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Chapter 3 | Derivatives
Figure 3.21 (a) One section of the racetrack can be modeled by the function f ( x ) = x 3 +3 x 2 + x . (b) The front corner of the grandstand is located at (−1.9, 2.8). 1. Physicists have determined that drivers are most likely to lose control of their cars as they are coming into a turn, at the point where the slope of the tangent line is 1. Find the ( x , y ) coordinates of this point near the turn. 2. Find the equation of the tangent line to the curve at this point. 3. To determine whether the spectators are in danger in this scenario, find the x -coordinate of the point where the tangent line crosses the line y =2.8. Is this point safely to the right of the grandstand? Or are the spectators in danger? 4. What if a driver loses control earlier than the physicists project? Suppose a driver loses control at the point (−2.5, 0.625). What is the slope of the tangent line at this point? 5. If a driver loses control as described in part 4, are the spectators safe? 6. Should you proceed with the current design for the grandstand, or should the grandstands be moved?
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