QUIZ 2 SAMPLE PROBLEMS
(1) Calculate the partial derivatives fx and fy of f(x, y) = ln(
x3−y3
1−y ).
(2) Find the equation of the tangent plane of f(x, y) = x2 + y2 + 3x + 2y + 1
at the point (1, 2).
(3) Let f(x, y) = x2y + sin(xy). Show that fxy = fyx.
(4) Evaluate the following limit, or show that it does not exist.
lim
(x,y)→(0,0)
x2y + y3
x2 + y2
.
(5) Evaluate the following limit, or show that it does not exist.
lim
(x,y)→(0,0)
2x3 + 3y2
x2 + y2
.
(6) The level curve ex−y + x2 − y = 1 defines y as a function of x around the
point P = (0, 0). Find dy/dx at P.
(7) A particle moves on a surface whose equation is
z = f(x, y) = (x− 1)2 + y2.
Assume that the x and y coordinates of the particle at time t are given by
x = 2 cos t and y = 2 sin t , where 0 ≤ t ≤ 2pi. Use the total derivative rule
to show that the rate of change of the height of the particle above the xy
plane is given by
dz
dt
= 4 sin t.
Hence find the maximum value of the height and the coordinates (x, y, z)
of the particle when the maximum occurs.
(8) The build up of minerals on the inner wall of a pipe has reduced its capacity
to carry liquid. If the pipe has an elliptical cross section with semimajor
axis 4 cm and semiminor axis 3 cm and the unwanted coating is 0.05 cm
thick, use differentials to estimate the reduction in cross sectional area.
(The area of an ellipse with semimajor axis a and semiminor axis b is piab.)
1
2 QUIZ 2 SAMPLE PROBLEMS
Answers
(1) fx =
3x2
x3−y3 and fy =
x3+2y3−3y2
(1−y)(x3−y3)
(2) z = 5x + 6y - 4
(3) fxy = 2x+ cos(xy)− xy sin(xy)
(4) 0
(5) Limit does not exist.
(6) 1/2
(7) Maximum z-value is 9, occuring at (−2, 0, 9) (when t = pi).
(8) 1.1cm2