We reviewed the derivatives of the six trigonmetric functions (Sec 3.5) and outlined some of the exercises in the WebWork set “Derivatives – Trigonometric.”
We then reviewed functions which are compositions of functions, and introduced the Chain Rule–a differentiation rule of finding the derivatives of such composite functions:
We went through on exercise from the “Derivatives – Chain Rule” WebWork set:
Here is a video which explains the Chain Rule in much the same way as I did in class, and goes through additional examples:
“Derivatives – Rates of Change” — extended to Monday, Oct 30
“Derivatives – Trigonometric” — due Friday, Nov 3
Topics
We introduced the derivatives of the trigonmetric functions (Sec 3.5).
We started by reviewing the definitions of the basic trig functions, sine and cosine, in terms of the unit circle, i.e., the circle of radius 1 centered at the origin (0,0); see Sec 1.3 for background on the trig functions. The other four trig functions–the tangent (tan), cosecant (csc), secant (sec) and cotangent (cot) are defined as ratios in terms of sin and cos:
The unit circle definition of sine and cosine leads to their graphs. Starting with the graph of y = sin x (the white curve below), we plotted the slopes of the tangent lines in yellow–we saw that it seems like the derivative sin x is cos x!
The proof that this is the case goes back to the limit definition of the derivative, applied to f(x) = sin x. We didn’t go through the details, which are presented in the textbook (which also contains a version of the graphs above):
Similar reasoning leads to the derivative of cos x:
From these two derivatives, we can find the derivatives of the other trig functions using the Quotient Rule (since they are all quotients involving sin and cos!):
You can do the exercises in the WebWork set “Derivatives – Trigonometric” using these six derivatives, together with the differentiation rules we learned previously (Power Rule, Product Rule, Quotient Rule).
We have studied how the derivative of position with respect to time gives instantaneous velocity.
To better understand this idea, I recommend reading this article by Cornell mathematician Steven Strogatz, which illustrates average and instantaneous velocities in the context of Usain Bolt’s world record setting 100-meter dash at the 2008 Olympics:
(Here is one line from the article, which should remind you of the limit definition of the derivative: “…the founders of differential calculus solved the riddle of instantaneous speed. Their intuitive solution was to define instantaneous speed as a limit — specifically, the limit of average speeds taken over shorter and shorter time intervals.”)
This class uses WeBWorK, an online homework system. Login information will be provided by your professor. For information about how to use the WeBWorK system, please see the WeBWorK Guide for Students.
The WeBWorK Q&A site is a place to ask and answer questions about your homework problems. HINT: To ask a question, start by logging in to your WeBWorK section, then click “Ask for Help” after any problem.
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