Practice Problems
Site: | Saylor Academy |
Course: | MA005: Calculus I |
Book: | Practice Problems |
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Date: | Monday, October 21, 2024, 10:57 PM |
Description
Work through the odd-numbered problems 1-43. Once you have completed the problem set, check your answers.
Problems
1. Label all of the local maximums and minimums of the function in Fig. 13. Also label all of the critical points.
Fig. 13
In problems 3-13, find all of the critical points and local maximums and minimums of each function.
15. Sketch the graph of a continuous function $\mathrm{f}$ so that
(a) , and the point is a relative maximum of .
(b) , , and the point is a relative minimum of .
(c) ,
is not differentiable at , and the point is a relative maximum of .
(d) , is not differentiable at , and the point is a relative minimum of .
(e) , and the point is not a relative minimum or maximum of .
(f) , is not differentiable at , and the point is not a relative minimum or maximum of .
In problems 17-25, find all critical points and local extremes of each function on the given intervals.
Fig. 15
29. Find the value for so the box in Fig. 17 has the largest possible volume? The smallest volume?
Fig. 17
31. Suppose you are working with a polynomial of degree on a closed interval.
(a) What is the largest number of critical points the function can have on the interval?
(b) What is the smallest number of critical points it can have?
(c) What
are the patterns for the most and fewest critical points a polynomial of degree on a closed interval can have?
33. Suppose and . What can we conclude about the point if
(a) for , and for ?
(b)
for , and for ?
(c) for , and for ?
(d) for , and for ?
35. is a continuous function, and Fig. 18 shows the graph of
(a) Which values of are critical points?
(b) At which values of is a local maximum?
(c) At which values of
is a local minimum?
Fig. 18
37. State the contrapositive form of the Extreme Value Theorem.
39. Imagine the graph of . Does have a minimum value for in the interval ?
(a)
(b)
(c)
(d)
(e)
41. Imagine the graph of . Does have a minimum value for in the following intervals?
(a)
(b)
(c)
(d)
(e)
43. Define to be the slope of the line through the points and , in Fig. 21 .
(a) At what value of is minimum?
(b) At what value
of is maximum?
Fig. 21
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.1-Finding-Maximums-and-Minimums.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.
Answers
1. Local maximums at , , , and . Global maximums at and .
Local minimums at , , , and .
Global minimum at .
3. so which is defined for all values of . when so is a critical number. There are no endpoints.
The only critical number is , and the
only critical point is which is the global (and local) minimum.
5. so which is defined for all values of . when so the values are critical numbers. There are no endpoints.
has local and global maximums at , and global and local minimums at .
7. so which is defined for all values of . when and so and are critical numbers. There are no endpoints. The only critical points are which is a local maximum and which is a local minimum. When the interval is the entire real number line, this function does not have a global maximum or global minimum.
9. so which is defined for all values of . when and so and are critical numbers. There are no endpoints. The only critical points are which is a local maximum and which is a local minimum. When the interval is the entire real number line, this function does not have a global maximum or global minimum.
11. so which is defined for all values of . is always positve (why?) so
is never equal to . There are no endpoints. The function
is always increasing and has no critical numbers, no critical points, no local or global maximums or minimums.
13. so which is defined for all values of . when so is a critical number. There are no endpoints. The only critical point is which is a local and global maximum. When the interval is the entire real number line, this function does not have a local or global minimum.
15. See Fig. 3.1P15
17. on so which is defined for all values of . when so is a critical number. The endpoints are and which are also critical numbers. The critical points are which is the local and global minimum, which is a local and global maximum, and which is a local maximum.
19. on so which is defined for all values of . when so is a critical number. The endpoints are and which are also critical numbers. The critical points are which is a local and global maximum, which is not a local or global maximum or minimum, and which is a local and global minimum.
21. on so which is defined for all values of . when and so these are critical numbers. The endpoints and are also critical numbers. The critical points are which is a local and global minimum on , the point which is a local and global maximum on , and the point which is a local and global minimum on .
23. so which is defined for all values of . when and in the interval so each of these values is a critical number. The endpoints and are also critical numbers. The critical points are which is a local minimum, which is a local and global maximum, and which is a local and global minimum. ( too, but is not in the interval ).
25. so which is defined for all values of . when but is not in the interval so is a not a critcal number. The endpoints and are critical numbers. The critical points are which is a local and global maximum, and which is a local and global minimum.
27.
A maximum is attained when .
29. for
cubic units is the largest volume.
Smallest volume is which occurs when and .
31. (a) . The endpoints and two values of for which .
(b) . The endpoints.
(c) At most . The endpoints and the interior points for which . At least . The
endpoints.
33. (a) local minimum at
(b) no extrema at
(c) local maximum at
(d) no extrema at
37. If f does not attain a maximum on or does not attain a mimimum on , then must have a discontinuity on .
39 (a) yes,
(b) no
(c) yes,
(d) no
(e) yes,
41. (a) yes,
(b) yes,
(c) yes,
(d) yes,
(e) yes,