Projects - Cengage Learning

C-60 Appendix C
y
C
R
L
Figure B
O
¨
A
P
x
to express the length of AP in terms of R, L, and u. Since, in Figure B, f(u)
equals the length of line segment OP we have
f(u) R cos u 2L 2 R 2 sin 2 u
How do the maximum and minimum value for f compare with your answers
from Exercise 1? Explain why this derivation would work when point C is in
the second quadrant.
Exercise 3 For given values of L and R, graph the indicated functions, for
0 u 4p, on the same set of axes and zoom in near the maximum, minimum,
and any other interesting points. Write short notes of your observations.
f(u) is the function derived in the previous exercise.
(a) Let L 20 cm and R 4 cm. Graph y L, y L R cos u, and y f(u).
(b) Let L 20 cm and R 10 cm. Graph y L, y L R cos u, and y f(u).
(c) Let L 20 cm and R 10 cm. Graph y 1.868 R R cos u, and y f(u).
(d) Let L 20 cm and R 10 cm. Graph y 1.868 R R cos u
0.1339 R cos 2u, and y f(u).
In parts (c) and (d) the functions graphed with f are the second and third partial
sums of the Fourier series for f. If you did the project on Fourier series at the
end of Section 8.3,* note how much better the third partial sum of the Fourier
series is as an approximation to f here than in Figure B of that project.
Finally, let’s apply the result of Exercise 2 to a typical automotive situation.
Exercise 4 Given that segment OC is rotating at 3000 revolutions per minute
(rpm) let g(t) be the x-coordinate of the point P at time t seconds and find a
formula for g(t). Hint: Find u in terms of t and substitute into the formula
for f(u). What would the formula be for k revolutions per minute?
*Precalculus: A Problems-Oriented Approach, 7th edition