Differentiation Rules: Apply It

  • Use basic rules to differentiate functions that involve adding, subtracting, scaling, and raising to powers
  • Apply specific rules to find derivatives of functions multiplied or divided by each other
  • Use a combination of rules to calculate derivatives for polynomial and rational functions

Racetrack Safety at the Formula One Grandstand

Formula One car races can be very exciting to watch and attract a lot of spectators. Formula One track designers have to ensure sufficient grandstand space is available around the track to accommodate these viewers. However, car racing can be dangerous, and safety considerations are paramount. The grandstands must be placed where spectators will not be in danger should a driver lose control of a car (Figure 3).

A photo of a grandstand next to a straightaway of a race track.
Figure 3. The grandstand next to a straightaway of the Circuit de Barcelona-Catalunya race track, located where the spectators are not in danger.

Safety is especially a concern on turns. If a driver does not slow down enough before entering the turn, the car may slide off the racetrack. Normally, this just results in a wider turn, which slows the driver down. But if the driver loses control completely, the car may fly off the track entirely, on a path tangent to the curve of the racetrack.

Suppose you are designing a new Formula One track. One section of the track can be modeled by the function [latex]f(x)=x^3+3x^2+x[/latex] (Figure 4). The current plan calls for grandstands to be built along the first straightaway and around a portion of the first curve. The plans call for the front corner of the grandstand to be located at the point [latex](-1.9,2.8)[/latex]. We want to determine whether this location puts the spectators in danger if a driver loses control of the car.

This figure has two parts labeled a and b. Figure a shows the graph of f(x) = x3 + 3x2 + x. Figure b shows the same graph but this time with two boxes on it. The first box appears along the left-hand side of the graph straddling the x-axis roughly parallel to f(x). The second box appears a little higher, also roughly parallel to f(x), with its front corner located at (−1.9, 2.8). Note that this corner is roughly in line with the direct path of the track before it started to turn.
Figure 4. (a) One section of the racetrack can be modeled by the function [latex]f(x)=x^3+3x^2+x[/latex]. (b) The front corner of the grandstand is located at [latex](-1.9,2.8)[/latex].