On a particular day in a city, the temperature is measured in degrees Celsius every 𝑡 hours. Using the given table, estimate the rate of change in the temperature at 𝑡 equals four.
From the table when 𝑡 is one, the temperature is 22, when 𝑡 is two the temperature is 24, when 𝑡 is four the temperature is 28, and when 𝑡 is six the temperature is 29. So we have been asked to estimate the rate of change in temperature at 𝑡 equals four. We recall a formula that calculates the rate of change between two points. The rate of change equals the change in 𝑦 over the change in 𝑥. This is a general rule that we are familiar with. But for our question, we’re dealing with time 𝑡 and temperature. So we adapt our formula to change in temperature over change in 𝑡.
So to use this formula, we need two sets of values so that we can work out the change. So which value should we use? Well, the most sensible option is to choose the values closest to four, either side of four. So what we’re going to do is approximate the rate of change between 𝑡 equals two and 𝑡 equals six which will give us an estimate for the rate of change at 𝑡 equals four. So the rate of change in the temperature is 29 minus 24 over six minus two. This is 5 over 4 or 1.25. So our estimate for the rate of change in the temperature at 𝑡 equals four is 1.25 degrees Celsius per hour.