Calculus: Questions 1
                                           Partial Differentiation
           The first six questions are from last year’s sheet 7 and are included here for revision. The other
           questions are new.
              1. Find all the first and second partial dervatives of
                 (a)    6x2 +4y(1−x)+(1−y)2                           (b)    sin(x2y)
                                                                                   2
              2. If f(x,y) = x2y2 with x = cost and y = sint, find df and d f by using partial differentiation.
                                                                          dt       dt2
              3. Using partial differentiation find dy where
                                                       dx
                 (a)    (x−1)y3+x2cosx=3                              (b)   cos(xy) = 0
              4. Find all the stationary points (i.e. points where fx = fy = 0) of
                                                     f(x,y) = ex+y(x2 +y2 −xy),
                 and find their natures (i.e. are they maxima, minima or saddle points?)
              5. Find all the stationary points of the function
                                                  f(x,y) = (x+y)4−x2 −y2−6xy
                 and identify their type.
              6. Show that the function
                                                  f(x,y) = x2y2 −2xy(x+y)+4xy
                 has stationary points at (1,1) and (2,0). Find the three other stationary points.
                 Identify the type of all the stationary points of this function.
              7. From lectures Find the stationary points of
                                                f(x,y) = 2x4 +8x2y2 −4(x2 −y2)+2
                 Identify the type of all the stationary points of this function.
              8. Find the stationary points of
                                                       f(x,y) = −x2 −y3 +12y2
                 Identify the type of all the stationary points of this function.
              9. How would you solve question 6 if you were told that you had to maximise f(x,y) with the
                 constraint x2 + y2 = 4?
                                                                                                                                                                                                                                                                             Solutions
                                                            1.                 (a) fx = 12x−4y, fy = 4(1−x)−2(1−y), fxx = 12, fxy = fyx = −4, fyy = 2.
                                                                               (b) f = 2xycos(x2y), f = x2cos(x2y), f                                                                                                                                                                                        =2ycos(x2y)−4x2y2sin(x2y),
                                                                                                         x                                                                                              y                                                                                       xx
                                                                                                    f                 =f =2xcos(x2y)−2x3ysin(x2y), f =−x4sin(x2y).
                                                                                                         xy                             yx                                                                                                                                                                           yy
                                                            2. df = ∂f dx + ∂f dy = 2xy2dx + 2x2ydy
                                                                            dt                       ∂x dt                                 ∂y dt                                                         dt                                              dt
                                                                              =2costsin2t×(−sint)+2cos2tsint×(cost) = −2sin3tcost+2sintcos3t
                                                                                                                                                                                                                                                                                                                                        
                                                                            d2f = ∂f d2x + ∂f d2y + ∂2f                                                                                                              dx 2 +2 ∂2f dxdy + ∂2f                                                                                                     dy 2
                                                                             dt2                           ∂x dt2                                   ∂y dt2                                   ∂x2                      dt                                       ∂x∂y dt dt                                                ∂y2                     dt
                                                                              =2xy2d2x +2x2yd2y +2y2dx2 +8xydxdy +2x2dyr
                                                                                                                          2                                                   2
                                                                                                                   dt                                                  dt                                                    dt                                                    dt dt                                                        dt
                                                                                                                                       2                                                                                                2                                                                                                                   2                                                             2
                                                                              =2costsin t×(−cost)+2cos tsint×(−sint)+2sin t×(−sint) +8costsint×(−sint)×
                                                                           (cost) +2cos2t×(cost)2 = 2cos4t−12cos2tsin2t+2sin4t.
                                                            3. Using the result that if f(x,y) = C then ∂f + ∂f dy = 0. Note that it is often better to leave
                                                                                                                                                                                                                                                                                           ∂x                        ∂y dx
                                                                           both x and y in your answers.
                                                                               (a) y3 +2xcosx−x2sinx+3(x−1)y2dy =0, or dy = −y3+2xcosx−x2sinx.
                                                                                                                                                                                                                                                                                        dx                                                   dx                                                       3(x−1)y2
                                                                               (b) −ysin(xy)−xsin(xy)dy = 0, or dy = −y/x.
                                                                                                                                                                                                                    dx                                                   dx
                                                            4. Stationary points at (x,y) = (0,0) and (−1,−1).
                                                                           (0,0) is a minimum.
                                                                           (−1,−1) are saddle point.
                                                                                                       -2.0                        -1.5                         -1.0                        -0.5                         0.0                         0.5                         1.0
                                                                                              1.01.01.0                                                                                                                                                                                    1.0
                                                                                              0.50.50.5                                                                                                                                                                                    0.5
                                                                                              0.00.00.0                                                                                                                                                                                    0.0
                                                                                            -0.5-0.5-0.5                                                                                                                                                                                   -0.5
                                                                                            -1.0-1.0-1.0                                                                                                                                                                                   -1.0
                                                                                            -1.5-1.5-1.5                                                                                                                                                                                   -1.5
                                                                                            -2.0-2.0-2.0                                                                                                                                                                                   -2.0
                                                                                                       -2.0-2.0-2.0                -1.5-1.5-1.5                 -1.0-1.0-1.0                -0.5-0.5-0.5                 0.00.00.0                   0.50.50.5                   1.01.01.0
                                                            5. Stationary points at (x,y) = (0,0), (1, 1) and (−1,−1).
                                                                                                                                                                                                                                                                    2 2                                                           2                    2
                                                                           (0,0) is a saddle point.
                                                                           (1, 1) and (−1,−1) are minima.
                                                                                2 2                                                           2                    2                                              
                                                                                                       -1.5                        -1.0                         -0.5                         0.0                         0.5                         1.0                         1.5
                                                                                              1.51.51.5                                                                                                                                                                                    1.5
                                                                                              1.01.01.0                                                                                                                                                                                    1.0
                                                                                              0.50.50.5                                                                                                                                                                                    0.5
                                                                                              0.00.00.0                                                                                                                                                                                    0.0
                                                                                            -0.5-0.5-0.5                                                                                                                                                                                   -0.5
                                                                                            -1.0-1.0-1.0                                                                                                                                                                                   -1.0
                                                                                            -1.5-1.5-1.5                                                                                                                                                                                   -1.5
                                                                                                       -1.5-1.5-1.5                -1.0-1.0-1.0                 -0.5-0.5-0.5                 0.00.00.0                   0.50.50.5                   1.01.01.0                   1.51.51.5
                                                            6. Stationary points at (x,y) = (0,0), (1,1), (2,2), (2,0) and (0,2).
                                                                           (0,0), (2,2), (2,0) and (0,2) are saddle points.
                                                                           (1,1) is a maximum.                                                                                                                    
                                                                                                                           -1                                 0                                 1                                2                                 3
                                                                                                    333                                                                                                                                                                                    3
                                                                                                    222                                                                                                                                                                                    2
                                                                                                    111                                                                                                                                                                                    1
                                                                                                    000                                                                                                                                                                                    0
                                                                                                 -1-1-1                                                                                                                                                                                    -1
                                                                                                                           -1-1-1                             000                               111                              222                               333
                                                            7. Stationary points at (x,y) = (0,0), (1,0), (−1,0).
                                                                           (0,0) is a saddle point, the others are minima.
                                                            8. Stationary points at (x,y) = (0,0), (0,8).
                                                                           (0,0) is a saddle point, (0,8) is a maximum.
                                                            9. See lectures