Chapter 2.6, Problem 52E

The graph shows the influence of the temperature T on the maximum sustainable swimming speed S of Coho salmon.

(a) What is the meaning of the derivative S'(T)? What are its units?

(b) Estimate the values of S'(15) and S'(25) and interpret them.

To determine

The derivative $S^{\prime }\left(T\right)$ and its unit.

Explanation of Solution

Given:

The influence of the temperature T on the maximum sustainable swimming speed S of coho salmon.

Calculation:

The maximum sustainable swimming speed S is depends on the influence of the temperature T.

The maximum sustainable swimming speed $S\left(T\right)$ is a function of the influence temperature T.

Note that, the derivative $f^{\prime }\left(x\right)$ is the instantaneous rate of change of the function $f\left(x\right)$ with respect to x

The derivative $S^{\prime }\left(T\right)$ is the rate of change of maximum sustainable speed with respect to temperature T.

The instantaneous rate of change is equal to $S^{\prime }\left(T\right)=\underset{\Delta x\to 0}{\lim }\frac{\Delta y}{\Delta t}\text{ here }y=S\left(T\right)$.

Here $\Delta y$ is measured in centimeter per second and $\Delta t$ is measured in centigrade.

Thus, the units are centimeter per second per centigrade.

Therefore, the unit are $\left(\text{cm/s}\right)/°\text{C}$.

To determine

To estimate: The value of $S^{\prime }\left(15\right)\text{and }{S}^{\prime }\left(25\right)$.

Explanation of Solution

Estimate the value of $S^{\prime }\left(15\right)$.

Form given Figure, it is observed that the tangent line at 15 is passing through the point $\left(16,24\right)\text{ and }\left(9,19\right)$.

The slope of the tangent line to the curve at 15 as follows,

$\begin{array}{c}m=\frac{19-24}{9-16}\\ =\frac{-5}{-7}\\ =0.7\end{array}$

Thus, the slope of the tangent line to the curve at 15 is, $m=0.7\left(\text{cm/s}\right)/°\text{C}$.

Note that, the slope of the tangent line to the curve at 15 is same $S^{\prime }\left(15\right)$.

Therefore, $m=0.7\left(\text{cm/s}\right)/°\text{C}$.

Estimate the value of $S^{\prime }\left(25\right)$.

Form given Figure, it is observed that the tangent line at 25 is passing through the point $\left(23.5,21\right)\text{ and }\left(24,20\right)$.

The slope of the tangent line to the curve at 25 as follows,

$\begin{array}{c}m=\frac{20-21}{24-23.5}\\ =\frac{-1}{0.5}\\ =-2\end{array}$

Thus, the slope of the tangent line to the curve at 25 is, $m=-2\left(\text{cm/s}\right)/°\text{C}$.

Note that, the slope of the tangent line to the curve at 25 is same as $S^{\prime }\left(25\right)$.

Therefore, $m=-2\left(\text{cm/s}\right)/°\text{C}$.

The derivative $S^{\prime }\left(15\right)=0.7\left(\text{cm/s}\right)/°\text{C}$ and $S^{\prime }\left(25\right)=-2\left(\text{cm/s}\right)/°\text{C}$ means that the swimming speed increasing at the rate of $0.7\left(\text{cm/s}\right)/°\text{C}$ as the temperature increasing post $15°\text{C}$ and the swimming speed decreasing at the rate of $2\left(\text{cm/s}\right)/°\text{C}$ as the temperature increasing post $25°\text{C}$.