Math 490
First Assignment Hints
Posted Thursday February 10, 2005.
Problem 3.
What is the equation of the function that is the top of the shaded region? The key property of this curve is the following:
All the points on the curve are the same distance from the top line of the square as they are from the center.
There are at least two ways to exploit this:
  1. A curve that consists of all points equidistant from a line and a point has a particular name; in particular, it's a conic section (see section 10.5 of Stewert).
  2. Name a point on the line; call it (x,y). Use the distance formula to find the distance from (x,y) to the line and to the center, equate these, and then solve for y.

Problem 4.
What's the derivative of f(x)? Ie, what is f'(x)? And once you have that, you can find a substitution that lets you evaluate the integral (don't forget to change the limits when you substitute!)

Problem 6.
If
f'(x) = 1 + [f(x)]2,
then what is
f'(g(x))?

Problem 7.
L'Hopital's Rule is a special case of Taylor series. Can you express the numerator and denominator as Taylor Series?