The Quantum Mechanical Postulates Numerical Problems P14.2 Which of the following functions are single-valued functions of the variable x? a. P14.5 2 27x sin—— a b. eve : 1-3sin*x c. : ene d. (a) Yes. The function we) has only a single value for a given value of x. a (b) No. The function ev has two values for a given value of x. (c) Yes. The function 1 - 3sin? x has only a single value for a given value of x. (d) Yes. The function e*”* has only a single value for a given value of x. Is the function (x? —1)/(x - 1) continuous at x = 1? Answer this question by evaluating f(1) and lim f(x). x71 It is not continuous Coe 1)/(x - 1) because f(1) _@+tDe-D is not defined. The expression cannot be simplified to = x +1 because it is not permissible to divide by zero. To calculate the limit, we use L’H6pital’s rule tim| £22 |= jim| FOU |_ im | > =2 x1] P14.8 g(x) x1] de(x)/dx x1, 1 Which of the following functions are acceptable wave functions over the indicated interval? a. e”~ 0 — 9, (c) eX —_ 100 < x < 100 is an acceptable wave function because it is finite over the interval, is defined at every point in the interval, and the first and second derivatives exist. (d) as 1 < x < e is an acceptable wave function because it is finite over the interval, is defined at ¥ every point in the interval and the first and second derivatives exist. 73 Copyright © 2013 Pearson Education, Inc.