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chapter10ProblemsAndSolutions

# chapter10ProblemsAndSolutions - Chapter 10 Trigonometric...

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Chapter 10 Trigonometric functions 10.1 Calculate the first derivative for the following functions. (a) y = sin x 2 (b) y = sin 2 x (c) y = cot 2 3 x (d) y = sec( x - 3 x 2 ) (e) y = 2 x 3 tan x (f) y = x cos x (g) y = x cos x (h) y = e - sin 2 1 x (i) y = (2 tan 3 x + 3 cos x ) 2 (j) y = cos(sin x ) + cos x sin x Detailed Solution: (a) dy dx = (cos x 2 ) · d dx ( x 2 ) = 2 x cos x 2 (b) dy dx = 2 sin x d dx (sin x ) = 2 sin x cos x = sin 2 x (c) dy dx = 2 cot 3 x d dx (cot 3 x ) = 2 cot 3 x ( - csc 2 3 x ) d dx ( 3 x ) = - 2 3 x - 2 3 (cot 3 x )(csc 2 3 x ) (d) dy dx = sec( x - 3 x 2 ) tan( x - 3 x 2 ) d dx ( x - 3 x 2 ) = (1 - 6 x ) sec( x - 3 x 2 ) tan( x - 3 x 2 ) (e) dy dx = 6 x 2 tan x + 2 x 3 sec 2 x v.2005.1 - September 4, 2009 1

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Math 102 Problems Chapter 10 (f) dy dx = 1 · cos x - x ( - sin x ) cos 2 x = cos x + x sin x cos 2 x (g) dy dx = 1 · cos x + x ( - sin x ) = cos x - x sin x (h) dy dx = e - sin 2 1 x d dx ( - sin 2 1 x ) = e - sin 2 1 x ( - 2 sin 1 x ) d dx (sin 1 x ) = e - sin 2 1 x ( - 2 sin 1 x )(cos 1 x )( d dx 1 x ) = 2 e - sin 2 1 x sin 1 x cos 1 x x 2 = sin 2 x x 2 e sin 2 1 x (i) dy dx = 2(2 tan 3 x + 3 cos x ) d dx (2 tan 3 x + 3 cos x ) = 2(2 tan3 x + 3 cos x )(2 sec 2 3 x · 3 - 3 sin x ) = 6(2 tan3 x + 3 cos x )(2 sec 2 3 x - sin x ) (j) dy dx = - sin(sin x ) · cos x +( - sin x ) · sin x +cos x · cos x = - sin(sin x ) · cos x +(cos 2 x - sin 2 x ) = - sin(sin x ) · cos x + cos 2 x 10.2 Take the derivative of the following functions. (a) f ( x ) = cos(ln( x 4 + 5 x 2 + 3)) (b) f ( x ) = sin( radicalbig cos 2 ( x ) + x 3 ) (c) f ( x ) = 2 x 3 + log 3 ( x ) (d) f ( x ) = ( x 2 e x + tan(3 x )) 4 (e) f ( x ) = x 2 radicalbig sin 3 ( x ) + cos 3 ( x ) Detailed Solution: (a) f prime ( x ) = - (4 x 3 + 10 x ) sin(ln( x 4 + 5 x 2 + 3)) ( x 4 + 5 x 2 + 3) (b) f prime ( x ) = (3 x 2 - 2 cos( x ) sin( x )) cos( radicalbig cos 2 ( x ) + x 3 ) (2 radicalbig cos 2 ( x ) + x 3 ) (c) f prime ( x ) = 6 x 2 + 1 x ln(3) v.2005.1 - September 4, 2009 2
Math 102 Problems Chapter 10 (d) f prime ( x ) = 4( x 2 e x + tan(3 x )) 3 (2 xe x + x 2 e x + 3 sec 2 (3 x )) (e) f prime ( x ) = 2 x radicalBig sin 3 ( x ) + cos 3 ( x ) + 3 x 2 (sin 2 ( x ) cos( x ) - cos 2 ( x ) sin( x )) 2 radicalbig sin 3 ( x ) + cos 3 ( x ) 10.3 Convert the following expressions in radians to degrees: (a) π (b) 5 π/ 3 (c) 21 π/ 23 (d) 24 π Convert the following expressions in degrees to radians: (e) 100 o (f) 8 o (g) 450 o (h) 90 o Using a Pythagorean triangle, evaluate each of the following: (i) cos( π/ 3) (j) sin( π/ 4) (k) tan( π/ 6) Detailed Solution: (a) 180 o (b) 300 o (c) 164 . 35 o (d) 4320 o or equivalently, 0 o (e) 5 π/ 9 (f) 2 π/ 45 (g) 5 π/ 2 (h) π/ 2 (i) 1 / 2 (j) 1 / 2 = 2 / 2 (k) 1 / 3 = 3 / 3 10.4 Graph the following functions over the indicated ranges: (a) y = x sin( x ) for - 2 π < x < 2 π (b) y = e x cos( x ) for 0 < x < 4 π . Detailed Solution: See Figure 10.1. (a) x y -2 π 2 π (b) x y Figure 10.1: Figure for solution to problem 10.4 v.2005.1 - September 4, 2009 3

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Math 102 Problems Chapter 10 10.5 Sketch the graph for each of the following functions: (a) y = 1 2 sin 3( x - π 4 ) (b) y = 2 - sin x (c) y = 3 cos 2 x (d) y = 2 cos( 1 2 x + π 4 ) Detailed Solution: -5 -4 -3 -2 -1 0 1 2 3 4 5 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Figure 10.2: Problem 10.5 (a), y = 1 2 sin 3( x - π 4 ) (a) Amplitude = 1 2 , period = 2 π 3 , phase shift = π 4 . See Figure 10.2. (b) Amplitude = 1, period = 2 π , phase shift = 0, reflected about the x -axis then shifted up by 2 units. See Figure 10.3.
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chapter10ProblemsAndSolutions - Chapter 10 Trigonometric...

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