Practice Problems
Site: | Saylor Academy |
Course: | MA005: Calculus I |
Book: | Practice Problems |
Printed by: | Guest user |
Date: | Saturday, September 7, 2024, 7:46 PM |
Description
Work through the odd-numbered problems 1-41. Once you have completed the problem set, check your answers.
Practice Problems
In problems 1 – 3, use the values in Table 1 to estimate the areas.
x | f(x) | g(x) | h(x) |
---|---|---|---|
0 | 5 | 2 | 5 |
1 | 6 | 1 | 6 |
2 | 6 | 2 | 8 |
3 | 4 | 2 | 6 |
4 | 3 | 3 | 5 |
5 | 2 | 4 | 4 |
6 | 2 | 5 | 2 |
1. Estimate the area between and
for
.
3. Estimate the area between and
for
.
5. Estimate the area of the island in Fig. 13.
In problems 7 – 17, sketch the graph of each function and find the area between the graphs of and
for
in the given interval.
In problems 19 – 21, use the values in Table 1 to estimate the average values.
19. Estimate the average value of on the interval
.
21. Estimate the average value of on the interval
.
In problems 23 – 31, find the average value of on the given interval.
33. Calculate the average value of on the interval
for
. What is the pattern?
35. Fig. 15 shows the number of telephone calls per minute at a large company. (a) Estimate the average number of calls per minute from 8 am to 5 pm. (b) From 9 am to 1 pm.
37. (a) How much work is done lifting a 20 pound bucket from the ground to the top of a 30 foot building with a cable which weighs 3 pounds per foot? (b) How much work is done lifting the same bucket from the ground to a height of 15 feet with the same cable?
39. (a) How much work is done lifting a 10 pound calculus book from the ground to the top of a 30 foot building with a cable which weighs 2 pounds per foot? (b) From the ground to a height of 10 feet? (c) From a height of 10 feet to a height of 20 feet?
41. How much work is done lifting an 60 pound injured child to the top of a 15 foot hole using a stretcher weighing 10 pounds and a cable which weighs 2 pound per foot?
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.8-First-Applications-of-Definite-Integrals.pdf This work is licensed under a Creative Commons Attribution 3.0 License.
Answers
1. between 11 (using left endpoints of intervals) and 6 (using right endpoints)
3. between 4 (using left endpoints of intervals) and 6 (using right endpoints)
5. Using left endpoint widths:
Right endpoint widths and average widths
give the same result
. All of these are reasonable methods for estimating the area of the island.
19. Estimate using midpoints of unit intervals: . About
.
21. Estimate using midpoints of unit intervals: . About 5.
33.
35.
37.
39.