BuyFind

Single Variable Calculus: Concepts...

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

Single Variable Calculus: Concepts...

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

Solutions

Chapter 3.8, Problem 32E

(a)

To determine

To find: The sensitivity of the body.

Expert Solution

Answer to Problem 32E

Sensitivity of the body is S=54.4x0.6(1+16x0.8+8x0.4).

The graph is plotted between x and y coordinates for both R and S is shown in figure (1).

Explanation of Solution

Given:

The experimental formula is shown below.

R=40+24x0.41+4x0.4 (1)

Calculation:

Calculate the sensitivity S.

S=dRdx

Sensitivity is defined to be the rate of change of the reaction with respect to x.

Substitute 40+24x0.41+4x0.4 for R in the above equation.

S=ddx(40+24x0.41+4x0.4)

Apply the quotient rule below.

(uv)'=u'vv'uv2

Substitute (40+24x0.4) for u and (1+4x0.4) for v in the above equation.

S=(40+24x0.4)'(1+4x0.4)(1+4x0.4)'(40+24x0.4)(1+4x0.4)2=9.6x0.6(1+4x0.4)1.6x0.6(40+24x0.4)(1+4x0.4)2=1.6x0.6(6(1+4x0.4)(40+24x0.4))(1+4x0.4)2=1.6x0.6(6+24x0.44024x0.4)(1+(4x0.4)2+2(1)(4x0.4))

S=54.4x0.6(1+16x0.8+8x0.4) (2)

Thus, the sensitivity of the body is 54.4x0.6(1+16x0.8+8x0.4).

(b)

To determine

To illustrate: The part (a) by graphing both R and S as functions of x and comment on the values of R and S at low levels of brightness.

Expert Solution

Explanation of Solution

Illustration:

Sketch the curve.

Calculate the value of R using the equation (1).

R=40+24x0.41+4x0.4

Substitute 0 for x in the equation (1).

R=40+24(0)0.41+4(0)0.4=40

Repeat the calculation of the value R for values of x from 0.2 till 10.

Tabulate the value of x and R as shown in table (1).

xR=40+24x0.41+4x0.4
0.0040.00
0.2016.96
0.4015.01
0.6013.98
0.8013.30
1.0012.80
1.2012.41
1.4012.10
1.6011.83
1.8011.61
2.0011.42
2.2011.24
2.4011.09
2.6010.95
2.8010.83
3.0010.72
3.2010.61
3.4010.52
3.6010.43
3.8010.35
4.0010.27
4.2010.20
4.4010.13
4.6010.06
4.8010.00
5.009.95
5.209.89
5.409.84
5.609.79
5.809.74
6.009.70
6.209.66
6.409.62
6.609.58
6.809.54
7.009.50
7.209.47
7.409.43
7.609.40
7.809.37
8.009.34
8.209.31
8.409.28
8.609.25
8.809.22
9.009.20
9.209.17
9.409.15
9.609.12
9.809.10
10.009.08

Calculate the value of S using the equation (2).

S=54.4x0.6(1+16x0.8+8x0.4)

Substitute 0 for x in the equation (1).

S=54.4(0)0.6(1+16(0)0.8+8(0)0.4)=0

Repeat the calculation of the value S for values of x from 0.2 till 10.

Tabulate the value of x  and S as shown in table (2).

xS=54.4x0.6(1+16x0.8+8x0.4)
0.000
0.20-14.856
0.40-6.6235
0.60-4.0713
0.80-2.8659
1.00-2.176
1.20-1.7342
1.40-1.4297
1.60-1.2083
1.80-1.041
2.00-0.9106
2.20-0.8064
2.40-0.7215
2.60-0.6512
2.80-0.592
3.00-0.5417
3.20-0.4984
3.40-0.4609
3.60-0.428
3.80-0.399
4.00-0.3733
4.20-0.3503
4.40-0.3298
4.60-0.3112
4.80-0.2944
5.00-0.2791
5.20-0.2651
5.40-0.2524
5.60-0.2406
5.80-0.2298
6.00-0.2198
6.20-0.2105
6.40-0.2019
6.60-0.1939
6.80-0.1864
7.00-0.1795
7.20-0.1729
7.40-0.1668
7.60-0.161
7.80-0.1556
8.00-0.1505
8.20-0.1456
8.40-0.1411
8.60-0.1367
8.80-0.1326
9.00-0.1288
9.20-0.1251
9.40-0.1215
9.60-0.1182
9.80-0.115
10.00-0.112

Graph:

Sketch the curve using table (1) and table (2) as shown in figure (1).

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 3.8, Problem 32E

Refer the figure (1).

For all the small values of x, we have R reaching a value near 40, which is quite high. So a small stimulus produces a large reaction which is something to except.

Comments:

At low level of brightness, the eye is more sensitive to slight changes than it is at higher level of brightness.

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