1S Summer Exam June 2000 - Calculus Dr Paul May

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1) Determine the following (1 mark each):

            a) dy/dx   if    y = 1000x²                                            b) dy/dr  if    y = 5r⁶ - r + 1

            c) dy/dx   if    y = 3cos x                                  d) dp/dq   if    p = 7e^-q

2) Differentiate the following functions with respect to x, and simplify the result where possible:

            a) y = (x + 8)(1 - 3x)               b) y = 2x ln x

            c)                        d) y = sin (3x² + 5)                               (8 marks)

3) A function which is often used to represent the form of an electronic wavefunction in certain atoms is:

                                    y = r² e^-r

a) This function has 3 stationary points.  One is at r = 0, and another at r = infinity.  Differentiate this function and thence determine the coordinates (r,y) of the remaining stationary point.                                                                                           (6 marks)

b) The second differential of this function is:

Use this to determine whether the stationary point you just found is a local maximum or minimum.                                                                                                         (2 Marks)

c) Hence sketch this function between r = 0 and r = infinity.                               (4 marks)

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Answers

1) [1mark for each].

a) dy/dx = 2000x                                 b) dy/dr = 30r⁵ - 1

c) dy/dx = -3sin x                                 d) dp/dq = -7e^-q

2) [2 marks each, 1 mark for differentiating correctly, 1 mark for simplifying it].

a) Product Rule: dy/dx = (x + 8).(-3) + (1 - 3x).1                  =          -6x - 23

b) Product Rule: dy/dx = 2x.(1/x) + (ln x).2                           =          2 + 2ln x

c) Quotient Rule: dy/dx =              =         

d) Function of a Function: dy/dx = cos (3x² + 5).6x   =          6x cos (3x² + 5)

3)  a)

   Product Rule: -r²e^-r + e^-r(2r)     [2 marks]        =          re^-r(2 - r)         [1 mark]

   For turning point,   re^-r(2 - r) = 0,   so either:  r = 0                \         r = 0,

                                                                                    e^-r = 0              \         r = infinity

                                                            or         (2 - r) = 0,       \         r = 2

   The last answer is the required one.    [2 marks]

 

   So the turning point is at (2, 0.54).       [1 mark]

 

b) Determine the sign of the second differential, d²y/dr².  Putting in the value of r = 2, we get d²y/dr² =   -0.27, which is -ve, so the t.p. is a local maximum.            [2 marks]

c)   Sketch: [4 marks].   Need to label axes correctly, get correct shape of graph, label t.p. correctly.

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