๐ Topic Summary
In calculus, displacement and total distance traveled are related concepts but have distinct meanings. Displacement refers to the change in position of an object and is a vector quantity, meaning it has both magnitude and direction. Total distance traveled, on the other hand, is the scalar quantity representing the sum of all distances the object has moved, regardless of direction. To find displacement, we evaluate the definite integral of the velocity function. To find total distance traveled, we integrate the absolute value of the velocity function over the given interval. ๐๐จ
๐ง Part A: Vocabulary
Match the terms with their correct definitions:
| Term |
Definition |
| 1. Velocity |
A. The integral of the absolute value of the velocity function. |
| 2. Displacement |
B. The rate of change of an object's position with respect to time. |
| 3. Total Distance Traveled |
C. A function whose derivative represents the velocity of an object. |
| 4. Position Function |
D. The change in position of an object. |
| 5. Integral |
E. A tool used to find the area under a curve, representing accumulation. |
โ๏ธ Part B: Fill in the Blanks
Fill in the missing words in the paragraph below:
The _________ is the change in an object's position, while the _________ is the sum of all distances traveled, regardless of direction. To find total distance traveled, you must integrate the _________ of the velocity function. This is because the integral of the velocity function gives _________ , which can be negative if the object changes direction.
๐ค Part C: Critical Thinking
Explain, in your own words, why it is necessary to integrate the absolute value of the velocity function to find the total distance traveled, but not for displacement. Provide an example to illustrate your point.