1873_solutions

# Click on the icon additional exercises on the change

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 x5 3 if 5 x7 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...
View Full Document

Ask a homework question - tutors are online