Explanation Given: The function is f ( x ) = ln ( x 2 + 1 ) and the value of a is 1 and the value of Δ x is 0.1 . Concept used: Error: The formula to calculate the error is as follows: Error = | Δ f − f ′ ( a ) Δ x | Percentage error: The formula to calculate the percentage error is as follows: Percentage error = | Error Actual value | ⋅ 100 Calculation: The given function is expressed as follow: f ( x ) = ln ( x 2 + 1 ) …. (1) The expression Δ f represents the change is the value of f ( x ) as the value of value of x is changed by Δ x . The value of Δ f is calculated as follows: Δ f = f ( a + Δ x ) − f ( a ) = ln ( ( a + Δ x ) 2 + 1 ) − ln ( a 2 + 1 ) = ln ( ( a + Δ x ) 2 + 1 a 2 + 1 ) To obtain the value of Δ f substitute 1 for a and 0.1 for Δ x in the above equation as shown below: Δ f = ln ( ( 1 + 0.1 ) 2 + 1 1 2 + 1 ) = ln ( 2.21 2 ) = ln ( 1.105 ) = 0.099 This implies that the value of Δ f is 0.099 . Differentiate equation (1) with respect to x as shown below: f ′ ( x ) = d ( ln ( x 2 + 1 ) ) d x = 1 x 2 + 1 ⋅ 2 x = 2 x x 2 + 1 Label the above equation as equation (2). f ′ ( x ) = 2 x x 2 + 1 …..(2) The expression to obtain the linear approximation of a function is as follows: f ( a + Δ x ) ≈ f ( a ) + f ′ ( a ) Δ x

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 4.1, Problem 12E

To determine

To find:

The value of Δf, error and the percentage error for the function f(x)=ln(x2+1) such that the value of a is 1 and the value of Δx is 0.1.

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Chapter 4.1, Problem 11E

Chapter 4.1, Problem 13E

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The following table shows total military and arms trade expenditure for a certain country in 2000, 2006, and 2012. Year t (year since 2000) 0 6 12 Military Expenditure C(t)($ billion) 380 540 660 (a) Compute the average rate of change of C(t) over the period 2006–2012 (that is, [6, 12]). Be sure to state the units of measurement. Interpret the average rate of change. This country's military expenditure increased at an average rate of about between 2006 and 2012. (b) Compute the average rate of change of C(t) over the period [0, 12]. Be sure to state the units of measurement. (Round your answer to two decimal places.) Interpret the average rate of change. This country's military expenditure increased at an average rate of about between 2000 and 2012.

The following table shows total military and arms trade expenditure for a certain country in 2000, 2006, and 2012. Year t (year since 2000) 0 6 12 Military Expenditure C(t)($ billion) 390 550 640 (a) Compute the average rate of change of C(t) over the period 2006–2012 (that is, [6, 12]). Be sure to state the units of measurement. HINT [See Example 1.] Interpret the average rate of change. This country's military expenditure increased at an average rate of about between 2006 and 2012. (b) Compute the average rate of change of C(t) over the period [0, 12]. Be sure to state the units of measurement. HINT [See Example 1.] (Round your answer to two decimal places.) Interpret the average rate of change. (Round your answer to two decimal places.) This country's military expenditure increased at an average rate of about between 2000 and 2012.

Please calculate H(x)

