Varun is typically a quiet guy. He keeps to himself and doesn’t talk much, unless it is about cars. He wants to own his own mechanic shop one day, so he spends his free time viewing videos on channels about automotive repair and how to start a business. He studies electricity since he believes that electric vehicles are going to become the main mode of transportation in the future. He senses that everything in the world is changing quickly, so he is always trying to learn more about what he loves to do. Nobody is asking him to do this work; he is taking the initiative himself because he has goals to achieve great things in the near future.

Question

Varun puts an old car that he is tuning up on a device called a dynamometer, or dyno for short. It measures the power and torque that is supplied to the wheels of the car. As the gas pedal is pressed and the engine spins faster, the amount of power and torque changes. The dyno records this information and displays it as a graph.

Part A
Question 1

Varun first looks at the curve for the power of the engine, measured in horsepower (hp). This measures the rate at which the engine is transferring energy to the wheels. Explain how Varun can determine if the curve is a function.

Question 2

Varun can think of 3 different types of functions that might match the power curve and writes them down in a list.

Linear
Exponential
Quadratic
Which of these types of functions most closely matches the power curve and why?

Question 3

What is the domain of the function that Varun should choose to model the power graph?

Question 4

Based on the graph, how can Varun determine if the power is changing at a constant rate?

Question 5

Write a function that estimates the relation between engine speed and power. Use P for power and s for rotational speed.

Question 6

Varun remembers another type of function: a piecewise linear function. Explain how this type of function could be an improvement over the one previously chosen.

Part B
Then Varun looks at the curve for the torque of the engine, measured in Newton-meters (N•m). This is a measure of how hard the wheels get twisted as the engine spins them, similar to how hard it is trying to unscrew a tight lid or screw.

Question 7

What type of function would make a good approximation to the torque curve? Pick one from the list below and explain why it is a good choice.

Linear
Exponential
Quadratic
Question 8

What is the approximate maximum torque value and what rpm value does it occur at? What key feature of the function does this correspond to for this type of function?

Answer 1

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.