Hello, I've been working on some homework for my Calculus class in university, and we really cant figure out the answers for the last few questions; any help would be greatly appreciated - a step by step with some explanation would be great!

Each question is worth 10 marks:

1. Using a result from the module, show that the equation

P(x)=x^{3}+ sin(x) - 30 = 0

has solution x^{*}with 3 ≤ x^{*}≤ 4 .

A lecturer decides to use the Newton-Raphson method 'x_{n+1}= g(x_{n}) to determine x^{*}. Write down and simplify the function g(x) which should be used. Using a calculator (and working to at least 4 decimal places), determine x_{1}, x_{2}, x_{3}, x_{4}, given that x_{0}= 3. For the value of x_{4}which you have found, determine P(x_{4}) .

2. Let

F(x) = x^{4}sin (π x) , G(x) = 4^{x}- 16 .

Determine the first and second derivatives of F and G. Determine the Taylor series of F and the Taylor series of G about x = 2, in each case up to and including the term (x - 2)^{2}. Use your answers to detemine

lim x->2 F(x) / G(x) .

3. Let

y = H(x) = 8x+5 / 2x-9 .

Determine the maximal domain of H, a formula for x in terms of y, and the corresponding range of H. Sketch the graph of H showing clearly any horizontal and/or vertical asymptotes and points where the graph meets x and/or y axes.

4. Determine the Taylor series of h(x) = ln(1 + 2x) about x=0, up to and including the term x^{3}. Use your answer to determine the following limits:

L_{1}= lim x->0+ ln(1 + 2x) / x ,

L_{2}= lim y->+∞ y ln(1 + 2 / y) ,

L_{3}= lim y->+∞ (1 + 2 / y)^{y}.

This is my first post on this forum so I'm sorry if everything isn't perfect, and thanks in advance for anyone who takes time to help me out! I'm more than happy to return the favour using my knowledge, thank you!

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