Math 121

Homework #1

Instructions. This is due on September 7 at the beginning of class. It is worth 20 points. Late papers lose 4 points if turned in by 8:00 Monday morning, plus 3 points per additional day late. You must work by yourself, but feel free to consult with me. Explain what you did and why you did it: good explanations are essential! Your paper should start with an introduction and finish with a conclusion. Start early, ask questions and do well.

Problems. In Figure 1.8, the final position of a knuckleball at time t=0.68 is shown as a function of the rotation rate w. The batter should decide by time t=0.4 whether or not to swing at the pitch. At t=0.4, the left/right position of the ball is given by

h(w) = 1/w - 5 sin(1.6w) / (8w²)

Graph h(w) and compare the graph to Figure 1.8. Conjecture the limit of h(w) as w®0. For w=0, is there any difference between what the batter sees at t=0.4 and what he tries to hit at t=0.68?

In a situation similar to that of example 1.6, the left/right position of a knuckleball is given by

P = 5 (1-cos(4wt)) / (8w²)

In example 1.6, we chose a specific t-value and evaluated the limit as w®0. While this gives some information about how hard the pitch is to hit, a clearer picture emerges if we look at a specific value of w and let t be the variable. In the formula for P, set w=10 and graph the resulting function

(1-cos40t) / 160

for 0 £ t £ 0.68. Imagine looking at the pitch from above and try to visualize a baseball starting at the pitcher at t=0 and reaching the batter at t=0.68. Repeat this with w=5 and w=1. Describe whether each pitch would be easy or hard to hit. Then by trial and error find the value of w that you think produces the most difficult pitch. Explain why you think that this pitch is hard to hit.