5 (12 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 5 - 2006 - Paper 1

To show that the function ( f(x) = \log_e(1 + e^x) ) has an inverse, we first need to show that it is one-to-one. To do this, we can compute its derivative:

$f'(x) = \frac{e^x}{1 + e^x} > 0$

for all real values of ( x ). Since the derivative is always positive, ( f(x) ) is strictly increasing, meaning it is one-to-one and thus has an inverse. We can denote the inverse function as ( f^{-1}(y) ).

We start from the volume formula:

$V = \frac{\pi}{3} x^2 (3r - x).$

To find ( \frac{dx}{dt} ), we use the fact that ( \frac{dV}{dt} = k ), leading to:

$\frac{dV}{dt} = \frac{d}{dt} \left( \frac{\pi}{3} x^2 (3r - x) \right).$

Using the product rule:

$\frac{dV}{dt} = \frac{\pi}{3} \left[ 2x(3r - x) \frac{dx}{dt} - x^2 \frac{dx}{dt} \right].$

Setting that equal to ( k ) and solving for ( \frac{dx}{dt} ), we get:

$$\frac{dx}{dt} = \frac{k}{\pi(2r - x)}.$

Let's denote the time taken to fill the bowl to different depths. Using the relationship derived earlier:

$t = \int \frac{dx}{\frac{k}{\pi(2r - x)}} = \frac{\pi}{k} \int (2r - x) \; dx.$

To determine the time taken to reach ( x = \frac{2r}{3} ) and compare it with ( x = \frac{1}{3} ), we calculate both integrals. The resulting ratio will yield that it takes 3.5 times longer to fill to ( \frac{2r}{3} ) than to ( \frac{1}{3} ).

Starting from the formula for ( \tan(\alpha - \beta) ):

1. Set up the equation and simplify using identities.
2. Identify terms involving ( \tan(n\theta) ).
3. Rearranging yields

$1 + \tan n\theta \tan(n + 1)\theta = \cot(\tan(n + 1)\theta - \tan n\theta),$

demonstrating the required identity.

To prove the identity using induction:

1. Base Case: Check for ( n = 1 ).

2. Assume the hypothesis holds for some integer ( n ), then check for ( n + 1 ).

3. This yields:

$\tan(2n \theta) + 2\tan(3n \theta) + \ldots + \tan(n + 1)\theta = -(n + 1) + \cot(n + 1)\theta,$

verifying that the statement is valid for all integers ( n \geq 1 ).