• Create direction fields for first-order differential equations
• Use a direction field to sketch solution curves
• Use Euler’s Method to find approximate solutions step by step

Coffee Temperature Analysis

You’re working as a barista at a local coffee shop and want to understand how quickly different beverages cool down to ensure the best customer experience. The rate at which a hot beverage cools follows Newton’s law of cooling, which can be modeled by the differential equation:

[latex]T'(t) = -k(T - A)[/latex]

where [latex]T(t)[/latex] is the temperature of the beverage at time [latex]t[/latex] (in minutes), [latex]k[/latex] is the cooling constant that depends on the beverage container and environment, and [latex]A[/latex] is the ambient room temperature.

For your analysis, you’re studying a specialty latte that starts at [latex]160°F[/latex] in a ceramic mug. The room temperature is [latex]72°F[/latex], and through experimentation, you’ve determined that [latex]k = 0.08[/latex]. This gives you the specific differential equation:

[latex]T'(t) = -0.08(T - 72)[/latex]

You want to create a direction field to visualize the cooling behavior and use Euler’s method to predict temperatures at specific times to help train other baristas about optimal serving times.