91Ó°ÊÓ

In calculus, derivative analysis helps us understand the behavior of a function, particularly its rate of change. When dealing with functions like sales approximations, it's important to know if they are increasing or decreasing, which is where the first derivative comes into play.
To find the first derivative of a function, such as the sales function, we apply differentiation rules like the power rule, which involves multiplying the exponent by the coefficient and then reducing the exponent by one. This results in a new function that shows how the original function changes at any given point.
For the sales function \( S(t) = 0.46 t^3 - 2.22 t^2 + 6.21 t + 17.25 \), the derivative is \( S'(t) = 1.38t^2 - 4.44t + 6.21 \). Here, \( S'(t) \) helps determine if the sales are rising or falling over time. Analyzing the sign of this derivative tells us whether the sales function is increasing or decreasing on the specified interval. If the entire first derivative is positive, as it is from the calculation, then the original function is increasing.
This simple yet profound analysis reveals the growth trend of sales over the specific timeline and is crucial for businesses planning future strategies.

The quadratic formula is a versatile and essential tool for finding the roots of quadratic equations, which are polynomials of degree two. A quadratic function in standard form is given by \( ax^2 + bx + c = 0 \). The roots or solutions of this equation are found through the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula is applicable when the expression inside the square root, called the discriminant \( b^2 - 4ac \), is evaluated. If the discriminant is positive, the quadratic equation has two distinct real roots. If it is zero, there is exactly one real root. However, if the discriminant is negative, as in our original problem, it indicates that no real roots exist and the parabola does not intersect the x-axis.
This result is particularly important because it helps us determine a function's range or verify the sign of the expression over a desired interval. In the context of our sales function's derivative \( S'(t) = 1.38t^2 - 4.44t + 6.21 \), a negative discriminant confirms that the derivative remains positive over its domain, implying a consistent increase in sales over time.

Increasing functions are functions whose output values rise as the input values increase. This property indicates a positive trend and is fundamental in fields like economics to understand growth metrics.
For a function to be increasing on an interval, its derivative should be positive across the entire interval. Derivative analysis, as seen earlier, helps confirm this. When the first derivative \( S'(t) \) of a function is consistently positive, as with the sales function \( S'(t) = 1.38t^2 - 4.44t + 6.21 \), it concludes that the function is strictly increasing over the given time period from \( t = 0 \) to \( t = 4 \).
Understanding whether a function is increasing is crucial for businesses and analysts. It helps in decision-making by providing insight into trends, enabling predictions, and allowing stakeholders to see the bigger picture. By knowing that the function is increasing, companies can confidently infer that sales will likely grow, which can influence investment and marketing strategies.