General Information for my section:

Basic Ground Rules:

Homework will be assigned on Thursdays, and will be due at Thursday's lecture the following week. You will almost always need to use the computer to do some of your homework.

Homework will contain instructions for reading. Make sure you do the reading before the class for which it is assigned. I will assume you have done so.

Make use of my office hours, Math Center, Maple Center, Electronic newsgroups (upenn.math.math104), Sunday Night Reviews (Sundays from 7-9 p.m. in DRL A1) etc..

Grading notes: At the end of the semester, everyone who has not withdrawn from the class will get a grade. Incompletes will not be given to avoid F's.

LATE WORK (homework, Maple, etc..) will NEVER be accepted.

Exams:

Class notes:

Homework and class notes:

Bonus problems and solutions:

• Bonus problem #1: Suppose that three points on the parabola y = x² have the property that their normal lines intersect at a common point. Show that the sum of their x-coordinates is 0. Is this if and only if?

• Bonus problem #2: Consider a square with side length L.  Let R be the region inside the square consisiting of all points that are closer to the center of the square than to any side of the square. What is the area of R?

• Bonus problem #3:  This is a classic but difficult problem.

  A cylindrical glass of radius r and height L is filled with water and then tilted until the water remaining in the glass exactly covers its base (as in the picture on the left below).

  (a) Determine a way to "slice" the water into parallel rectangular cross sections and then set up a definite integral for the volume of the water in the glass.

  (b) Determine a way to "slice" the water into parallel cross-sections that are trapezoids and then set up a definite integral for the volume of the water.

  (c) Find the volume of water in the glass by evaluating one of the integrals in part (a) or part (b).

  (d) Find the volume of water in the glass from purely geometric considerations.

  (e) Suppose the glass is tilted until the water covers half the base, as in the picture on the right below. In what direction can you "slice" the water into triangular cross sections? Rectangular cross-sections? Cross-sections that are segments of circles? Find the volume of water in the glass.

  Here are two pictures that may help:

  

• Bonus problem #4: Evaluate
  
• Bonus problem #5: Evaluate:
  
• Bonus problem #6: This problem gives you an opportunity to calculate the escape velocity of a rocket leaving the earth. It is an interesting combination of some physics, the idea of differential equations, and limits. Let's work this one together this week!
  According to Newton's Law of Universal Gravitation, the gravitational force on an object of mass mthat has been projected vertically upward from the earth's surface is where x=x(t) is the object's distance above the sruface at time t, R is the radius of the earth, andg is the acceleration due to gravity at the surface of the earth. Also, by Newton's Second Law (the F=ma one),

  .

  (a) Suppose a rocket is fired vertically upward with an initial velocity . Let hbe the maximum height above the surface reached by the object. Show that

  .

  (Hint: By the Chain Rule, . )

  (b) Calculate . This limit is called the escape velocityfor the earth.

  (c) Use R = 3960 miles and g = 32 ft/ to calculate in feet per second and in miles per second.

• Bonus problem #7: A dog sees a rabbit running in a straight line across an open field and gives chase. In a rectangular coordinate system, assume:

  (i) The rabbit is at the origin and the dog is at the point (L,0) at the instant the dog first sees the rabbit.

  (ii) The rabbit runs up the y-axis and the dog always runs straight for the rabbit.

  (iii) The dog runs at the same speed as the rabbit.

                                       

  (a) Explain why the dog's path is the graph of the function y = f(x), where y satisfies the differential equation:

  

  (b) Determine the solution of the equation in part (a) that satisfies the initial conditions y = y' = 0 when x = L. (Hint:  Let z = dy/dx in the differential equation and solve the resulting first-order equation to find z. Then integrate z to find y.)

  (c) Does the dog ever catch the rabbit?

  (d) What if you replace hypothesis (iii) by

   (iii)' The dog runs twice as fast as the rabbit.

  You have to adjust the differential equation for the path of the dog (put a "2" in the right place). Do this, and solve the differential equation. At what point does the dog catch the rabbit?

  (Hint: So you know the slope of the dog's path (it's along the line from wherever the dog is to wherever the rabbit is), and you know the dog and the rabbit traverse the same distance (i.e., arclength) in the same amount of time. How do these observations help you to construct the differential equation?)

• Bonus problem #8:

  A two-parter:

  (a) Show that tan (x/2) = cot(x/2) - 2 cot(x)

  (b) Find the sum of the series:
  Part (a) looks like a trig identity...can you prove it? How can you use it to prove part (b) ?

• Bonus problem #9:

  Another two-parter:

  (a) Find the sum of the series

  

  where the terms are the reciprocals of the positive integers whose only prime factors are 2s and 3s.

  (b) Consider the series whose terms are the reciprocals of the positive integers that can be written in base-10 notation without using the digit 0. Show that this series is convergent and that its sum is less than 90.

• Bonus problem #10:

  (a) Show that for ,

  

  if the left side lies between and .

  (b) Show that

  

  (c) Deduce the following formula of John Machin (1680-1751):

  

  (d) Use the Maclaurin series for arctan to show that arctan(1/5) is between 0.197395559 and 0.197395562.

  (e) Show that arctan(1/239) is between 0.004184075 and 0.004184077.

  (f) Deduce that, correct to seven decimal places, is 3.1415927.

  Machin used this method in 1706 to find correct to 100 decimal places. (Of course, by now, with computers, has been calculated to more than a billion decimal places.)

Maple worksheets from class:

Solutions to homework and in-class problems