Chapter 4 Solutions

Calculus: Early Transcendentals

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.1 - Prob. 13E Ch. 4.1 - Prob. 14E Ch. 4.1 - Prob. 15E Ch. 4.1 - Prob. 16E Ch. 4.1 - Prob. 17E Ch. 4.1 - Prob. 18E Ch. 4.1 - Prob. 19E Ch. 4.1 - Prob. 20E Ch. 4.1 - Prob. 21E Ch. 4.1 - Prob. 22E Ch. 4.1 - Prob. 23E Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E Ch. 4.1 - Prob. 30E Ch. 4.1 - Prob. 31E Ch. 4.1 - Prob. 32E Ch. 4.1 - Prob. 33E Ch. 4.1 - Prob. 34E Ch. 4.1 - Prob. 35E Ch. 4.1 - Prob. 36E Ch. 4.1 - Prob. 37E Ch. 4.1 - Prob. 38E Ch. 4.1 - Prob. 39E Ch. 4.1 - Prob. 40E Ch. 4.1 - Prob. 41E Ch. 4.1 - Prob. 42E Ch. 4.1 - Prob. 43E Ch. 4.1 - Prob. 44E Ch. 4.1 - Prob. 45E Ch. 4.1 - Prob. 46E Ch. 4.1 - Prob. 47E Ch. 4.1 - Prob. 48E Ch. 4.1 - Prob. 49E Ch. 4.1 - Prob. 50E Ch. 4.1 - Prob. 51E Ch. 4.1 - Prob. 52E Ch. 4.1 - Prob. 53E Ch. 4.1 - Prob. 54E Ch. 4.1 - Prob. 55E Ch. 4.1 - Prob. 56E Ch. 4.1 - 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Prob. 3E Ch. 4.8 - Prob. 4E Ch. 4.8 - Prob. 5E Ch. 4.8 - Prob. 6E Ch. 4.8 - Prob. 7E Ch. 4.8 - Prob. 8E Ch. 4.8 - Prob. 9E Ch. 4.8 - Prob. 10E Ch. 4.8 - Prob. 11E Ch. 4.8 - Prob. 12E Ch. 4.8 - Prob. 13E Ch. 4.8 - Prob. 14E Ch. 4.8 - Prob. 15E Ch. 4.8 - Prob. 16E Ch. 4.8 - Prob. 17E Ch. 4.8 - Prob. 18E Ch. 4.8 - Prob. 19E Ch. 4.8 - Prob. 20E Ch. 4.8 - Prob. 21E Ch. 4.8 - Prob. 22E Ch. 4.8 - Prob. 23E Ch. 4.8 - Prob. 24E Ch. 4.8 - Prob. 25E Ch. 4.8 - Prob. 26E Ch. 4.8 - Prob. 27E Ch. 4.8 - Prob. 28E Ch. 4.8 - Prob. 29E Ch. 4.8 - Prob. 30E Ch. 4.8 - Prob. 31E Ch. 4.8 - Prob. 32E Ch. 4.8 - Prob. 33E Ch. 4.8 - Prob. 34E Ch. 4.8 - Prob. 35E Ch. 4.8 - Prob. 36E Ch. 4.8 - Prob. 37E Ch. 4.8 - Prob. 38E Ch. 4.8 - Prob. 39E Ch. 4.8 - Prob. 40E Ch. 4.8 - Prob. 41E Ch. 4 - Prob. 1CRE Ch. 4 - Prob. 2CRE Ch. 4 - Prob. 3CRE Ch. 4 - Prob. 4CRE Ch. 4 - Prob. 5CRE Ch. 4 - Prob. 6CRE Ch. 4 - Prob. 7CRE Ch. 4 - Prob. 8CRE Ch. 4 - Prob. 9CRE Ch. 4 - Prob. 10CRE Ch. 4 - Prob. 11CRE Ch. 4 - Prob. 12CRE Ch. 4 - 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The value of Δ f , error and the percentage error for the function f ( x ) = ln ( x 2 + 1 ) such that the value of a is 1 and the value of Δ x is 0.1 .

Solution Summary: The author calculates the value of Delta f, the error and the percentage error for the function.

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To find:

The value of Δf, error and the percentage error for the function f(x)=ln(x2+1) such that the value of a is 1 and the value of Δx is 0.1.

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Chapter 4 Solutions

The value of Δ f , error and the percentage error for the function f ( x ) = ln ( x 2 + 1 ) such that the value of a is 1 and the value of Δ x is 0.1 .

Solution Summary: The author calculates the value of Delta f, the error and the percentage error for the function.

Videos

To find: The value of Δf, error and the percentage error for the function f(x)=ln(x2+1) such that the value of a is 1 and the value of Δx is 0.1.

Want to see the full answer?

Chapter 4 Solutions

To find:

The value of Δf, error and the percentage error for the function f(x)=ln(x2+1) such that the value of a is 1 and the value of Δx is 0.1.