Practice Problems
Site: | Saylor Academy |
Course: | MA005: Calculus I |
Book: | Practice Problems |
Printed by: | Guest user |
Date: | Tuesday, October 22, 2024, 4:41 AM |
Description
Work through the odd-numbered problems 1-49. Once you have completed the problem set, check your answers.
Problems
29. Find a point on the graph of so the tangent line to at goes through the origin.
31. Rumor. The percent of a population, , who have heard a rumor by time is often modeled for some positive constant A. Calculate how fast the rumor is spreading, .
In problems 33 – 41, find a function with the given derivative.
Problems 43 – 47 involve parametric equations.
43. At time minutes, robot is at and robot is at .
(a) Where is each robot when and ?
(b) Sketch the path each robot follows during the first minute.
(c) Find the slope of the tangent line, , to the path of each robot at minute.
(d) Find the speed of each robot at minute.
(e) Discuss the motion of a robot which follows the path for 20 minutes.
45. For the parametric graph in Fig. 9, determine whether and are positive, negative or zero when and .
(b) Find , the tangent slope , and speed when and .
(The graph of is called a cycloid and is the path of a light attached to the edge of a rolling wheel with radius ).
49. Describe the path of a robot whose location at time is
(c) Give the parametric equations so the robot will move along the same path as in part (a) but in the opposite direction.
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.6-Some-Applications-of-the-Chain-Rule.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.
Answers
29. . Let . Then we must satisfy with and so and .
(e) This robot moves on the same path , but it moves to the right and up for about 1.57 minutes, reverses its direction and returns to its starting point, then continues left and down for another 1.57 minutes, reverses, and continues to oscillate.
(a) graph
If , the motion is oscillatory along the -axis. If , the motion is oscillatory along the -axis.