(a) To determine To define: The number e . Answer The number e is lim x → 0 e x − 1 x = 1 . Explanation From the definition, e is the number such that lim x → 0 e x − 1 x = 1 . (b) To determine To estimate: The values of the limits corrected to two decimal places. Answer The value of the limits lim h → 0 2.7 h − 1 h and lim h → 0 2.8 h − 1 h are approximately 0.99 and 1.03. The value of e must be between 2.7 and 2.8. Explanation Result used: If the exponential function f ( x ) = b x is differential at 0, then it is differentiable everywhere and f ′ ( x ) = f ′ ( 0 ) b x , where f ′ ( 0 ) = lim h → 0 b h − 1 h . Calculation: Obtain the value of the limit lim h → 0 2.7 h − 1 h . h 2.7 h − 1 h 0.1 1.04 0.01 1.00 0.001 0.99 0.0001 0.99 0.00001 0.99 Therefore, the value of limit lim h → 0 2.7 h − 1 h is approximately 0.99. Obtain the value of the limit lim h → 0 2.8 h − 1 h . h 2.8 h − 1 h 0.1 1.08 0.01 1.03 0.001 1.03 0.0001 1.03 0.00001 1.03 Therefore, the value of the limit lim h → 0 2.8 h − 1 h is approximately 1.03. Obtain the value of e . Consider the exponential functions 2.7 x and 2.8 x . Substitute 2.7 for b in f ( x ) = b x and apply the Result, f ′ ( x ) = f ′ ( 0 ) ( 2.7 ) x , f ′ ( 0 ) = lim h → 0 2.7 h − 1 h Substitute 2.8 for b in f ( x ) = b x and apply the Result, f ′ ( x ) = f ′ ( 0 ) ( 2.8 ) x , f ′ ( 0 ) = lim h → 0 2.8 h − 1 h Since lim h → 0 2.7 h − 1 h ≈ 0.99 and lim h → 0 2.8 h − 1 h ≈ 1.03 , the respective derivative function is f ′ ( x ) = 0.99 ( 2.7 ) x and f ′ ( x ) = 1.03 ( 2.8 ) x . From the Result , the simplest differential formula exists when f ′ ( 0 ) = 1 for all possible values of b in f ( x ) = b x . Since f ′ ( 0 ) b = 2.7 < 1 < f ′ ( 0 ) b = 2.8 , there is a number b between 2.7 and 2.8 for which f ′ ( 0 ) = 1 and that number b is called as e . Therefore, the number e must be between 2.7 and 2.8.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

4th Edition

ISBN: 9781337687805

Author: James Stewart

Publisher: Cengage Learning

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Textbook Question

Chapter 3.1, Problem 1E

(a) How is the number e defined?

(b) Use a calculator to estimate the values of the limits

lim h → 0 2.7 h − 1 h and lim h → 0 2.8 h − 1 h

correct to two decimal places. What can you conclude about the value of e?

(a)

Expert Solution

To determine

To define: The number e.

Answer to Problem 1E

The number e is limx→0ex−1x=1.

Explanation of Solution

From the definition, e is the number such that limx→0ex−1x=1.

(b)

Expert Solution

To determine

To estimate: The values of the limits corrected to two decimal places.

Answer to Problem 1E

The value of the limits limh→02.7h−1h and limh→02.8h−1h are approximately 0.99 and 1.03.

The value of e must be between 2.7 and 2.8.

Explanation of Solution

Result used:

If the exponential function f(x)=bx is differential at 0, then it is differentiable everywhere and f′(x)=f′(0)bx, where f′(0)=limh→0bh−1h.

Calculation:

Obtain the value of the limit limh→02.7h−1h.

h	2.7h−1h
0.1	1.04
0.01	1.00
0.001	0.99
0.0001	0.99
0.00001	0.99

Therefore, the value of limit limh→02.7h−1h is approximately 0.99.

