# The derivative S ′ ( T ) and its unit.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2.6, Problem 52E

(a)

To determine

## The derivative S′(T) and its unit.

Expert Solution

### 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(T) is a function of the influence temperature T.

Note that, the derivative f(x) is the instantaneous rate of change of the function f(x) with respect to x

The derivative S(T) is the rate of change of maximum sustainable speed with respect to temperature T.

The instantaneous rate of change is equal to S(T)=limΔx0ΔyΔt    here y=S(T).

Here Δy is measured in centimeter per second and Δt is measured in centigrade.

Thus, the units are centimeter per second per centigrade.

Therefore, the unit are (cm/s)/°C.

(b)

To determine

Expert Solution

### Explanation of Solution

Estimate the value of S(15).

Form given Figure, it is observed that the tangent line at 15 is passing through the point (16,24) and (9,19).

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

m=1924916=57=0.7

Thus, the slope of the tangent line to the curve at 15 is, m=0.7(cm/s)/°C.

Note that, the slope of the tangent line to the curve at 15 is same S(15).

Therefore, m=0.7(cm/s)/°C.

Estimate the value of S(25).

Form given Figure, it is observed that the tangent line at 25 is passing through the point (23.5,21) and (24,20).

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

m=20212423.5=10.5=2

Thus, the slope of the tangent line to the curve at 25 is, m=2(cm/s)/°C.

Note that, the slope of the tangent line to the curve at 25 is same as S(25).

Therefore, m=2(cm/s)/°C.

The derivative S(15)=0.7(cm/s)/°C and S(25)=2(cm/s)/°C means that the swimming speed increasing at the rate of 0.7(cm/s)/°C as the temperature increasing post 15°C and the swimming speed decreasing at the rate of 2(cm/s)/°C as the temperature increasing post 25°C.

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