Practice Problems
Site: | Saylor Academy |
Course: | MA005: Calculus I |
Book: | Practice Problems |
Printed by: | Guest user |
Date: | Friday, February 4, 2022, 8:51 PM |
Description
Work through the odd-numbered problems 1-35. Once you have completed the problem set, check your answers.
Problems
1. In Fig. 9 , find the location of the number(s) "c" which Rolle's Theorem promises (guarantees).
Fig. 9
For problem 3, verify that the hypotheses of Rolle's Theorem are satisfied for each of the functions on the given intervals, and find the value of the number(s) " " which Rolle's Theorem promises exists.
5. Suppose you toss a ball straight up and catch it when it comes down. If is the height of the ball at time
, then what does Rolle's Theorem say about the velocity of the ball? Why is it easier to catch a ball which someone
on the ground tosses up to you on a balcony, than for you to be on the ground and catch a ball which someone on a balcony tosses down to you?
7. If , then
and
but
is never equal to
. Why doesn't this function violate Rolle's Theorem?
9. If I take off in an airplane, fly around for awhile and land at the same place I took off from, then my starting and stopping heights are the same but the airplane is always moving. Doesn't this violate Rolle's theorem which says there is an instant
when my velocity is ?
11. Use the corollary in problem 10 to justify the conclusion that the only root of is
. (Suggestion: What could you conclude about
if
had another root?)
In problems 13-15, verify that the hypotheses of the Mean Value Theorem are satisfied for each of the functions on the given intervals, and find the value of a number(s) " " which Mean Value Theorem guarantees.
17. If , then
and
but
is never equal to
. Why doesn't this function violate the Mean Value Theorem?
In problem 19, you are a traffic court judge. In each case, a speeding ticket has been given and you need to decide if the ticket is appropriate.
19. The driver in the next case heard the toll taker and says, "Your Honor, my average velocity on that portion of the toll road was only 17 miles per hour, so I could not have been speeding. I don't deserve a ticket".
21. Find a function so that
and
.
23. Find values for and
so that the graph of the parabola
is
(a) tangent to the line at the point
(b) tangent to the line
at the point
(c) tangent to the parabola at the point
25. Sketch the graphs of several members of the "family" of functions whose derivatives always equal . Give a formula which defines every function in this family.
27. Assume that a rocket is fired from the ground and has the upward velocity shown in Fig. 11. Estimate the height of the rocket when , and
seconds.
Fig. 11
29. Use the following information to determine an equation for , and
.
31. Define to be the area bounded by the
-axis, the line
, and a vertical line at
(Fig. 13).
(a) Find a formula for ?
(b) Determine
Fig. 13
33. Define to be the area bounded by the
-axis, the line
, and a vertical line at
(Fig. 15).
(a) Find a formula for ?
(b) Determine
Fig. 15
In problem 35, we have a list of numbers , and the consecutive differences between numbers in the list are
35. If and the difference between consecutive numbers in the list is always 3, find a formula for
?
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.2-Mean-Value-Theorem.pdf This work is licensed under a Creative Commons Attribution 3.0 License.
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
.