This problem has difficulty distinguishing between a correct answer and an incorrect answer that is close numerically. Often, this is not a big issue, but here the incorrect answer that many students get reflects a serious conceptual error.
Let me quote an e-mail from my colleague, Margaret Symington. I don't know that there is an easy way to resolve this issue without just re-writing the whole problem. I would appreciate any good ideas.
"The problem is:
The axis of a light in a lighthouse is tilted. When the light points east, it
is inclined upward at x degree(s). When it points north, it is inclined upward
at y degree(s). What is its maximum angle of elevation?
If x and y are the angles of inclination that are given, a "cheap" answer is:
While that has some merit in that it stems from recalling how a gradient is
built out of its component directional derivatives, the component directional derivatives are not x and y, but tan(x) and tan(y)... and we have to convert
from degrees to radians and back to degrees. So the correct answer is
But for small angles (between 1 and 8 degrees) that are not close to being
equal, these two functions of x and y are very close. To see what was going
on I plotted the two functions on three different domains. On the smallest
domain you can barely distinguish the two near x=y=10."
multivariate problem: setVmultivariable6Gradient/ur_vc_6_17.pg
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