Obtain the value of the limit limh→02.8h−1h.

h	2.8h−1h
0.1	1.08
0.01	1.03
0.001	1.03
0.0001	1.03
0.00001	1.03

Therefore, the value of the limit limh→02.8h−1h is approximately 1.03.

Obtain the value of e.

Consider the exponential functions 2.7x and 2.8x.

Substitute 2.7 for b in f(x)=bx and apply the Result,

f′(x)=f′(0)(2.7)x, f′(0)=limh→02.7h−1h

Substitute 2.8 for b in f(x)=bx and apply the Result,

f′(x)=f′(0)(2.8)x, f′(0)=limh→02.8h−1h

Since limh→02.7h−1h≈0.99 and limh→02.8h−1h≈1.03, the respective derivative function is f′(x)=0.99(2.7)x and f′(x)=1.03(2.8)x.

From the Result, the simplest differential formula exists when f′(0)=1 for all possible values of b in f(x)=bx.

Since f′(0)b=2.7<1<f′(0)b=2.8, there is a number b between 2.7 and 2.8 for which f′(0)=1 and that number b is called as e.

Therefore, the number e must be between 2.7 and 2.8.

Chapter 2, Problem 17P

Chapter 3.1, Problem 2E

Chapter 3 Solutions

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

Ch. 3.1 - (a) How is the number e defined? (b) Use a...Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E

