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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Reasearchers measured the average blood alcohol concentration C(t) of eight men starting one hour after consumption of30 mL of ethanol (corresponding to two alcoholic drinks).

t (hours) 1.0 1.5 2.0 2.5 3.0
C(t) (mg/mL) 0.33 0.24 0.18 0.12 0.07

(a) Find the average rate of change of C with respect to 1 over each time interval:

(i) [l.0, 2.0]

(ii) [l.5, 2.0]

(iii) [2.0, 2.5]

(iv) [2.0, 3.0]

In each case, include the units.

(b) Estimate the instantaneous rate of change at t = 2 and interpret your result What are the units?

(a)

To determine

To find: The average rate of change of C(t) over each time interval.

Explanation

Given:

Let C(t) be represents the concentration of blood alcohol at time t.

Formula used:

The average rate of change of C with respect to t on the interval [t1,t2] is,

ΔCΔt=C(t2)C(t1)t2t1 (1)

Calculation:

Section (i)

Obtain the average rate of change of C(t) on the interval [1.0,2.0].

Substitute 1.0 for t1 and 2.0 for t2 in equation (1),

ΔCΔt=C(2.0)C(1.0)2.01.0

Consider the given table, the values of C(1.0)=0.33 and C(2.0)=0.18.

ΔCΔt=0.180.332.01.0=0.151=0.15

Thus, the average rate of change of C(t) on the interval [1.0,2.0] is 0.15 mg/mLh_.

Section (ii)

Obtain the average rate of change of C(t) on the interval [1.5,2.0].

Substitute 1.5 for t1 and 2.0 for t2 in equation (1),

ΔCΔt=C(2.0)C(1.5)2.01.5

Consider the given table, the values of C(1.5)=0.24 and C(2.0)=0.18.

ΔCΔt=0.180.242.01.5=0.060.5=0.12

Thus, the average rate of change of C(t) on the interval [1

(b)

To determine

To estimate: The instantaneous rate of change at t=2 and state the interpretation.

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