(a) Explanation Given Data: The given function is as follows, y ( t ) = 4 sin 2 t (1) Values of t are 0 , 0.001 , 0.01 , 0.1 and 1 . Calculation: A linear function is a simple function composed of constant terms and simple variable without exponent. When argument of sine function is close to zero then sine function behaves as linear function and now value of linear function is equal to its argument. So linear approximation of function y ( t ) is as follows, y ( t ) l i n e a r ≈ 4 ( 2 t ) y ( t ) l i n e a r = 8 t (2) The expression for percentage change in given function is as follows, Δ y ( t ) = y ( t ) l i n e a r − y ( t ) y ( t ) × 100 (3) Here, Δ y ( t ) is percentage change in given function, y ( t ) is given function and y ( t ) l i n e a r is linear approximation f function. For t = 0 , Substitute 0 for t in equation (1), y ( 0 ) = 4 sin 2 ( 0 ) = 4 sin 0 = 4 ( 0 ) = 0 Substitute 0 for t in equation (2), y ( 0 ) l i n e a r = 8 ( 0 ) = 0 As value of y ( 0 ) l i n e a r is equal to value of y ( 0 ) . which means percentage change in the value of function is 0 . For t = 0.001 , Substitute 0.001 for t in equation (1), y ( 0.001 ) = 4 sin 2 ( 0.001 ) = 4 sin 0.002 = 4 ( 0.00199 ) = 0.00799 Substitute 0.001 for t in equation (2), y ( 0.001 ) l i n e a r = 8 ( 0.001 ) = 0.008 Substitute 0.00799 for y ( 0.001 ) and 0.008 for y ( 0.001 ) l i n e a r in equation (3), Δ y ( 0.001 ) = 0.008 − 0.00799 0.00799 × 100 = 0.00001 0.00799 × 100 = 0.125 % So percentage change in given function for t = 0.001 is 0.125 % . For t = 0.01 , Substitute 0.01 for t in equation (1), y ( 0.01 ) = 4 sin 2 ( 0.01 ) = 4 sin 0.02 = 4 ( 0.0199 ) = 0.0799 Substitute 0 (b) To determine For which values of t value of linear function is approximately equal to y ( t ) .

9th Edition

ISBN: 9780073545516

Author: Hayt, William H. (william Hart), Jr, Kemmerly, Jack E. (jack Ellsworth), Durbin, Steven M.

Publisher: Mcgraw-hill Education,

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Chapter 5, Problem 2E

(a)

To determine

Evaluate linear function for given values of t.

(b)

To determine

For which values of t value of linear function is approximately equal to y(t).

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Chapter 5, Problem 1E

Chapter 5, Problem 3E

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