Part A
Question 1: To determine if the curve for the power of the engine is a function, Varun can use the Vertical Line Test. This test states that if a vertical line intersects the curve at more than one point, then the relation is not a function. If every vertical line drawn through the graph of the power curve intersects it at most once, then it qualifies as a function.
Question 2: The most appropriate function type to model the power curve is likely Exponential or Quadratic. The power output of an engine typically increases rapidly with engine speed up to a certain point and then may plateau or decrease, resembling the behavior of a quadratic function. An exponential function might be less applicable since it suggests continuous growth without a maximum point. Therefore, Quadratic functions tend to model such relationships well, particularly because they can depict a curve that rises to a maximum and then declines.
Question 3: The domain of Varun's chosen function to model the power graph should be the set of all possible engine speeds within the operational limits of the vehicle, usually starting from 0 up to the maximum RPM of the engine. Therefore, the domain can be expressed as: \[ D = [0, \text{max_RPM}] \]
Question 4: Varun can determine if the power is changing at a constant rate by analyzing the slope of the curve. If the slope (the first derivative of the function) is constant, the power is changing at a constant rate. If the slope changes (either increasing or decreasing), then the power is not changing at a constant rate—indicating variable acceleration or performance.
Question 5: A simple function to estimate the relation between engine speed (s) and power (P) could be written as: \[ P(s) = -a(s - b)^2 + c \] where \( a \) represents the rate at which power increases, \( b \) is the engine speed at which power is maximized, and \( c \) is the maximum attainable power.
Question 6: A piecewise linear function could improve the model because it allows Varun to represent sections of the power curve with linear segments. This means he can capture the variations more accurately, especially if the power delivery is not smooth across all engine speeds (i.e., there might be shifts in performance at various RPM ranges). It allows for more flexible modeling of real-world scenarios, such as certain RPM ranges where power output may drop suddenly or increase sharply.
Part B
Question 7: A Quadratic function would make a good approximation for the torque curve. This is because torque, like power, typically increases with engine speed to a point (the peak torque) before it begins to decrease. The shape of a quadratic function can effectively represent this behavior, with a distinct maximum point reflecting the torque peak, which aligns well with how an engine performs.
Question 8: The approximate maximum torque value can usually be found at a specific RPM point where the curve peaks. For example, if the maximum torque occurs at around 4,000 RPM (this is just a numerical example—varies in actual graphs), the maximum torque could be something like 300 N•m, depending on the design of the engine. This peak corresponds to the vertex of the quadratic function, indicating the maximum point of torque output, which is often where the engine operates most efficiently during acceleration.