| MAT 294 The Mathematics of Change I |
Fall 2005 Assignments |
| Context |
Comments |
|
| The rate of change of height
with respect to time (1) |
Very good symbolic
descriptions. Good explanation of which approximations are
underestimates and which are overestimates. Included units. Note that
the contextual description of the error is not correct (it is the bound
on the error). Good use of graphs and diagrams of the situation. |
pdf |
| The rate of change of height with respect to time (2) | The write-up is accurate but is missing symbolic representations for some of the questions. Also a diagram of the situation is missing. | pdf |
| The rate of change of area with
respect to radius
(1) |
The numbers in section 7 are the approximations, and the errors are described correctly, but U should not be described as ΔArea/Δradius. Some of the quantities are not described with respect to the circle, for example the overestimate for the rate is the are of the outer ring divided by the width of that ring. | pdf |
| The rate of change of area with respect to radius (2) | Includes a description of
which approximations are underestimates and which are overestimates. |
pdf |
| The rate of change of gravitational force with respect to distance (1) | The diagrams are correct but should be explained. In #4, Δd should go to zero, not 230 km. The point of interest is 230 km. In #5, it says we are approximating FG, but it should say dF/dd. Note that using d as a variable causes some confusion with this notation (having "dd" in the denominator). Part 6 is not clear - maybe not correct. Parts 7 and 8 and the accompanying figures are very good. | pdf |
| The rate of change of gravitational force with respect to distance (2) | The justification for the
over and underestimates should be based on the fact that the force
decreases slower and slower as the objects are separated. |
pdf |
| The rate of change of height with respect to volume (1) | Note that the first page says
the unknown value is 1.5, but should say that the unknown value is the
instantaneous rate at h=1.5. Also the approximations are teh ratios
listed next to the line "error = ..." The errors are the absolute value
terms listed next to the line "bound = ... ". The errors are
incoreectly describe as |1.5-1.6| below this. The bound is described
correctly at the bottom of the first page, accompanied by a good
number-line picture to illustrate this. |
pdf |
| The rate of change of height with respect to volume (2) | Note that the rate does not
depend on whether the bottle is draining or filling since any ΔV will
correspond to the same Δh regardless of how the water is changing. In the
write-up, V1 and V2 are called "average rates of change" but they
actually represent "changes in volume." In the diagram they refer to
the slices of volume shown above and below h=1.5. The error is not the
difference between the average rates, but is the difference between the
average and instantaneous rates. The error is expressed correctly
symbollically. |
pdf |