Practice Problems
Work through the odd-numbered problems 1-35. Once you have completed the problem set, check your answers.
Answers
5. Rolle's Theorem asserts that the velocity will equal
at some point between the time the ball is tossed and the time it comes back down. The ball is not moving as fast when it reaches the balcony from below.
7. The function does not violate Rolle's Thm. because the function does not satisfy the hypotheses of the theorem: is not differentiable at
, a point in the interval
.
9. No. The velocity is not the same as the rate of change of altitude, since altitude is only one of the components of position. Rolle's Theorem only says there was a time when my altitude was not changing.
11. Since has no real roots. If
for a value of
other than 2, then by the corollary from Problem 8, we would have an immediate
contradiction.
13. (a) ,
,
.
implies that
.
(b) ,
.
implies that
.
15. (a) ,
implies
so
.
(b) ,
,
.
so any
between
and
will do.
17. The hypotheses are not all satisfied since does not exist at
which is between
and
.
19. Guilty. All we know is that at some point, but this does not prove that the motorist "could not have been speeding".
23. (a) . We need
and
so
and
and
.
(b)
and
so
and
and
.
(c) and
. There is no such
.
The point
is not on the parabola
25. , a family of "parallel" curves for different values of
.
27. . Assuming the rocket left the ground at
, we have
.