Problem 59
Question
A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function $$S(t)=-0.00003237 t^{5}+0.0009037 t^{4}-0.008956 t^{3} +0.03629 t^{2}-0.04458 t+0.4074$$ where \(t\) is measured in years since August of \(1993 .\) Estimate the times when sugar was cheapest and most expensive during the period \(1993-2003 .\)
Step-by-Step Solution
Verified Answer
The sugar was cheapest in 1999 and most expensive in 1993.
1Step 1: Find the Derivative of the Function
To find when sugar was cheapest or most expensive, we must locate the local minima and maxima of the function. First, find the derivative of \( S(t) \). This derivative \( S'(t) \) will help us identify where the slope is zero, indicating potential points of minimum or maximum price.
2Step 2: Calculate the Derivative
The derivative \( S'(t) \) is calculated as follows:\[S'(t) = -0.00016185t^4 + 0.0036148t^3 - 0.026868t^2 + 0.07258t - 0.04458\]
3Step 3: Solve for Critical Points
Set \( S'(t) = 0 \) to find the critical points:\[-0.00016185t^4 + 0.0036148t^3 - 0.026868t^2 + 0.07258t - 0.04458 = 0\]Solve this quartic equation to find values of \( t \). This can usually be done using numerical methods or a graphing calculator.
4Step 4: Evaluate Critical Points
Evaluate \( S(t) \) at the critical points found in Step 3, as well as at the endpoints of the given period \( t = 0 \) (1993) and \( t = 10 \) (2003). Compare these values to find the lowest and highest to determine when the sugar price was cheapest and most expensive.
5Step 5: Compare Values to Determine Extremes
Compare the calculated \( S(t) \) values from Step 4. The smallest value indicates the time when sugar was cheapest, and the largest value indicates when sugar was most expensive. Ensure to specify the years by adding \( t \) to 1993.
Key Concepts
Critical PointsDerivativeLocal Minima and MaximaPolynomial FunctionNumerical Methods
Critical Points
Critical points are essential in determining where a function reaches its minimum or maximum values. These are the points on a graph where the slope of the function is zero or undefined. In simpler terms, a critical point occurs where the derivative of the function equals zero, meaning the tangent to the curve is horizontal.
These points are potential locations for local minima or maxima. In the context of the sugar price model, finding the critical points involves solving for where the derivative of the function, \( S'(t) \), equals zero. This will give us the times when the price may have been cheapest or most expensive between 1993 and 2003.
Remember, once you find a critical point, you must analyze it further to determine whether it is a minimum, maximum, or a point of inflection where the function changes behavior but without reaching a min or max.
These points are potential locations for local minima or maxima. In the context of the sugar price model, finding the critical points involves solving for where the derivative of the function, \( S'(t) \), equals zero. This will give us the times when the price may have been cheapest or most expensive between 1993 and 2003.
Remember, once you find a critical point, you must analyze it further to determine whether it is a minimum, maximum, or a point of inflection where the function changes behavior but without reaching a min or max.
Derivative
The derivative of a function gives you the rate of change at any point along that function. This rate of change is steep when the function is rising or falling quickly and zero when the function levels out at a flat point. Calculating the derivative is crucial for finding local minima and maxima.
For polynomial functions like our sugar price model, the derivative can be computed by applying the power rule. This involves reducing the power of each term by one and multiplying by the original exponent.
For instance, the derivative of \( S(t) = -0.00003237 t^{5} \) would be \( -0.00016185 t^{4} \). By applying this rule to each term, we derive \( S'(t) \), which pinpoints the rate at which the sugar price changes over time.
This derived function, \( S'(t) \), is then used to find points where \( S(t) \) might have peaked or troughed.
For polynomial functions like our sugar price model, the derivative can be computed by applying the power rule. This involves reducing the power of each term by one and multiplying by the original exponent.
For instance, the derivative of \( S(t) = -0.00003237 t^{5} \) would be \( -0.00016185 t^{4} \). By applying this rule to each term, we derive \( S'(t) \), which pinpoints the rate at which the sugar price changes over time.
This derived function, \( S'(t) \), is then used to find points where \( S(t) \) might have peaked or troughed.
Local Minima and Maxima
Local minima and maxima are the points in a function where it reaches a low or high value within a specific interval. In other words, a local maximum occurs where the function value is greater than the surrounding points, and a local minimum occurs where the function value is less than the surrounding points.
These points are vital in optimization problems because they indicate the best or worst possible scenario within a certain range. After finding critical points by setting the derivative to zero, we determine which of these points correspond to local minima and maxima.
By examining the second derivative or using a first-derivative test (checking the sign change of \( S'(t) \)), we can verify the nature of each critical point—whether it’s a minimum or a maximum.
This information supports decisions or predictions, such as finding the cheapest and most expensive years for sugar prices in our problem.
These points are vital in optimization problems because they indicate the best or worst possible scenario within a certain range. After finding critical points by setting the derivative to zero, we determine which of these points correspond to local minima and maxima.
By examining the second derivative or using a first-derivative test (checking the sign change of \( S'(t) \)), we can verify the nature of each critical point—whether it’s a minimum or a maximum.
This information supports decisions or predictions, such as finding the cheapest and most expensive years for sugar prices in our problem.
Polynomial Function
Polynomial functions like \( S(t) \) are expressions made up of terms consisting of variables raised to whole-number exponents. They can model a wide range of practical situations because of their flexibility and simplicity in calculation.
The given function, \( S(t) = -0.00003237 t^{5} + 0.0009037 t^{4} - 0.008956 t^{3} + 0.03629 t^{2} - 0.04458 t + 0.4074 \), is a fifth-degree polynomial.
The given function, \( S(t) = -0.00003237 t^{5} + 0.0009037 t^{4} - 0.008956 t^{3} + 0.03629 t^{2} - 0.04458 t + 0.4074 \), is a fifth-degree polynomial.
- It has real coefficients, making it straightforward to analyze for real values of \( t \).
- The highest exponent, 5, determines the nature of the polynomial's behavior, impacting how many local minima and maxima it might have.
Numerical Methods
Numerical methods are essential for dealing with complex equations that are difficult to solve analytically. These mathematical tools provide approximate solutions that are often good enough for practical purposes, especially in real-world applications like economics or engineering.
In our example, we have a quartic equation from setting the derivative to zero. Solving such equations directly can be challenging, and numerical methods, such as the Newton-Raphson method or graphing calculator tools, become very useful.
In our example, we have a quartic equation from setting the derivative to zero. Solving such equations directly can be challenging, and numerical methods, such as the Newton-Raphson method or graphing calculator tools, become very useful.
- These methods iteratively refine guesses to zero in on an accurate solution within a desired level of precision.
- They allow us to estimate critical points in cases where the algebra may be too complex to simplify directly.
Other exercises in this chapter
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