MATH1151 week3c3 - Time allowed 15 minutes 1(2 marks Use logarithmic differentiation to find d dx x sin x for x> 0 2(2 marks By expanding x − y

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Unformatted text preview: Time allowed: 15 minutes. 1. (2 marks) Use logarithmic differentiation to find d dx x sin x for x > 0. 2. (2 marks) By expanding ( x − y ) 2 prove that x 2 + y 2 ≥ 2 xy for all real numbers x, y . Hence, or otherwise, determine the minimum value of y = x 6 + 1 x 6 . 3. (1 mark) Find the exact value of sin − 1 sin 200 π 3 . 4. (5 marks) (a) Given that cosh( A + B ) = sinh A sinh B + cosh A cosh B , find a formula for cosh 2 x . By differentiation or otherwise, find a formula for sinh 2 x . (b) Using the results of part (i), express cosh 3 x as a cubic polynomial in cosh x . Hence, or otherwise, find cosh 3 x . Week 3 Friday 11-12 Class 1. Let y = x sin x . Then ln y = sin x ln x , so d dx ln y = sin x × 1 x + cos x × ln x . But d dx ln y = d dy (ln y ) dy dx = 1 y dy dx . ∴ dy dx = y sin x × 1 x + cos x × ln x and d dx x sin x = x sin x 1 x sin x + (cos x )(ln x ) . 2. ( x − y ) 2 = x 2 + y 2 − 2 xy However, ( x − y ) 2 ≥ 0 for all real x, y....
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MATH1151 week3c3 - Time allowed 15 minutes 1(2 marks Use logarithmic differentiation to find d dx x sin x for x> 0 2(2 marks By expanding x − y

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