91Ó°ÊÓ

Polynomial functions are expressions built from variables and constants using addition, subtraction, multiplication, and non-negative integer exponentiation of variables. In the exercise about Lake Tahoe, the level of the lake is represented by a polynomial function of degree 3:
\[L(d) = (-5.345 \times 10^{-7})d^3 + (2.543 \times 10^{-4})d^2 - 0.0192d + 6226.192\]This equation is made up of several terms:

• A cubic term \((-5.345 \times 10^{-7})d^3\), which indicates the function is a cubic polynomial with a leading coefficient of \(-5.345 \times 10^{-7}\).
• A quadratic term \((2.543 \times 10^{-4})d^2\) that adds curvature to the graph.
• A linear term \(-0.0192d\), contributing to the slope of the function.
• A constant term \((6226.192)\) representing the lake level when \(d = 0\).

Polynomial functions like this can model complex phenomena, such as lake levels, over a period of time.

The derivative of a function reflects how the function's output value changes as its input changes. In calculus, finding the derivative is a crucial step in analyzing the behavior of polynomial functions like the one modeling Lake Tahoe.
For the function \(L(d)\), we use the power rule to calculate its derivative \(L'(d)\):
\[L'(d) = \frac{d}{dd}(-5.345 \times 10^{-7})d^3 + \frac{d}{dd}(2.543 \times 10^{-4})d^2 - \frac{d}{dd}0.0192d\]Using the power rule \((nx^{n-1})\), the derivative becomes:

• \(L'(d) = -1.6035 \times 10^{-6}d^2 + 5.086 \times 10^{-4}d - 0.0192\)

Finding the derivative enables us to locate critical points, helping determine where the lake level changes its behavior.

Critical points occur where the derivative of a function is zero or undefined. These are the points where the function's graph may change direction, indicating potential maxima, minima, or points of inflection.
In our exercise, we need to find the critical points of the lake level function \(L(d)\) by setting its derivative \(L'(d)\) to zero:
\[ -1.6035 \times 10^{-6}d^2 + 5.086 \times 10^{-4}d - 0.0192 = 0 \]Solving this quadratic equation will provide the values of \(d\) where the lake level might have relative extrema (maximum or minimum points).By identifying these critical points, we can investigate their corresponding lake levels and determine how the lake behaved between the given times.

Relative extrema are the highest or lowest points in a specific section of a function's graph, specifically known as local maxima and minima. When examining the lake level, we calculate the values of \(L(d)\) at the critical points and endpoints of the interval.
The comparison will help identify where the lake level reaches local highs and lows between \(d = 1\) and \(d = 304\).

• A relative maximum occurs where \(L(d)\) is higher than at neighboring points.
• A relative minimum is where \(L(d)\) is lower than nearby points.

To decide if these are maxima or minima, we compare their corresponding lake levels. This helps us describe the behavior of Lake Tahoe's water level over the specified timeframe and assess whether it fell below the federally mandated limit.