MATH1151 week3c2 - x = 1 2 sin 2 x = 2 cos 2 x 1....

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Time allowed: 15 minutes. 1. (2 marks) On the same axes, sketch the graphs of y = | 2 x 7 | and y =3 x 2. Hence, or otherwise, solve | 2 x 7 | =3 x 2. 2. (2 marks) Show that f ( x )= x 3 +6 x 2 +15 x + 3 has an inverse function. 3. (4 marks) Determine a, b, c, d such that sin π 8 = ± a + b 2 and cos π 8 = ± c + d 2. 4. (2 marks) Find d dx cosh(sin x ). Week 3 Tuesday 10-11 Class
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1. | 2 x 7 | =3 x 2 2 x 7=3 x 2or (2 x 7) = 3 x 2 2 x 5=3 x or 5 x =9 x = 5or x = 9 5 . However, when x = 5 , | 2 x 7 | = 17 and 3 x 2= 17, so x = 9 5 is the only solution. Figure 1: y = | 2 x 7 | and y =3 x 2 2. f ( x )= x 3 +6 x 2 +15 x +3 f ± ( x )=3 x 2 +12 x +15 =3( x 2 +4 x +5) = 3(( x +2) 2 +1) 0 for all x R . Since the curve is monotonic increasing, it is one-to-one. [Also, since f ( x ) is a cubic polynomial, it is onto.] Thus, it has an inverse function. Week 3 Tuesday 10-11 Class
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3. According to the double angle formulae, cos 2
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Unformatted text preview: x = 1 2 sin 2 x = 2 cos 2 x 1. Substituting x = 8 gives: cos 4 = 1 2 sin 2 8 sin 2 8 = 1 2 1 2 2 sin 8 = 1 2 2 4 (taking the positive square root, since 8 is in quadrant 1). a = 1 2 and b = 1 4 . Substituting x = 8 gives: cos 4 = 2 cos 2 8 1 cos 2 8 = 1 2 2 2 + 1 cos 8 = 2 4 + 1 2 (taking the positive square root, since 8 is in quadrant 1). c = 1 2 and d = 1 4 . 4. d dx cosh(sin x ) = d dx cosh( u ) where u = sin x = du dx d du cosh( u ) = cos x sinh( u ) = cos x sinh(sin x ) . Week 3 Tuesday 10-11 Class...
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MATH1151 week3c2 - x = 1 2 sin 2 x = 2 cos 2 x 1....

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