Remarks. (i) For problem 16, the space C([a,b]) was defined at the end of Chapter IV (p. 90). (ii) Problem 19 is a version of Taylor's Theorem that turns out to be more useful than the one we proved in class, even though the formula for the remainder (the last term) isn't as "pretty". (iii) Hint for problem 20: Apply the MVT to each component function $f_k$. A difficulty you will have to deal with is that on each subinterval, the $c$'s the MVT gives you will in general be different for each $f_k$.) (iv) In problem 22, I also want you to use your calculator to figure out an approximation to Euler's constant. (v) Do problem 23 without resorting to theorems from courses you've taken on differential equations.