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10Cquiz1sol

# 10Cquiz1sol - x = cos x for x near 0 Ans f x = cos x ⇒ f...

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Quiz 1 for MATH 10 C Solution, 2010 1 . Suppose that x measures the time (in hours) it takes for a student to complete an exam. All students are done within two hours and the density function for x is p ( x ) = cx 3 if 0 < x < 2, 0 otherwise. (a) Find the value of c . Ans. Note the property of a density function that Z - p ( x ) dx = 1. Since p ( x ) is only defined when 0 < x < 2, Z 2 0 p ( x ) dx = Z 2 0 cx 3 dx = h c 4 x 4 i 2 0 = c 4 · 16 = 4 c = 1 c = 1 4 . (b) What is the mean time for students to complete the exam? Ans. Note that if a quantity has density function p ( x ) , then Mean Value of the quantity = Z - xp ( x ) dx . Thus, the mean time for students to compete the exam is Z 2 0 xp ( x ) dx = Z 2 0 x · 1 4 x 3 dx = Z 2 0 1 4 x 4 dx = 1 20 x 5 2 0 = 32 20 = 8 5 = 1.6 (hr) (c) Compute the median of this distribution. Ans. Note that a median T is such a value that satisfies Z T - p ( x ) dx = 1 2 . So, Z T 0 1 4 x 3 dx = 1 16 x 4 T 0 = T 4 16 = 1 2 2 T 4 = 16 T 4 = 8 T = 2 3/4 1.682

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2 . Construct the Taylor polynomial 1 of degree 6 approximating the function f
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Unformatted text preview: x ) = cos x for x near 0. Ans. f ( x ) = cos x ⇒ f ( ) = 1 f ( x ) =-sin x ⇒ f ( ) = f 00 ( x ) =-cos x ⇒ f 00 ( ) =-1 f 000 ( x ) = sin x ⇒ f 000 ( ) = f ( 4 ) ( x ) = cos x ⇒ f ( 4 ) ( ) = 1 f ( 5 ) ( x ) =-sin x ⇒ f ( 5 ) ( ) = f ( 6 ) ( x ) =-cos x ⇒ f ( 6 ) ( ) =-1 Thus, the Taylor polynomial of degree 6 approximating f ( x ) = cos x for x near 0 is P 6 ( x ) = 1-1 2 x 2 + 1 4! x 4-1 6! x 6 . 1 Note : Taylor polynomial of degree n approximating f ( x ) for x near 0 is f ( x ) ≈ P n ( x ) = f ( ) + f ( ) x + f 00 ( ) 2! x 2 + f 000 ( ) 3! x 3 + f ( 4 ) ( ) 4! x 4 + ··· + f ( n ) ( ) n ! x n...
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