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ProblemSet0

# ProblemSet0 - x-5 2(a Find the x,y coordinates of the local...

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Physics 1116 : Math Problem Set 0 This problem set is meant to test the math skills necessary for Physics 1116. We will not collect and grade this problem set, but you will have a quiz early next week based on these problems. 1. Show, using a Taylor series expansion, that for small angles θ 1 (where θ is expressed in radians) that sin θ θ and that cos θ 1 - x 2 2 . Calculate the discrepancy between the true values of cos and sin and this “small angle approximation” for values of θ = 0 . 5 rad, 0.1 rad, and 0.01 rad (in degrees, this corresponds to about 29 , 6 and 0 . 6 ). Now include the θ 3 term in the Taylor series for sin θ and see how much of a difference this makes in the approximation for these values. 2. Consider the function
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Unformatted text preview: x-5) 2 . (a) Find the ( x ,y ) coordinates of the local minimum y (circled on the graph) (b) Determine the coeﬃcients A , B , and C of the second-order polynomial A + B ( x-x ) + C ( x-x ) 2 which can best approximate the function y around y . 3. Consider a string wrapped around a spool, which is rolling and unwinding on a table. The speed at which the spool rolls on the table ( A m/s, in meters / second) is equal to the total length of string ( A m, in meters) that has been unwound on the table. At T = 0, the length of string unwound on the table is L = 1 m. Find the length of string L on the table as a function of time T . 40 20 10 5 y = 8 ln(x) + (x - 5) 2 1...
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