13. Let v > 0 be a true value, and a > 0 be an approximation of v. Prove that |t| is invariant with respect to scalar multiplication. In other words, for scalar c > 0, the absolute relative true error for ca approximating cv is equal to |t|, the absolute relative true error for a approximating v. 14. Prove that if |t| 0.5 10m, then the true value v and the approximate value a are equal at the first m significant digits. Hint 1: you may assume v > 0 and a > 0. Hint 2: use the previous exercise. 15. Prove or disprove the converse of the statement in the previous exercise. In other words, if the true value and approximation agree in the first m digits, is it necessarily true that |t| 0.5 10m? 16. Suppose your approximation yields an absolute relative true error of 0.003%. How many sig- nificant digits of your approximation are guranteed to be accurate. 17. Suppose x 0 and y 0 are approximate values with respective true errors 1 and 2. Determine the true error inherent in the product xy. Compare your answer with the answer f x 1 + f y 2, where f (x, y) = xy. 18. Consider a sequence of numbers xn, n 0, that satisfies the equation axn + bxn1 = 0, where a, b 6 = 0, for all n 1. Show that this equation is satisfied by xn = (b/a)n. 19. Suppose xn is an inceasing sequence of numbers with the property that (xn xn1)/xn = c, where 0 < c < 1 is a constant. In other words, the relative approximation error is constant. Show that xn does not converge. Hint: use the previous exercise. 20. The formula for strain S on a longitudinal bar is given by S = F/(AE), where F is the applied force, A is the cross-sectiional area, and E is Young's modulus. If F = 50 0.50 N, A = 0.2 0.002 m2, and E = 210 109 1 109 Pa, determine a first-order approximation of the maximum error in measuring S.Compare your approximation with the actual maximum true error