I am given a table that shows the temperature of the water and the days, which looks like this:
t(days) W(t) (in Celcius)
0 20
3 31
6 28
9 24
12 22
15 21
the temperature of the water in a pond is a differentiable function W of time t. The table above shows the water temperature recorded every 3 days over a 15-day period.
Use data from the table to find an approximation for W'(12). Show the computations that lead to your answer.
I am not sure about what to do. I've tried graphing it, but I didn't know what to do from there. What other methods can I use?
w'= deltaW/delta time= (21-24)/6 in degreesC per day.
If you graph it, approximate the tangent between day 9 and day 15.
so the final answer would be 1/2degrees celcius at time=12?
I meant -1/2.
To find an approximation for W'(12), we need to calculate the rate of change of temperature with respect to time at t = 12.
From the given table, we can see that the temperature at t = 12 is 22°C and at t =9 is 24°C. The time interval between these two points is 12 - 9 = 3 days.
To approximate W'(12), we can use the formula for average rate of change:
W'(12) ≈ (W(12) - W(9)) / (12 - 9)
Substituting the values from the table, we get:
W'(12) ≈ (22 - 24) / 3
Simplifying, we get:
W'(12) ≈ -2 / 3
So the approximation for W'(12) is -2/3 degrees Celsius per day.