Assessed Problem Sheet

Demonstrate all your workings – marks will not be awarded for ‘answer only’ responses.

1.      If  and  

         (a)          From first principles, find the derivative of .

                    Use an appropriate, labelled sketch in your answer.          

                           (5 marks)

(b)         Find the derivative of .     (4 marks)

(c)         Find the derivative of           (4 marks)


(a)            Determine the integral           (4 marks)

(b)         Show that the integral  = 0

             where A is a constant.           (3 marks)

(c)         Using a sketch of the function to help you, determine the area enclosed between y = 0 and the line

              in the range         (5 marks)

3.        If

Find an expression for   in terms of  and , and write this in its simplest form.

                    (5 marks)

4.      (a)    

                Find                                                                  (2 marks)


                Show that                                                                    (2 marks)

5.      Determine the integrals


                  Give your answer in the form of a single natural logarithm.

                                  (4 marks)


                           (5 marks)

6.      (a) find the particular solution of

         , if the curve passes through the point (1,1).

                                  (3 marks)

         (b) find the general solution of the differential equation

                                (4 marks)

End of questions

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