Problem 13 - Calculus Concepts

Question

Coal Prices The average price paid by the synfuel industry for a short ton of coal between 2002 and 2005 can be modeled as $$ p(t)=1.2 t^{2}-6.1 t+39.5 \text { dollars } $$ where \(t\) is the number of years since the beginning of \(2000 .\) a. Use the limit definition of the derivative to develop a formula for the rate of change of the price of coal used by the synthetic fuel industry. b. How quickly was the price of coal used by the synthetic fuel industry growing in the middle of \(2003 ?\)

Step-by-Step Solution

Answer

The price was growing at 2.3 dollars per year in mid-2003.

Step 1: Understanding the Problem

We are given a polynomial function \(p(t)\) that models the price of coal over time. Our task is to find the rate of change of this price with respect to time using the derivative, and then evaluate this rate at the midpoint of 2003.

Step 2: Derivative Using the Limit Definition

The limit definition of a derivative for function \(f(t)\) is given by \[f'(t) = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}.\] Here, our function is \(p(t) = 1.2t^2 - 6.1t + 39.5\). So, we need to find \[ p'(t) = \lim_{h \to 0} \frac{(1.2(t+h)^2 - 6.1(t+h) + 39.5) - (1.2t^2 - 6.1t + 39.5)}{h}.\] Simplifying: \[p(t+h) = 1.2(t^2 + 2th + h^2) - 6.1(t+h) + 39.5.\]Thus, \[p(t+h) - p(t) = 1.2(t^2 + 2th + h^2) - 1.2t^2 - 6.1(t+h) + 6.1t.\]Cancelling terms gives \[= 1.2(2th + h^2) - 6.1h.\]\[= (2.4t + 1.2h - 6.1)h.\]Passing the limit:\[p'(t) = \lim_{h \to 0} \frac{(2.4t + 1.2h - 6.1)h}{h} = \lim_{h \to 0} (2.4t + 1.2h - 6.1).\] \[= 2.4t - 6.1.\] Therefore, the derivative \(p'(t) = 2.4t - 6.1\).

Step 3: Evaluating the Derivative at 2003.5

The year 2003.5 corresponds to \(t = 3.5\), considering \(t = 0\) in the year 2000. Substituting \(t = 3.5\) into \(p'(t)\): \[p'(3.5) = 2.4 \times 3.5 - 6.1 = 8.4 - 6.1 = 2.3.\] Thus, the rate of change of the coal price at this time is \(2.3\) dollars per year.

Key Concepts

Limit Definition

The limit definition is a fundamental concept in calculus. It helps calculate the derivative, which represents the rate of change of a function. To grasp this, consider a basic function. The derivative is essentially the slope of the tangent line at any point on the function's curve. The limit definition for a derivative of a function \(f(t)\) is: \[f'(t) = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}.\]- **The essence**: This formula computes the slope by closing the gap \(h\) between two points on the function until they almost merge.- **Why use limits?**: By shrinking \(h\), we essentially approach an exact derivative, providing the instantaneous rate of change at any given point on the function.In this exercise, applying this concept to find the price derivative of coal requires us to substitute and simplify; doing so gives us a neat expression for the rate at any year \(t\).

Polynomial Function

Polynomial functions are expressions consisting of terms with variables raised to non-negative integer powers. They come in various degrees, depending on the highest power present. In the assignment, the polynomial function used is \(p(t) = 1.2t^2 - 6.1t + 39.5\), which models how coal prices change over time.- **Components**: - **Coefficient**: This is the number in front of the variable, like \(1.2\) and \(-6.1\) here. - **Variable**: Usually denoted as \(t\) or \(x\), representing the input value. - **Constant term**: The term without a variable, like \(39.5\). - **Characteristics**: They are smooth and continuous, making them predictable.- **Application**: In synthetic fuel industry, this specific function helps visualize and compute trends like price increases, a realistic measure to project costs over years.

Rate of Change

The rate of change tells us how a quantity varies with another. In calculus, it is encapsulated by the derivative. In this context, it helps to know how fast the price of coal changes over time for the synthetic fuel industry.- **Derivative role**: Here, \(p'(t) = 2.4t - 6.1\) gives the rate at which coal prices change per year.- **Understanding \(p'(3.5)\)**: Plugging \(t = 3.5\) (halfway through 2003) into \(p'(t)\), we see the price was changing at \(2.3\) dollars/year.- **Implication**: This information is valuable for financial planning and analysis within the industry. By predicting trends, businesses can better handle cost fluctuations.

Synthetic Fuel Industry

The synthetic fuel industry refers to sectors producing fuels through chemical processes, often using coal or biomass as feedstock. It's an alternative to traditional fossil fuels, usually seen as a cleaner or more sustainable option. - **Connection to calculus**: Understanding how fuel prices vary is essential to economic modeling and budgeting within this domain. - **Significance of polynomial models**: Calculating derivatives of price functions, as done here, enables stakeholders to derive actionable insights. - **Industry challenges**: With coal as a primary input, price volatility can impact production costs and overall market viability. Thus, using derivatives to gauge price trends helps in strategic decision-making and fostering industry resilience.