Assign.#3Sol. - Pawel Lorek, University of Ottawa, MAT...

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Pawe±l Lorek, University of Ottawa, MAT 1330, Fall 2008 Assignment 3, due November 26, 19:00 in class Student Name Student Number DGD 1 (Monday) DGD 2 (Thursday) Problem 1: [2 points] Find the Taylor polynomials of degree 3 and 5 of the function f ( x )=sin( x ) around x 0 =0 . Let’s recall the taylor polynomial matching ²rst n derivatives of function f ( x )a t x = x 0 : P n ( x )= f ( x 0 )+ f ! ( x 0 )( x x 0 f ! ( x 0 ) 2 ( x x 0 ) 2 + f (3) ( x 0 ) 3! ( x x 0 ) 3 + ... + f ( n ) n ! ( x x 0 ) n , where f ( k ) ( x 0 ) denotes k th derivative of f ( x ) evaluated at x = x 0 . We have to calculate ²rst 5 derivatives of f ( x x ): f ! ( x )=(s in( x )) ! =cos( x ) ,f ! (0) = 1 f ! ( x )=(cos( x )) ! = sin( x ) ! (0) = 0 f (3) ( x )=( sin( x )) ! = cos( x ) (3) (0) = 1 f (4) ( x cos( x )) ! = ( sin( x )) = sin( x ) (4) (0) = 0 , f (5) ( x x )) ! x ) (5) (0) = 1 . Thus P 3 ( x )=1+0 · ( x 0) + 0 2 ( x ) 2 + 1 3! ( x x 0 ) 3 =1 1 6 x 3 , P 5 ( x · ( x 0)+ 0 2 ( x ) 2 + 1 3! ( x x 0 ) 3 + 0 4! ( x 0) 4 + 1 5! ( x 0) 5 1 6 x 3 + 1 120 x 5 , where we used: 3! = 1 · 2 · 3=6and5 !=1 · 2 · 3 · 4 · 5 = 120. Problem 2: [6 points] A cow grows from 10kg to 150kg in weight in 7 years. Suppose that the weight is a di³erentiable function of time. (a) Why must the weight have been exactly 20kg at some time? (b) Why must the weight gain been exactly 20kg/year at some time? (c) Draw a graph of the weight versus time where the weight of 20kg occurs at 1 year and the rate of 20kg/year occurs at 5 years. (Note that the solution to part (c) is not unique.) Let w ( t ) denotes the weight of the cow at time t (measured in years). From the formulation of the problem, we have: w (0) = 10 and w (7) = 150 (kg). a) From Intermediate Value Theorem we know that for every c : w (0) = 10 c 150 = w (7) there exists t 0 such that w ( t 0 c . In particular, if we take c =20then the theorem implies that at some time t 0 the cow weights 20kg, i.e. w ( t 0 ) = 20. 1
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b) Mean Value Theorem states, that for diFerentiable function w ( t )oninterva l[ a, b ] there exists at least one point such that the derivative (slope of tangent) is equal to the slope of the secant line connecting ( a, w ( a )) and ( b, w ( b )).
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This note was uploaded on 01/25/2011 for the course MAT 1330 taught by Professor Dumitriscu during the Spring '08 term at University of Ottawa.

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Assign.#3Sol. - Pawel Lorek, University of Ottawa, MAT...

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