1) Refer to the hyperbolic graph in the margin, and the text that accompanies it, on Page 261 of your text.
Use the exponential definitions of the Sinh and Cosh functions to find the exact coordinates of the point P when t = ln 8. For that value of t, how large is the corresponding shaded area in this graph?
2a) Let , the Hyperbolic Tangent function.
Given that , use Chain Rule to help find .
b) Use your calculator (or computing software) to help find the approximate values of when . (report each correct to four decimal places)
3a) Let , the inverse Hyperbolic Sine function.
Write down the “ln” form of this function, then use it to show what the exact value of g(x) is when x = -1.
b) Identify g’(x), then find the exact value of g’(x) when x = -1.
Sample Solution
Religion and spirituality were extremely important to almost all 18th century societies, and those of the South Pacific were no exception. The Polynesians had many gods, with many different names and attributes, to whom the practise of making human sacrifices was not uncommon. Religion was similar across most of Polynesia, and centred around the sacred site of Marae Taputapuatea on Raiatea. Everything changed with the arrival of the arrival of Europeans, who brought with them Christianity. From a Western perspective, the adoption of Christianity in the Pacific can be seen as positive, as it encouraged peace amongst warring villages. When missionaries began to make the journey to the South Pacific in the late 18th century, their sole aim was ‘civilising’ the indigenous population, and set out to change almost every aspect of their lives. People’s clothing became more conservative, the ancient art of body tattooing fell out of use and great religious artworks (for Pacific religions had mostly been based around icons) were dismissed as idols, and burnt. Marae Taputapuatea, which had been a site for worship for millennia, was allowed to fall into disrepair as the focus of South Pacific religion turned its attention towards Rome and the West. As one Maori man put it, “When the Westerners arrived, the Polynesians learnt to build houses from lime and limewash and build churches for worshipping God. We started praying inside the Church. We let go of the gods, the marae, the ways of our ancestors, all of it” (2). Missionaries had particularly struggled on New Zealand, failing to ‘save’ a single soul in their first fifteen years there. The South Pacific languages had no written form, so to communicate ideas and messages, they used artwork. Lots of artwork was based around the gods and important rituals, usually in the form of wooden statues, although tattooing and scarification were also popular, particularly amongst the Maori. The Maori were unique in their practise of facial tattoos, or moku, a practice which continues today. Each design has a genealogy – M>
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