Problem 257 - Calculus Volume 3

Question

The temperature \(T\) at a point \((x, y)\) is \(T(x, y)\) and is measured using the Celsius scale. A fly crawls so that its position after \(t\) seconds is given by \(x=\sqrt{1+t}\) and \(y=2+\frac{1}{3} t,\) where \(x\) and \(y\) are measured in centimeters. The temperature function satisfies \(T_{x}(2,3)=4\) and \(T_{y}(2,3)=3\). How fast is the temperature increasing on the fly's path after 3 sec?

Step-by-Step Solution

Answer

The temperature increases at 2°C per second after 3 seconds.

Step 1: Understand the problem

We need to find how quickly the temperature is increasing with respect to time as the fly moves. Since temperature is a function of two variables, this requires the chain rule to differentiate with respect to time.

Step 2: Differentiate x and y with respect to t

The fly's path is given by the equations \(x=\sqrt{1+t}\) and \(y=2+\frac{1}{3}t\). We take the derivatives of these functions with respect to \(t\): \[\frac{dx}{dt} = \frac{1}{2\sqrt{1+t}}\] \[\frac{dy}{dt} = \frac{1}{3}\].

Step 3: Set up the chain rule for temperature change

The chain rule for the temperature increasing on the fly's path is given by \[\frac{dT}{dt} = \frac{\partial T}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial T}{\partial y} \cdot \frac{dy}{dt}\]. Substitute the given partial derivatives at the point \(x=2, y=3\): \(\frac{\partial T}{\partial x} = 4\) and \(\frac{\partial T}{\partial y} = 3\).

Step 4: Evaluate x and y at t=3

To compute the derivatives at \(t=3\), find \(x\) and \(y\) at this time: \[x(3) = \sqrt{1+3} = 2\] \[y(3) = 2 + \frac{1}{3} \times 3 = 3\]. Thus, at \(t=3\), the point is \((2,3)\).

Step 5: Substitute and calculate the rate of change

We already have all the derivatives and values at \(t=3\), so substitute in the chain rule formula. \[\frac{dT}{dt} = 4 \cdot \frac{1}{2 \cdot \sqrt{1+3}} + 3 \cdot \frac{1}{3}\] \[= 4 \cdot \frac{1}{4} + 3 \cdot \frac{1}{3}\] \[= 1 + 1 = 2\]. Thus, the temperature is increasing at \(2\text{ degrees Celsius per second}\) when \(t=3\).

Key Concepts

Partial Derivatives

Partial derivatives are a key concept in calculus. They help us understand how a function changes when we vary one variable while keeping other variables constant.
It's like observing how the temperature changes by moving a fly in just one direction at a time. For instance, in our problem, temperature depends on both x and y, so we consider:

• \( \frac{\partial T}{\partial x} \): This represents the rate at which temperature changes as we move along the x-axis, keeping y constant.
• \( \frac{\partial T}{\partial y} \): This indicates how temperature changes along the y-axis, with x held constant.

These derivatives are crucial in applying the chain rule, which allows us to track how temperature changes along a specific path.

Temperature Function

The temperature function is a representation of how temperature varies at different points in a region.
This function, \( T(x, y) \), describes temperature in terms of the coordinates \( (x, y) \). In our exercise, the specific partial derivatives at point \((2,3)\) are given as \( T_x = 4 \) and \( T_y = 3 \).
These values tell us the directional sensitivity of temperature. A positive derivative, like the ones here, implies temperature rises if the fly crawls in the direction.The main role of this function in the problem is to provide specific rates of change at certain coordinates, which are combined with the fly's movement to determine how temperature changes over time.

Path of a Particle

The path of a particle represents how an object's position changes over time.
In this case, the fly moves in a defined path described by the equations \(x=\sqrt{1+t}\) and \(y=2+\frac{1}{3}t\). The path's equations allow us to calculate the rate of movement regardless of time.Understanding this path is essential because:

• It tells us how fast the fly moves in the x and y directions using the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \).
• It helps compute the exact position of the fly at any given time, like at \( t=3 \).
• These rates are plugged into the chain rule to find out how fast the temperature changes as the fly moves.

This understanding connects the fly's motion with the rate of temperature change, a critical aspect highlighted in this particle path problem.