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(Solved): Question 7 (a) Verify by differentiation that \int (x)/(2\sqrt(2+x))dx=(1)/(3)(x-4)\sqrt(2+x)+K wher ...
Question 7 (a) Verify by differentiation that
\int (x)/(2\sqrt(2+x))dx=(1)/(3)(x-4)\sqrt(2+x)+Kwhere
Kis a constant. [6 marks] (b) Consider the function
f(x)=x^(3)-3x^(2). (i) Determine the critical number(s) of
f. [2 marks] (ii) Find the intervals where
fis increasing. [3 marks] (iii) Find the intervals where
fis decreasing. [3 marks] (iv) Find the local extrema of
f. [2 marks] (v) Sketch the graph of
f, clearly indicating the intercepts. [5 marks] (c) Use L'Hôpital's rule to evaluate
\lim_(x->3)((1)/(x)-(1)/(3))/(x^(2)-9)[4 marks]
