In the lead example of this chapter, an experiment was described to determine the spring constant k in Hooke's law: E(l) k{l E). The function F is the force required to stretch the spring I units, where the constant E = 5.3 in. is the length ofthe unstretched spring. a. Suppose measurements are made ofthe length /, in inches, for applied weights F(l), in pounds, as given in the following table. FU) / 2 7.0 4 9.4 6 12.3 Find the least squares approximation for k.b. Additional measurements are made, giving more data: F(l) / 3 8.3 5 11.3 8 14.4 10 15.9 Compute the new least squares approximation for k. Which of (a) or (b) best fits the total experimental data?

MATH152 | Amy Austin | Week 1 Section 5.5 | Integration by Substitution • U-Substitution – Unwinding the Chain Rule • The Substitution Rule [in math terms, and then more simply explained]: o If u=g(x) is a differentiable function, then ∫ f(g(x))g’(x)dx = ∫ f(u)du o ∫ f(g(x))g’(x)dx ▪ u = g(x); where u is the differentiable expression ▪ du = g’(x)dx; where du is the derivative of ‘u’ ▪ Substitute ‘u’ and ‘du’ into the given integral to get ∫ f(u)du • Steps to Integrate using U-Substitution 1. Examine the integral a. Look for an expression and its derivative if possible b.