full transcript

Worked Solution & Example Answer:Evaluate $$\lim_{x \to 0} \frac{\sin 3x}{x}$$ - HSC - SSCE Mathematics Extension 1 - Question 1 - 2002 - Paper 1

Step 1

Evaluate $$\lim_{x \to 0} \frac{\sin 3x}{x}$$

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Answer

To evaluate the limit $\lim_{x \to 0} \frac{\sin 3x}{x}$ , we can use L'Hôpital's Rule as it is in the indeterminate form $\frac{0}{0}$ . We differentiate the numerator and denominator:

$\text{Numerator: } \frac{d}{dx}(\sin 3x) = 3\cos 3x$ $\text{Denominator: } \frac{d}{dx}(x) = 1$

Thus, the limit becomes:

$\lim_{x \to 0} \frac{3\cos 3x}{1} = 3\cos(0) = 3$ .

Step 2

Find $$\frac{d}{dx}\left(3x^2 \ln x\right)$$ for $$x > 0$$

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Answer

To differentiate the function $f(x) = 3x^2 \ln x$ , we use the product rule, which states that if $f(x) = u(x)v(x)$ , then $f'(x) = u'(x)v(x) + u(x)v'(x)$ .

Let $u = 3x^2$ and $v = \ln x$ :

Differentiate $u$ : $u' = 6x$
Differentiate $v$ : $v' = \frac{1}{x}$

Applying the product rule:

$f'(x) = 6x \ln x + 3x^2 \cdot \frac{1}{x} = 6x \ln x + 3x$ .

Step 3

Use the table of standard integrals to evaluate $$\int_{0}^{\frac{\pi}{3}} \sec 2x \tan 2x \, dx$$

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Answer

To evaluate the integral $\int \sec 2x \tan 2x \, dx$ , we recognize that the integral of $\sec(x)$ is $\ln |\sec(x) + \tan(x)| + C$ . Thus:

Using the substitution $u = 2x$ , then $du = 2dx$ or $dx = \frac{du}{2}$ . Change the limits accordingly:

When $x = 0$ , $u = 0$ .
When $x = \frac{\pi}{3}$ , $u = \frac{2\pi}{3}$ .

Now, $\int_{0}^{\frac{\pi}{3}} \sec 2x \tan 2x \, dx = \frac{1}{2}\int_{0}^{\frac{2\pi}{3}} \sec u \tan u \, du$ , which equals $\frac{1}{2}\left[\ln |\sec u + \tan u| \right]_{0}^{\frac{2\pi}{3}}$ .

Calculating this gives: $\frac{1}{2}\left(\ln |\sec(\frac{2\pi}{3}) + \tan(\frac{2\pi}{3})| - \ln |\sec(0) + \tan(0)|\right).$

Step 4

State the domain and range of the function $$f(x) = 3\sin^{-1}\left(\frac{x}{2}\right)$$

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Answer

The function $f(x) = 3\sin^{-1}\left(\frac{x}{2}\right)$ is defined for the input values of $x$ that satisfy the condition of the arcsine function, where $-1 \leq \frac{x}{2} \leq 1$ . Therefore:

The domain of $f(x)$ is: $-2 \leq x \leq 2$ .

The range of $f(x)$ can be determined by considering the output of $\sin^{-1}(y)$ , which produces values in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$ . Thus, multiplying by 3 gives:

The range of $f(x)$ is: $[-\frac{3\pi}{2}, \frac{3\pi}{2}]$ .

Step 5

The variable point $$(3t,2t)$$ lies on a parabola. Find the Cartesian equation for this parabola.

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Answer

We can express the points as $x = 3t$ and $y = 2t$ . To eliminate the parameter $t$ :

Solve for $t$ in terms of $x$ : $t = \frac{x}{3}$ .
Substitute $t$ back into the equation for $y$ : $y = 2\left(\frac{x}{3}\right) = \frac{2x}{3}$ .

Thus, the Cartesian equation relating $x$ and $y$ is: $y = \frac{2}{3}x$ .

Step 6

Use the substitution $$u = 1 - x^2$$ to evaluate $$\int_{\frac{3}{2}}^{2} \frac{2x}{(1 - x^2)^2} \, dx$$

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Answer

Using the substitution $u = 1 - x^2$ gives us:

Then, differentiate: $du = -2x \, dx$ or $dx = -\frac{du}{2x}$ .
Change the limits:
- When $x = \frac{3}{2}$ , $u = 1 - \left(\frac{3}{2}\right)^2 = 1 - \frac{9}{4} = -\frac{5}{4}$ .
- When $x = 2$ , $u = 1 - 2^2 = 1 - 4 = -3$ .

Thus, the integral becomes: $-\int_{-\frac{5}{4}}^{-3} \frac{2x}{u^2} \left(-\frac{du}{2x}\right) = \int_{-3}^{-\frac{5}{4}} \frac{1}{u^2} \, du$ .

Which evaluates to: $\left[-\frac{1}{u}\right]_{-3}^{-\frac{5}{4}} = \left(-\frac{1}{-\frac{5}{4}} + \frac{1}{-3}\right) = \frac{4}{5} - \frac{1}{3} = \frac{12}{15} - \frac{5}{15} = \frac{7}{15}$ .

Evaluate $$\lim_{x \to 0} \frac{\sin 3x}{x}$$ - HSC - SSCE Mathematics Extension 1 - Question 1 - 2002 - Paper 1

Worked Solution & Example Answer:Evaluate $$\lim_{x \to 0} \frac{\sin 3x}{x}$$ - HSC - SSCE Mathematics Extension 1 - Question 1 - 2002 - Paper 1

Evaluate $$\lim_{x \to 0} \frac{\sin 3x}{x}$$

Find $$\frac{d}{dx}\left(3x^2 \ln x\right)$$ for $$x > 0$$

Use the table of standard integrals to evaluate $$\int_{0}^{\frac{\pi}{3}} \sec 2x \tan 2x \, dx$$

State the domain and range of the function $$f(x) = 3\sin^{-1}\left(\frac{x}{2}\right)$$

The variable point $$(3t,2t)$$ lies on a parabola. Find the Cartesian equation for this parabola.

Use the substitution $$u = 1 - x^2$$ to evaluate $$\int_{\frac{3}{2}}^{2} \frac{2x}{(1 - x^2)^2} \, dx$$

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