The accompanying graph shows the total distance s traveled by ...

Question

The accompanying graph shows the total distance s traveled by a bicyclist after t hours.

b. Estimate the bicyclist’s instantaneous speed at the times t=1/2, t=2, and t=3.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a car travels along a highway and its distances after T hours is shown. Estimate the car's instantaneous velocity at T equal to 1 hour. Below the problem, we're given a graph that has our distances in miles plotted on the Y axis, and our time T in hours plotted on the x-axis. We also have a curve plotted on this graph, and this curve begins at the origin and increases somewhat linearly. Until about T equal to 1.5. Then the slope of this curve decreases until it reaches a slope of 0, around T equal to 2. Then the slope increases again until about T equal to 2.75, and then our function increases linearly until T equal to 4. We're also given 4 possible choices as our answers. For choice A, we have 26 MPH, for choice B, we have 30 MPH. For choice C, we have 36 MPH, and for choice D, we have 40 MPH. So we need to find in velocity of our car at T equal to 1 hour. And if we look at a graph, we see that our function here is increasing somewhat linearly between T equal to 0.5 hours and T equal to 1.5 hours. So we'll place two points at the end points of that time interval. And if we look at the points that we've plotted on the graph, we'll see that when T is equal to 0.5 hours. Our distance traveled is approximately 15 miles. We'll have a point at 0.5.15. And at the other end point, when T is equal to 1.5 hours, our car has traveled approximately 45 miles. So we'll have another point at 1.5.45. Now, to get the instantaneous velocity at T equal to 1 hour of a car, we're going to set that instantaneous velocity equal to the slope of the sequent line between these two points. Since our function is somewhat linear, that means that the instantaneous velotte is going to be equal to the slope of this line. And so our speed V is going to be equal to the slope of our secret line, which is going to be our change in our y values, which is 45 minus 15, and that quantity is going to be divided by the change in our x values, which is 1.5 minus 0.5. And when we evaluate this expression, this is going to be equal to 30, and this has units of MPH since our distance was measured in miles and our time was measured in hours. So this is the answer that we are looking for for this problem. And if we compare our answer to our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

The accompanying graph shows the total distance s traveled by a bicyclist after t hours.

b. Estimate the bicyclist’s instantaneous speed at the times t=1/2, t=2, and t=3.

Key Concepts

Instantaneous Speed

Derivative

Graph Interpretation

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