MA103 Winter 2012 Final - MA103 Winter 2012 Final Examination[4 marks Page 1 of 12 1 Find the following limits x2 1 x(a lim x 2 x[4 marks(b lim x2 sin[4

# MA103 Winter 2012 Final - MA103 Winter 2012 Final...

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MA103, Winter 2012 - Final Examination Page 1 of 12 1. Find the following limits:(a)limx→∞x2+ 1-x[4 marks](b) limx0x2sin2x(Hint: Squeeze theorem) [ 4 marks ] (c)limx→∞(2 +x)1/x(Hint: L’Hospital’s rule) [ 4 marks ] Over
MA103, Winter 2012 - Final Examination Page 2 of 12 2. Show that the functionf(x) =Zcosx0pt2+ sin2t dtis decreasing in the interval0,π2.[4 marks]3. Follow the steps below to show that 0612Z31lnxxdx6e-1.(You may use the approximate values:e2.7182818,e-10.367879441andln 31.0986123.)(a) Find the critical numbers off(x) =lnxxin [1,3].[5 marks](b) Use theclosed interval methodto find the absolute maximum and absolute minimumvalues off(x) in [0,2], and show that 0612Z3lnxxdx6e-1.[6 marks] 1
MA103, Winter 2012 - Final Examination Page 3 of 12 4. Follow the steps to sketch the graph of the functionf(x) =e-x22(a) Find all intercepts ofy=f(x) (bothxandy-intercepts, if they exist) [ 2 marks ] (b) Find all asymptotes ofy=f(x), if they exist. [ 2 marks ]