quiz4 - cos = 1 12 x . Hence, = x 12 cos x =3 , = 2 3 = 1 2...

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MATH 125 Spring 2008 – Quiz 4 Name 1. (8 points) A ladder 12 ft. long rests against a vertical wall. The bottom of the ladder slides away from the wall at a speed of 3 ft./sec. How fast is the angle between the top of the ladder and the wall changing when the angle is 2 π 3 radians? Express your answer in proper units. 2. (7 points) Use the linear approximation at x = 27 to estimate 3 26. Is your answer under or over estimate? Give your answer as fraction. 1
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3. (5 points) Show that for any a and b | cos a - cos b | ≤ | a - b | . (Hint: Try to use MVT to show ± ± ± cos b - cos a b - a ± ± ± 1 . ) 2
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Brief Answer: 1. Let the distance from bottom of the ladder and the wall be x feet, and the angle between the top of the ladder and the wall be θ . We need to find dt = θ 0 =? Equation is sin θ = x 12 . By chain rule
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Unformatted text preview: cos = 1 12 x . Hence, = x 12 cos x =3 , = 2 3 = 1 2 radian/sec . 2. Set f ( x ) = 3 x . The linearization of f at x = 27 is then L ( x ) = f (27) + f (27)( x-27) = 3 + 1 27 ( x-27) . Hence, f (26) L (26) = 3 + 1 27 (26-27) = 80 27 . It is over estimate, since the tangent line will be above the curve of f . 3. It is obvious if a = b . In the rest, we will prove when b 6 = a . Let f ( x ) = cos x . Then (a) f is continuous. (b) f is continuous. By MVT, there exits c ( a,b ) such that f ( c ) = f ( b )-f ( a ) b-a . So sin c = cos b-cos a b-a . It implies cos b-cos a b-a = | sin c | 1 . 3...
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quiz4 - cos = 1 12 x . Hence, = x 12 cos x =3 , = 2 3 = 1 2...

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