This section mostly dealt with rates of change. While I have been familiar with rates of change for years as a result of many math classes, I didn't realize how useful they are in real life. The paragraph regarding instantaneous rates of change in velocity was really interesting, as was the example that used the average rate of change to determine the speed of sound in air. I think its really interesting how math and science, more specifically physics in this problem go together.
Problems:
1. Estimate the instantaneous velocity of a particle traveling a distance of s(t) = t^2 + 4t at t=5, using intervals to the left and right of t=5.
To solve this I used 4.99 to approach 5 from the left and 5.01 to approach 5 on the right. I set my equations up to look like this:
[4.99,5]: 5^2 + 2(5) - (4.99)^2 - 2(4.99) all divided by 0.01.
[5,5.01]: 5.01^2 + 2(5.01) - 5^2 - 2(5) all divided by 0.01.
The reason I divided each by 0.01 is because that is the difference between the intervals in both equations. The answers that I got were 11.99 for the first equation and 12.01 for the second equation. This means that when t=5, the instantaneous rate of change is approximately 12.
2. Estimate the average rate of change of cos x, at [π/6, π/4].
cos(π/4) - cos(π/6)
----------------- = -0.60701
(π/4-(π/6)
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