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Q5. The Taylor Series is defined using the following summation: f(n) (a) n! f(x) = -(x-a)n n=0 where f(n) (a) represents the nth order derivative of f(x) with an input of a, the central point of the series. a) Write the first five terms in the Taylor Series for f(x) = In(√x), with a central point of a = 1. (7) b) Use this series to estimate the value of In(√2). Give your estimation to six decimal places. (2) (9 marks)

Q4. Consider the curve y = x³ - ²x + 4. a) Using calculus, find the coordinates for the turning point(s) of the curve. State clearly whether they are local maximum or minimum points. (5) (5) b) Find the equations of the tangent and normal lines to the curve when x = 1. (10 marks)

Q3. Consider the function, f(x) = (2x - 3)25 a) Using the Binomial Theorem, write the first five terms of this binomial expansion. State the full value of each coefficient (without rounding). (5) b) Calculate the coefficient for the central term(s) in this binomial expansion. State your answer(s) with an accuracy of 6 significant figures. (7) (12 marks)

Q2. The nth term in a geometric sequence is defined by the following rule: 6 fn 47-1 a) Using this rule, write five consecutive terms in the sequence, starting with the first. b) Calculate the sum of the infinite series defined by this rule. (3) (3) (6 marks)

Q1. For each of the following functions, find the first three derivatives: a) f(x) = x+ + 4x² - ² b) g(x)=2x-3 + x² c) h(x) = cos(2x) sin() (3) (3) (7) (13 marks)