Hints and comments on WeBWorK problems
WeBWorK instructions
- Problem 1
Archimedes would have done this problem easily without using
calculus, using Archimedes' principle for the surface area
of a spherical surface (or, more generally, for a slice of
finite thickness of a spherical shell). Imagine a sphere
inscribed in a cylinder -- crudely a raquetball in a can that
contains only one raquetball. Archimedes observed that if you
cut a (very thin) slice of the can using planes perpendicular
to its axis of symmetry then the surface area of the slice of
the spherical shell equals the surface area of the slice of the
cylindrical shell. (Draw a sketch and calculate the surface
area of each thin slice and see that that they agree.) He then
used this principle to calculate the surface area of a sphere
(or any slice) using a simple formula for the surface area of a
slice of a cylinder.
- Problem 6.
Archimedes would have done this problem easily without using calculus,
using Archimedes' principle for the volume of a solid sphere
(or, more generally, for a slice of finite thickness of a sphere).
Imagine a sphere inscribed in a cylinder --
crudely a raquetball in a can that contains only one
raquetball. Imagine the two-napped cone consisting of
rays from the center of the ball to the rims of the can.
Archimedes observed that if you slice the can along
any plane perpendicular to its axis of symmetry then the
cross-sectional area of the sphere plus the cross-sectional
area of the cone equals the cross-sectional area of the cylinder.
He used this principle to calculate the volume of a sphere
(or any slice) using the formulas for the volume of a
cylinder and of a cone. (Remark: The formula for the volume of a cone is
the same as for the volume of a pyramid, one third the base times
the height. To get a formula for the volume of a pyramid,
observe that by dimensional/scaling arguments it must be *some*
fixed constant times the base times the height; to get that constant,
slice a cube into pyramids.)
- Problem 1.
A Riemann sum is just the sum of the volumes of
a bunch of boxes. For this problem the height
of each box is the value of the integrand sampled
at the center of each square.
- Problem 3.
This problem can be done by straightforwardly setting up
integrals, but it can be made much easier if you
see how to rescale the axes.
- Problem 6.
Think signed volume. Look for symmetry.
- Problem 8.
Symmetry makes life simpler.
- Problem 10.
Use symmetry to get rid of the absolute value.
- Problems 1–2.
To search for extrema on the boundary curves
you can either parametrize or use Lagrange
multipliers.
- Problems 3.
This is a constrained optimization problem
(i.e. a ripe candidate for Lagrange multipliers).
- Problem 4.
This is an example of a linear programming
problem,
i.e. a problem where you must find the extremes of a
linear function
on a polygon. (Polygons result when the
constraints are linear inequalities.)
Linear functions are flat and so cannot have extrema
unless they are constant. This is why the extrema
always occur on the corners.
- Problem 5.
Notice that the function f is not
linear. Therefore, you cannot just test f
at the corners of the triangle.
You can deal with the boundaries either by
parametrizing them or by using Lagrange multipliers.
- Problem 7.
This problem is trying to find the
closest point of a plane to the origin.
- Problem 10.
Recall that the Taylor series
approximation T
of the function f
expanded around the origin is
just the polynomial whose partial
derivatives agree with those of f:
T = f0
+x fx
+y fy
+(x2/2) fxx
+xy fxy
+(y2/2) fyy
+(x3/6) fxxx
+(x2y/2) fxxy
+(x y2/2) fxyy
+(y3/6) fyyy
+ ...,
where the partial derivatives are all evaluated at zero.
It is extremely easy to check this result.
All you have to do is to check that each partial
derivative of the expansion agrees with the
corresponding partial derivative of the function.
For example, if in the generic expansion above
you want to check the fxxy
term, just differentiate
(x2y/2) fxxy
twice with respect to x
and once with respect to y
(remembering that fxxy
stands for fxxy(0,0)
and is therefore just a constant), and you will
discover that
Txxy(0,0)= fxxy(0,0),
as required.
- Problems 6–7.
The tangent to the
curve of intersection of two surfaces is the
intersection of their tangent planes.
- Problem 8.
The tangent hyperplane of f(x,y,z) at point P
is just the linear approximation:
w = f0 + df,
where f0 = f(P) and
where the differential df is
df = fxdx
+ fydy
+ fzdz
and dx:= x-x0, etc.
- In general, it is important to distinguish
between the differential df of a function f
and the linear approximation f0+df.
- Problem 1,7.
If you need a review of quadric surfaces,
study §12.6 in the text or check out
Wikipedia page on quadric surfaces
- Problem 4-5.
If you need a review of cylindrical and spherical
coordinates (hopefully you have at least seen cylindrical
coordinates before, since they are just polar coordinates
plus a "z" coordinate), study
the definitions and equations relating
rectangular, cylindrical, and spherical coordinates
in §15.6.
- Problem 6.
Please note that this problem is just asking for the
composition of functions. You are not asked to
differentiate anything.
- Problem 12.
Limits of multivariable functions are discussed in
section 14.2.
For practice showing that limits do not
exist study example 5 on page 980 and problems
35-42 in the section 14.2 exercises.
To show that a limit does exist,
you need to show that you get the same value regardless
of the path by which you approach the origin.
Polar coordinates can be helpful. In this case you
have to show that you get the same limit as r goes to
zero regardless of what happens to theta.
- Problem 5.
The answer you are asked to enter is admittedly a no-brainer.
So you are on the honor system for this problem.
How would you check your answer?
- Problems 8, 9, and 10
involve equations of planes
and equations and parameterizations of lines.
They are excellent exercises in careful logical reasoning.
- Problem 8. Hints:
- the cross product of the normal vectors of two intersecting
planes points in the direction of their line of intersection.
- to get symmetric equations for a line, solve for the parameter t.
- Problem 9.
Hints:
- What two points are on the line?
- If you know two points on a line, you have its direction vector.
- What initial point does their parametrization assume?
- Problem 10.
Hints:
- If you have a point on a plane and a normal vector
you can write down the equation of the plane.
- If you have two nonparallel vector that lie in a plane
then their cross product is perpendicular to the plane.
- If you know two points in a plane, then their difference
is a vector in the plane.