The graph shows the influence of the temperature t on the maximum sustainable swimming speed s of coho salmon. (b) estimate the values of s '(5) and s '(25).

From the graph of S, we want to have information about S' (the derivate of S)

For S'(5) we need to look at the beginning of the curve, we can see that S(5) is a value around the 15cm/s, and it increases after that point, so S'(5) must be a positive number, this means that the neighborhood of T = 5°C, the speed will increase as the temperature increases.

For S'(25) we can see that the graph decreases after that point, so S'(25) is a negative number, this means that after the 25°C, if we keep increasing the temperature we will have a decrease in the speed.

We can make an estimation about the values:

now, the average rate of change in an interval can be calculated as:

S' = (Y2 - Y1)/(X2 - X1)

S is almost linear between T = 5°C and 10°C and we have that

S(5) = 15cm/s and S(10) = 20cm/s

so we can find the average rate of change in that interval as:

S' = (20cm/s - 15cm/s)/(10°C  - 5°C) = 1(cm/s°C)

So we can estimate that S'(5) is around 1 (cm/s°C)

We can do something similar for S'(25)

we can estimate that S(20) = 25cm/s and S(25) = 20cm/s

so the average rate of change is:

S' = (20cm/s - 25cm/s)/(25°C - 20°C) = -1 (cm/s°C)

So we can expect that S'(25) is around -1 (cm/s°C)

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