# The Mathematics of Coffee Cooling

Take freshly made coffee off the plate warmer and it begins to cool. In my book, Everyday CalculusI describe a mathematical model that describes the temperature T of the coffee minutes after being taken off the warming plate. The result is equation (5) in the book:

$T(t)=75+85e^{-0.318t}. \$

This function is graphed below (this is Figure 2.3 in Everyday Calculus), along with the tangent line to the graph at the point where the black dot is. Feel free to explore the graph on your own, and/or follow the guidelines below the graph.

Graph controls: You can pan by clicking and dragging the graph. You can also zoom in/out using your mouse wheel, or by clicking the arrow below the wrench, or by adjusting the bounds of the graph by clicking on the wrench. Finally, click on any dots on the graph to see the coordinates of those points.

## What’s Going On In This Graph?

The slope of the tangent line (the red dashed line on the graph) is the derive T'(a), where is the x-value of the black dot (click on the black dot to see that point’s coordinates). The value of T'(a) is shown in pane 4 on the left. The derivative tells us how the temperature is changing. Move around the black dot and you’ll see that all slopes are negative, which means that the temperature is decreasing; this confirms our experience with cooling coffee.

Now, click on the circles next to panes 5 and 7 on the left. You’ll notice that a red curve has appeared near the bottom of the scree, along with a black dot on that curve. Click on that black dot to show it’s coordinates. The red curve is the derivative function T'(t) (Figure 2.4 in Everyday Calculus):

$T'(t)=-27.03e^{-0.318t}. \$

At each t-value, T'(t) returns the slope of the tangent line to T(t). Confirm this by dragging the black dot on the blue curve (the graph of T(t)) and comparing the values in pane 4 to the y-values of the black dot on the red graph.

Feel free to continue exploring functions and their derivatives by changing the formula in pane 1. Enjoy!