Evaluate $ \lim_{x \to 0} \frac{\sin 3x}{x} $. Find $ \frac{d}{dx}(3x^2 \ln x) $ for $ x > 0 $. Use the table of standard integrals to evaluate $ \int_{0}^{\frac{\pi}{2}} \sec 2x \tan 2x \, dx $. State the domain and range of the function $ f(x) = 3\sin^{-1}(\frac{x}{2}) $. The variable point $ (3, 2) $ lies on a parabola. Find the Cartesian equation for this parabola. Use the substitution $ u = 1 - x^2 $ to evaluate $ \int_{2}^{3} \frac{2x}{(1 - x^2)^2} \, dx $.

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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

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

Evaluate $ \lim_{x \to 0} \frac{\sin 3x}{x} $. Find $ \frac{d}{dx}(3x^2 \ln x) $ 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, we can use L'Hôpital's Rule since it gives an indeterminate form of ( \frac{0}{0} ). Using the derivative, we have:

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

Therefore, the answer is 3.

Step 2

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

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Answer

Using the product rule, we differentiate ( 3x^2 ) and ( \ln x ):

( \frac{d}{dx}(3x^2) = 6x )
( \frac{d}{dx}(\ln x) = \frac{1}{x} )

Applying the product rule:

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

Thus, the derivative is ( 3x + 6x\ln x ).

Step 3

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

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Answer

Using the substitution ( u = 2x ), we have: ( du = 2dx ) or ( dx = \frac{du}{2} ).

Thus, the integral becomes:

$\int \sec u \tan u \cdot \frac{1}{2} \, du = \frac{1}{2} \int \sec u \tan u \, du = \frac{1}{2} \sec u + C.$

Evaluating between limits, changing the limits accordingly:

$\left[ \frac{1}{2} \sec u \right]_{0}^{\pi} = \frac{1}{2}(\sec \pi - \sec 0) = \frac{1}{2}(-1 - 1) = -1.$

So, the result of this integral is -1.

Step 4

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

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Answer

The function ( f(x) = 3\sin^{-1}(\frac{x}{2}) ) has a domain given by the argument of the inverse sine function. Thus, ( -2 \leq x \leq 2 ) is the domain.

The range can be determined by analyzing the output of the function:

$f(x) \in [3\sin^{-1}(-1), 3\sin^{-1}(1)] = [-\frac{3\pi}{2}, \frac{3\pi}{2}].$

Hence, the domain is ([-2, 2]) and the range is ([-\frac{3\pi}{2}, \frac{3\pi}{2}]).

Step 5

Find the Cartesian equation for the parabola given that the variable point $ (3, 2) $

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Answer

Assuming a standard form of the parabola opening upwards:\n( y = ax^2 + bx + c ) and substituting the point ( (3, 2) ):

Replace ( x ) with 3 and ( y ) with 2 to form: $2 = 9a + 3b + c.$

To get more information about the parabola, we may need additional points or conditions, which are needed to determine constants ( a, b, c ).

Step 6

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

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Answer

Using the substitution ( u = 1 - x^2 ), we get:

( du = -2x , dx ) or ( dx = -\frac{du}{2x} ).

The integral changes to:

$= -\int \frac{2x}{u^2} \cdot -\frac{du}{2x} = \int \frac{1}{u^2} \, du = -\frac{1}{u}.$

We need to evaluate this from the new limits:

When ( x = 2, u = 1 - 2^2 = -3; ) When ( x = 3, u = 1 - 3^2 = -8. )

Thus, the integral can be expressed as:

$\left[-\frac{1}{u} \right]_{-3}^{-8} = -\frac{1}{-8} - (-\frac{1}{-3}) = \frac{1}{3} - \frac{1}{8}.$

Calculating gives:

$= \frac{8 - 3}{24} = \frac{5}{24}.$

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

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

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

Find the Cartesian equation for the parabola given that the variable point \( (3, 2) \)

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

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