IX Math 337-001 Due May 4th, 2015 1. Let f ( x) x 3 for x [0,1]. Given n N , let Pn be the partition of [0,1] defined as 1 2 n Pn {0 x0 , x1 , x2 ,..., xn 1}. n n n a) Find U ( f , Pn ) and L( f , Pn ). 1 b) Use part a) to calculate x dx. 3 0 2. Let f be integrable on [a,b] and suppose that m f ( x) M for all x [a, b]. Show that b m(b a) f ( x)dx M (b a). a x 3. If F ( x) 1 t 2 dt ( x 0), find F ' (3). 0 4. Page 270, Exercise 32.3. 5. Page 289, Exercise 33.4. 6. Page 289, Exercise 33.5. 7. Page 297, Exercise 34.11 8. Page 290, Exercise 33.8 b) only