. The power to propel an airplane (or even a bird) forward at a velocity, v, (hence the velocity is positive) is P = Av^3 +(BL^2/ v) where A and B are positive constants specific to the particular aircraft (or bird) and L is the lift, the upward force supporting the weight of the plane (or bird). What is the implied domain of this function? Think about lower and upper bounds for velocity.
..? How is this helpful
To determine the implied domain of the given function P = Av^3 + (BL^2 / v), we need to consider the lower and upper bounds for the velocity, v.
First, let's consider the lower bound. Since the velocity is described as positive, it implies that the velocity cannot be less than zero since the concept of velocity is tied to direction and a negative velocity would mean moving in the opposite direction. Thus, the lower bound for the velocity is 0.
Next, let's consider the upper bound. There is no specific information given about an upper limit for the velocity. Therefore, in the absence of further information or constraints, we can assume that the velocity could potentially be infinite, meaning there is no upper bound.
In summary, the implied domain of the function is when the velocity, v, is greater than or equal to zero.
To determine the implied domain of the given function, we need to consider the restrictions and limitations on the variables involved. In this case, we are dealing with the velocity (v) of the airplane or bird.
First, let's analyze the different components of the function:
1. Av^3: This term represents the power required for propulsion and it is multiplied by v^3. Since v is raised to the power of 3, it indicates that the power is a cubic function of velocity.
2. (BL^2/v): This term represents the power required for lift and it involves a division by velocity (v). This term is inversely proportional to velocity.
Considering these components, we can determine the implied domain as follows:
1. For the term Av^3, there are no explicit restrictions on the input velocity (v). Therefore, the implied domain for this term is (-∞, +∞), which means it is defined for all real numbers.
2. For the term (BL^2/v), since it involves a division by velocity (v), we need to check for any potential division by zero. If v = 0, it would result in undefined values. To avoid this, we need to ensure that v ≠ 0.
Therefore, the implied domain for this term is (-∞, 0) U (0, +∞), which means it is defined for all real numbers except for zero.
Combining the ranges for both terms, we can conclude that the overall implied domain of the function is (-∞, 0) U (0, +∞), which means the function is defined for all real numbers except zero.
since v is the forward velocity, v > 0
I guess the upper bound would depend on the available power for existing engines.