Science:Math Exam Resources/Courses/MATH103/April 2014/Question 01 (c) ii
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Question 01 (c) ii 

The graph below depicts the velocity of a car traveling along a straight road as a function of time, . Initial time is .
Determine whether the following statement is True, False, or Indeterminate: The car returns to its initial position when . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

What does the (signed) area between the graph and the axis represent? 
Hint 2 

If is the velocity at time , then is the distance covered during the time interval . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. The car’s distance from its starting point at time is given by the total area between the graph and the axis accumulated by time (where regions of the graph underneath the axis count towards negative area). We can see from the graph that the area accumulated over the time interval is the same as that accumulated in the interval , but the latter is counted as the negative of the former. Thus, the total area accumulated over is . The statement is true. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. The car’s distance from its starting point at time is given by the integral . When , we get
since The statement is true. 