Work individually or in groups of two or three. The assignment is due (a printed copy,
typed) in class on Wednesday, 29 November.
Choose one of the two following options.
Option 1 Prove the following statement using rectangles and the de nition of the de nite
integral: If n is any of the numbers 0;1;2;::: and if a;b are any real numbers,
Z b
a
xndx = b
n+1
n + 1
an+1
n + 1:
You are not allowed to use either part of the Fundamental Theorem of Calculus.
Option 2 Suppose that f is a function and that (a;b) is a point on the graph of f. Suppose
also that there is a circle C such that the point (a;b) is also on the circle. Let us also assume
that at the point (a;b), the circle and the function f have the same rst derivative and the
same second derivative. What is the radius of the circle? Your answer may be in terms of
one or more of the quantities a;b;f0(a), and f00(a).