- You have a homework due today.
- Your quiz is returned.
- Some of you are using symmetry very well -- congratulations!
- a. Some things to think about:
- The sign of the slope of $f$ tells you the the sign of the derivative of $f$
- the size of the slope of $f$ tells you the size of the derivative of $f$
- So the steepest slopes will be peaks (or pits) of the derivative. (Try to get those lined up.)
- Your derivative should be odd, since the function is even.
Take a look at the C-I pair below.
- b. Now use this strategy of part a. in part b, as well as symmetry.
- Today we'll mostly just do some applications. Recap:
- The derivatives of trig functions rely on some basic
understanding of the circle.
- Radian measure: $RADIANS=\frac{2\pi}{360}DEGREES$. In calculus, we almost always use degrees.
- Length of a sector, related to angle subtending
- Definitions of the trig functions
We will derive the derivative of the sine function from the limit definition of the derivative (although we can see the derivative graphically above).
We'll need two important identities in the course of this:
- The Pythagorean theorem: $\sin^2(x)+\cos^2(x)=1$
- The sine of a sum: $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$
- By the way, there is one more important trig identity to remember:
The cosine of a sum: $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$
And from these identities we can deduce all the other ones -- all of the relationships between sine and cosine.
Then the derivative of the cosine will be derived in two different ways:
- by relating cosine to sine graphically, and
- by the straight limit definition of the derivative.
- Examples:
- Derivative of tangent
- Fourth derivative of sine, cosine
- #7, p. 146
- #26
- #35
- #49 and 50