Ch. 3.1 - Prob. 11E Ch. 3.1 - Prob. 12E Ch. 3.1 - Prob. 13E Ch. 3.1 - Prob. 14E Ch. 3.1 - Prob. 15E Ch. 3.1 - Prob. 16E Ch. 3.1 - Prob. 17E Ch. 3.1 - Prob. 18E Ch. 3.1 - Prob. 19E Ch. 3.1 - Prob. 20E Ch. 3.1 - Prob. 21E Ch. 3.1 - Prob. 22E Ch. 3.1 - Prob. 23E Ch. 3.1 - Prob. 24E Ch. 3.1 - Prob. 25E Ch. 3.1 - Prob. 26E Ch. 3.1 - Prob. 27E Ch. 3.1 - Prob. 28E Ch. 3.1 - Prob. 29E Ch. 3.1 - Prob. 30E Ch. 3.1 - Prob. 31E Ch. 3.1 - Prob. 32E Ch. 3.1 - Prob. 33E Ch. 3.1 - Prob. 34E Ch. 3.1 - Prob. 35E Ch. 3.1 - Prob. 36E Ch. 3.1 - Prob. 37E Ch. 3.1 - Prob. 38E Ch. 3.1 - Prob. 39E Ch. 3.1 - Prob. 40E Ch. 3.1 - Prob. 41E Ch. 3.1 - Prob. 42E Ch. 3.1 - Prob. 43E Ch. 3.1 - Prob. 44E Ch. 3.1 - Prob. 45E Ch. 3.1 - Prob. 46E Ch. 3.1 - Prob. 47E Ch. 3.1 - Prob. 48E Ch. 3.1 - Prob. 49E Ch. 3.1 - Prob. 50E Ch. 3.1 - Prob. 51E Ch. 3.1 - Prob. 52E Ch. 3.1 - Prob. 53E Ch. 3.1 - Prob. 54E Ch. 3.1 - Prob. 55E Ch. 3.1 - Prob. 56E Ch. 3.1 - Draw a diagram to show that there are two tangent...Ch. 3.1 - Prob. 58E Ch. 3.1 - 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Write the composite function in the form f(g(x))....Ch. 3.4 - Write the composite function in the form f(g(x))....Ch. 3.4 - Prob. 3E Ch. 3.4 - Write the composite function in the form f(g(x))....Ch. 3.4 - Prob. 5E Ch. 3.4 - Prob. 6E Ch. 3.4 - Prob. 7E Ch. 3.4 - Prob. 8E Ch. 3.4 - Prob. 9E Ch. 3.4 - Prob. 10E Ch. 3.4 - Prob. 11E Ch. 3.4 - Prob. 12E Ch. 3.4 - Prob. 13E Ch. 3.4 - Prob. 14E Ch. 3.4 - Prob. 15E Ch. 3.4 - Prob. 16E Ch. 3.4 - Prob. 17E Ch. 3.4 - Prob. 18E Ch. 3.4 - Prob. 19E Ch. 3.4 - Prob. 20E Ch. 3.4 - Prob. 21E Ch. 3.4 - Prob. 22E Ch. 3.4 - Prob. 23E Ch. 3.4 - Prob. 24E Ch. 3.4 - Prob. 25E Ch. 3.4 - Prob. 26E Ch. 3.4 - Prob. 27E Ch. 3.4 - Prob. 28E Ch. 3.4 - Prob. 29E Ch. 3.4 - Prob. 30E Ch. 3.4 - Prob. 31E Ch. 3.4 - Prob. 32E Ch. 3.4 - Prob. 33E Ch. 3.4 - Prob. 34E Ch. 3.4 - Prob. 35E Ch. 3.4 - Prob. 36E Ch. 3.4 - Prob. 37E Ch. 3.4 - Prob. 38E Ch. 3.4 - Prob. 39E Ch. 3.4 - Prob. 40E Ch. 3.4 - Prob. 41E Ch. 3.4 - Prob. 42E Ch. 3.4 - Prob. 43E Ch. 3.4 - Prob. 44E Ch. 3.4 - Prob. 45E Ch. 3.4 - Prob. 46E Ch. 3.4 - Prob. 47E Ch. 3.4 - Prob. 48E Ch. 3.4 - Prob. 49E Ch. 3.4 - At what point on the curve y=1+2x is the tangent...Ch. 3.4 - Prob. 51E Ch. 3.4 - Prob. 52E Ch. 3.4 - A table of values for f, g, f, and g is given. (a)...Ch. 3.4 - Let f and g be the functions in Exercise 63. (a)...Ch. 3.4 - Prob. 55E Ch. 3.4 - Prob. 56E Ch. 3.4 - Prob. 57E Ch. 3.4 - Prob. 58E Ch. 3.4 - Prob. 59E Ch. 3.4 - Prob. 60E Ch. 3.4 - Prob. 61E Ch. 3.4 - Prob. 62E Ch. 3.4 - Prob. 63E Ch. 3.4 - Prob. 64E Ch. 3.4 - Prob. 65E Ch. 3.4 - Prob. 66E Ch. 3.4 - Prob. 67E Ch. 3.4 - Find the 1000th derivative of f(x) = xex.Ch. 3.4 - The displacement of a particle on a vibrating...Ch. 3.4 - If the equation of motion of a particle is given...Ch. 3.4 - Prob. 71E Ch. 3.4 - Prob. 72E Ch. 3.4 - The motion of a spring that is subject to a...Ch. 3.4 - Prob. 74E Ch. 3.4 - Prob. 75E Ch. 3.4 - Prob. 76E Ch. 3.4 - Prob. 77E Ch. 3.4 - The table gives the US population from 1790 to...Ch. 3.4 - Prob. 79E Ch. 3.4 - Prob. 80E Ch. 3.4 - Prob. 81E Ch. 3.4 - Prob. 82E Ch. 3.4 - Prob. 83E Ch. 3.4 - Prob. 84E Ch. 3.4 - Prob. 85E Ch. 3.4 - Prob. 86E Ch. 3.4 - Prob. 87E Ch. 3.4 - Prob. 88E Ch. 3.4 - Prob. 89E Ch. 3.4 - Prob. 90E Ch. 3.4 - Prob. 91E Ch. 3.4 - Prob. 92E Ch. 3.4 - Prob. 93E Ch. 3.4 - Prob. 94E Ch. 3.5 - 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Prob. 39E Ch. 3.5 - Prob. 40E Ch. 3.5 - Prob. 41E Ch. 3.5 - Prob. 42E Ch. 3.5 - Prob. 43E Ch. 3.5 - Two curves are orthogonal if their tangent lines...Ch. 3.5 - Show that the ellipse x2/a2 + y2/b2 = 1 and the...Ch. 3.5 - Prob. 46E Ch. 3.5 - Prob. 47E Ch. 3.5 - Prob. 49E Ch. 3.5 - (a) Where does the normal line to the ellipse x2 ...Ch. 3.5 - Prob. 51E Ch. 3.5 - Prob. 52E Ch. 3.5 - Prob. 53E Ch. 3.5 - Prob. 54E Ch. 3.5 - The Bessel function of order 0, y = J(x),...Ch. 3.5 - The figure shows a lamp located three units to the...Ch. 3.6 - Prob. 1E Ch. 3.6 - Prob. 2E Ch. 3.6 - Prob. 3E Ch. 3.6 - Prob. 4E Ch. 3.6 - Prob. 5E Ch. 3.6 - Prob. 6E Ch. 3.6 - Prob. 7E Ch. 3.6 - Prob. 8E Ch. 3.6 - Prob. 9E Ch. 3.6 - Prob. 10E Ch. 3.6 - Prob. 11E Ch. 3.6 - Prob. 12E Ch. 3.6 - Prob. 13E Ch. 3.6 - Prob. 14E Ch. 3.6 - Prob. 15E Ch. 3.6 - Prob. 16E Ch. 3.6 - Prob. 17E Ch. 3.6 - Prob. 18E Ch. 3.6 - Prob. 19E Ch. 3.6 - Prob. 20E Ch. 3.6 - Prob. 21E Ch. 3.6 - Prob. 22E Ch. 3.6 - Prob. 23E Ch. 3.6 - Prob. 24E Ch. 3.6 - 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Prob. 20E Ch. 3.8 - Prob. 21E Ch. 3.8 - Prob. 22E Ch. 3.8 - Prob. 23E Ch. 3.8 - Prob. 24E Ch. 3.8 - The table shows how the average age of first...Ch. 3.8 - Refer to the law of laminar flow given in Example...Ch. 3.8 - Prob. 28E Ch. 3.8 - Prob. 29E Ch. 3.8 - The cost function for a certain commodity is C(q)...Ch. 3.8 - Prob. 31E Ch. 3.8 - Prob. 32E Ch. 3.8 - Patients undergo dialysis treatment to remove urea...Ch. 3.8 - Invasive species often display a wave of advance...Ch. 3.8 - Prob. 35E Ch. 3.9 - Prob. 1E Ch. 3.9 - Prob. 2E Ch. 3.9 - Prob. 3E Ch. 3.9 - Prob. 4E Ch. 3.9 - Prob. 5E Ch. 3.9 - Prob. 6E Ch. 3.9 - Prob. 7E Ch. 3.9 - Prob. 8E Ch. 3.9 - Prob. 9E Ch. 3.9 - Prob. 10E Ch. 3.9 - Prob. 11E Ch. 3.9 - Prob. 12E Ch. 3.9 - Prob. 13E Ch. 3.9 - Prob. 14E Ch. 3.9 - Prob. 15E Ch. 3.9 - Prob. 16E Ch. 3.9 - Prob. 17E Ch. 3.9 - Prob. 18E Ch. 3.9 - Prob. 19E Ch. 3.9 - Prob. 20E Ch. 3.9 - Prob. 21E Ch. 3.9 - Prob. 22E Ch. 3.9 - Prob. 23E Ch. 3.9 - Prob. 24E Ch. 3.9 - Prob. 25E Ch. 3.9 - Prob. 26E Ch. 3.9 - 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(a) How is the number e defined? (b) Use a calculator to estimate the values of the limits lim h → 0 2.7 h − 1 h and lim h → 0 2.8 h − 1 h correct to two decimal places. What can you conclude about the value of e ?

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