Video Transcript
An object moves along a straight line. On the graph, the blue line shows the displacement 𝑑 of the object from its starting position over time 𝑡. Which of the three dashed lines is a tangent to the blue line at 𝑡 equals four seconds? Is it (a) the orange line, (b) the red line, or (c) the purple line?
Let’s first recall the definition of a tangent. The tangent is a straight line that touches a curve and has the same slope as the curve at the point where they touch. So let’s now look at our graph which has time 𝑡 on the horizontal axis and displacement 𝑑 on the vertical axis. We are asked to find the tangent at a time of four seconds. So let’s first find 𝑡 equals four seconds on the horizontal axis. Now working upwards from the correct position on the horizontal axis, we find the blue line. And we see that all three of the dashed lines are touching the blue line at this point. So the question we need to answer is, which one of those dashed lines has the same slope as the blue line at this point?
Now, if we start with the orange line, we can see that just before 𝑡 equals four seconds, it is below the blue line and just after 𝑡 equals four seconds, it is above the blue line. Its slope is much steeper than that of the blue line, and so that cannot be the tangent. Next, let’s look at the red line. Just before 𝑡 equals four seconds, the red line is above the blue line, and just after 𝑡 equals four seconds, it is below the blue line. Its slope is much shallower than the blue line, and therefore, that also cannot be the tangent.
Now let’s look at the purple line. We can see that just before 𝑡 equals four seconds, the purple line is just above the blue line. And then just after 𝑡 equals four seconds, it remains just above the blue line. It looks like it has about the same slope as the blue line around this point. Therefore, the tangent to the blue line at 𝑡 equals four seconds is (c) the purple line.
