MATH 124: Calculus and Analytic Geometry.
Fall 2008

• The information sheet

• Chapter 1

• A proof that cos(pi/5) is exactly one half of the golden ratio.

• Limits in Excel

• Chapter 2

• Exam 1. The histogram of grades

• Chapter 3

• A movie related to the first assignment.

• Another movie related to the first assignment.

• The black curve in this movie is not a parabola.

• One more movie.

• Another parabola related movie. Can you make a conjecture based on this movie?

• From a sine to many sines, a movie, or in color.

• A trig related movie.

• A variation on the preceding movie.

• Assignment 1: solutions.

• Assignment 1: The histogram of grades.

• Exam 2: Solutions. Notice that I gave two versions of Problem 3 on Exam 2. The second version, without a solution, is here. Solving this version of Problem 3 is a good practice for Problem 3 in the homework below.

• Exam 2, the homework part. This is the final version of the homework.
This animation might help with Problem 3.

• Chapter 4

• Exam 2. The histogram of grades.

• The histogram of current grades.

• Assignment 2.

• Cut out the yellow disk. Make an incision into the disk along the thicker black line. Now you can make many yellow cones using this disk. Find one that has the maximum volume.

• An animation of Problem 28 in Section 4.3. Notice that in this animation the height-to-width ratio is very small (in fact, it is tiny). Therefore the actual angle of the orange and the red line is distorted. In reality the angle between these lines is very small. This is a plot of the tangent of that angle. The maximum angle is approximately 0.102896 radians and the maximum angle is attained at x approximately equal to 0.87328.

Here is a similar problem in which the corresponding lines can be orthogonal: f(x) = x^2 exp(2-x), x > 0. In this animation the height-to-width ratio is 1. Therefore the angles that we see are the actual angles between the orange and the red line. The orange and the red lines are orthogonal at x approximately 2.37555 and at x approximately 3.37683. The smallest angle between these two right angles is attained at x approximately 2.79007. This angle is not much smaller than the right angle; it is approximately equal to 1.4537, or 0.92545 (Pi/2), that is 92.545% of the right angle.

Please enjoy each of the three ways of solving Problem 28.
• The first: Find the point where the orange and the red line have the same slope.
• The second: Find the orange line with the maximum slope.
• The third: Find the red line which passes through the origin.

• Am I the only one who find these things (pictures 1, 2, 3, a movie) fascinating? The pictures and the animation are inspired by the following standard (and easy?) exercise in precalculus: Given a line and a point not on that line, find a line which goes through the given point and is orthogonal to the given line. So, it seems natural to replace the given line in this exercise with a parabola.
Given a parabola and a point not on that parabola, find lines (how many?) that go through the given point and are orthogonal to the given parabola.
A more specific problem would be as follows:
Consider the parabola y = x^2. Identify the set of all points P in the xy-plane with the following property: There exist exactly two lines that are orthogonal to the parabola and go through the point P.
Or, even more specifically:
Consider the point P = (4,7/2). Find all lines through this point that are orthogonal to the parabola.

• Exam 3. The histogram of grades.

• Exam 3: Solutions.

• Chapter 6

• Here is a solution of the last problem on Assignment 2.

• A talked to a colleague who is also teaching Math 124 on Friday. She mentioned the following famous optimization problem:  Consider a long rectangular piece of paper. Assume that its with is 1 unit. Determine the minimum crease length that can be obtained by folding the lower right-hand corner of this piece of paper to some point on the opposite edge of the paper. (Try with a letter size paper.)
Here is my solution. Try on your own before looking at the solution. My hint is: Look for similar triangles.

• Here is the histogram of the final grades.

• This is my take on the final exam.