Problem 75 - Calculus and its applications

Question

A lab technician controls the temperature \(T\) inside a kiln. From an initial temperature of 0 degrees Celsius \(\left({ }^{\circ} \mathrm{C}\right),\) he allows the temperature to increase by \(2{ }^{\circ} \mathrm{C}\) per minute for the next 60 minutes. After the 60 th minute, he allows the temperature to cool by \(3^{\circ} \mathrm{C}\) per minute. If \(t\) is the number of minutes, the temperature \(T\) is given by \(T(t)=\left\\{\begin{array}{ll}2 t, & \text { for } t \leq 60, \\ 300-3 t, & \text { for } t>60 .\end{array}\right.\) Find \(\lim _{t \rightarrow 6^{-}} T(t), \lim _{t \rightarrow 60^{+}} T(t),\) and \(\lim _{t \rightarrow 60} T(t) .\)

Step-by-Step Solution

Answer

All three limits are 120.

Step 1: Understand the Problem

We have a piecewise function for temperature, where the temperature increases by \(2^{\circ}\) C per minute for \(t \leq 60\) and decreases by \(3^{\circ}\) C per minute for \(t > 60\). We need to find the limits at and around the time \(t = 60\).

Step 2: Calculate \(\lim \limits_{t \to 60^{-}} T(t)\)

For \(t \leq 60\), the temperature is given by the function \( T(t) = 2t \). To find \(\lim \limits_{t \to 60^{-}} T(t)\), substitute \(t = 60\) into this function: \[ \lim_{t \to 60^{-}} T(t) = 2 \times 60 = 120. \]

Step 3: Calculate \(\lim \limits_{t \to 60^{+}} T(t)\)

For \(t > 60\), the temperature function changes to \(T(t) = 300 - 3t\). To find \(\lim \limits_{t \to 60^{+}} T(t)\), substitute \(t = 60\) into this function: \[ \lim_{t \to 60^{+}} T(t) = 300 - 3 \times 60 = 120. \]

Step 4: Calculate \(\lim \limits_{t \to 60} T(t)\)

We have previously found that both \(\lim_{t \to 60^{-}} T(t) = 120\) and \(\lim_{t \to 60^{+}} T(t) = 120\). Since these two one-sided limits are equal, the two-sided limit exists and is: \[ \lim_{t \to 60} T(t) = 120. \]

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions that have different rules or formulas applied to various parts of their domain. They "piece together" different functions to create one comprehensive function.
These functions are common in real-world scenarios where conditions change, like the temperature control in a kiln. In this case, the kiln's temperature is given by two separate expressions.
Here's how it works:

• For the first part, when the time \(t\) is 60 minutes or less, the temperature increases by \(2^{\circ} \text{C}\) per minute. The formula here is \(T(t) = 2t\).
• For the second part, when time \(t\) is more than 60 minutes, the temperature decreases by \(3^{\circ} \text{C}\) per minute, changing the formula to \(T(t) = 300 - 3t\).

Understanding how to switch between these parts based on \(t\) is crucial for using piecewise functions effectively. In essence, you are blending multiple functions based on the context or time period specified.

Continuity

Continuity in mathematics means a function's graph has no breaks, jumps, or holes at a point. For a function to be continuous at a certain point, its limit from the left must equal its limit from the right, and both must equal the function's value at that point.
For piecewise functions, checking continuity involves ensuring that the transitions between pieces don't create a break.
In this kiln example:

• The temperature function \(T(t)\) at \(t = 60\) is critical. We need \(\lim_{t \to 60^{-}} T(t)\), \(\lim_{t \to 60^{+}} T(t)\), and the value \(T(60)\) to be the same.
• The given solution shows that as \(t\) approaches 60 from both sides, the limit for both expressions (\(2t\) and \(300 - 3t\)) arrives at \(120\).
• Thus, \(T(60)\) is also \(120\), confirming continuity at \(t = 60\).

Continuity ensures the kiln's temperature change is smooth and without sudden jolts, highlighting the importance of this property in real-life applications.

Limit Calculation

Calculating limits, especially near points of interest or changing conditions, helps understand a function's behavior. Limits describe the anticipated value a function approaches as the input nears a certain point.
In this example, we are interested in the limits around \(t = 60\):

• The left-hand limit \(\lim_{t \to 60^{-}} T(t)\) is calculated using the formula for \(t \leq 60\), giving \(2 \times 60 = 120\).
• The right-hand limit \(\lim_{t \to 60^{+}} T(t)\) uses the formula for \(t > 60\), gives \(300 - 3 \times 60 = 120\).
• Since both side limits result in \(120\), the two-sided limit \(\lim_{t \to 60} T(t)\) exists and equals \(120\).

This process of calculating limits ensures precision in understanding transitions, whether they are mathematical or physical, like adjusting kiln temperature with precision to avoid sudden shifts.