MechE 132, Problem Set #8

MechE 132, Problem Set #8 Version 1 Spring, 2014, UC Berkeley Below are 4 problems. I will post more problems over the course of the weekend, but this should be good to get started. These are due on Friday, April 11, at 5PM. 1. This purpose of this problem is to remind you how first-order Taylor series (ie., linear approximation) are used to approximate a function of many variables. The temperature in a particular 3-dimensional solid is a function of position, and is known to be T(x, y, z) = 42 + (x − 2)2 + 3 (y − 4)2 − 5 (z − 6)2 + 2yz (a) Find the first order approximation (linearization) of the temperature near the location (¯x = 4, y¯ = 6, z¯ = 0). Use δx, δy and δz as your deviation variables. (b) What is the maximum error between the actual temperature and the first order approximation formula for |δx| ≤ 0.3, |δy| ≤ 0.2, |δz| ≤ 0.1? Solve this numerically, by simply sampling a dense grid over the 3-dimesional cube, and determining the maximum error. (c) More generally, suppose that ¯x ∈ R, ¯y ∈ R, ¯z ∈ R. Find the first order approximation of the temperature near the location (¯x, y, ¯ z¯).