1873_solutions

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Unformatted text preview: rval p, q within which f steps. Since f is a step function on the interval p, q , it follows from Exercise 1 that f is a step function on a, b . 3. Give an example of a step function on the interval 0, 2 that does not step within any regular partition of 0, 2 . Solution: We define fx  0 if 0 1 if x 2 2 x 2 Now if P is any regular partition of the interval 0, 2 then, since the irrationality of 2 makes it impossible to find integers n and j such that 2j 2  0 n we know that 2 can’t be a point of P. In other words, the number 2 must be in one of the open intervals of P and f fails to be constant in that interval. 4. Explain why a step function must always be bounded. Suppose that f is a step function on an interval a, b . Choose a partition P of a, b such that f steps with P. We express P in the form x 0 , x 1 , , x n . Since f is constant in each subinterval x j 1 , x j , the range of f must be a finite set and therefore f is bounded. 5. Prove that if f and g are step functions on an interval a, b then so are their sum f  g and their product fg. Hint: You can find a proof of this assertion in the section on linearity of integration of step functions. 6. Prove that if f and g are step functions then so are their sum f  g and their product fg. Choose an interval a, b such that both of the functions f and g take the value 0 at every number in R a, b . We deduce from Exercise 2 that both f and g are step functions on the interval a, b and it follows from Exercise 5 that f  g and fg are step functions on a, b and we conclude that these 286 functions are step functions. 7. Prove that a continuous step function on an interval must be constant on that interval. Suppose that f is a continuous step function on an interval a, b and choose a partition P  x 0 , x 1 , , x n of a, b within which f steps. If c is the constant value of f on the interval x 0 , x 1 then, since f is continuous at x 0 and at x 1 we have f x 0  f x 1  c. Therefore, since f is continuous at x 1 , the number c must also be the constant value of f on x 1 , x 2 . Continuing in this way we see that f has the constant value c throughout the interval a, b . Exercises on Integration of Step Functions 1. Given that x that P is the partition if x x2 x  if x  3 2, 1, 2, 4, 5, 7, 10 of the interval 3 , 2, 10 , that 0 if x  1 1 if 1 x 2 1 if 2  x  4 fx  2 if 4 x5 3 if 5 x7 0 if x 7 3 2 1 0 -2 2 4 6 8 10 -1 and that Q is the refinement of P given by Q  2, 1, 2, 3, 4, 5, 6, 7, 10 work out the sums  P, f,  and  Q, f,  and verify that they are equal to each other. We have  P, f,   0 1 1 1 2 1 1 16 2  2 25 16  3 49 25  0 100 49  77 and  Q, f,    1 J , 3  0 1 1 1 2 1 13 2 1 16 9  2 25 16  3 49 25  0 100 49 3 0 1 1 1 2 1 13 2 1 16 9  2 25 16  3 49 25  0 100 49 1 16 9 1 16 2 19 and since 19 3 13 2 we have  Q, f,    P, f,   77. 287 2. Given that f is the function whose graph appears in the figure, evaluate Þ Ý Ý f. 3 2 1 0 -2 2 4 6 8 10 -1 We sum the function f over...
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