SUNY Geneseo Department of Mathematics
Monday, October 5
Math 221 02
Fall 2020
Prof. Doug Baldwin
In the book’s example of the Squeeze Theorem, where does the idea of bounding x cos x by |x| and -|x| come from? To some extent it’s a clever idea that doesn’t have any simple rule for finding, but one way it might come about is to realize that since cos x ≤ 1, x cosx should be less than or equal to x (or -x for negative values of x), and similarly since -1 ≤ cos x, -x (x for negative values of x) should be less than or equal to cos x. Then someone, especially someone looking for things to squeeze x cos x between, could cleverly notice that the positive branches of those inequalities are |x|, and the negative branches are -|x|:
Problem set 4 is now available, on derivatives and related things. Work on it this week, and grade towards the end of this week or the beginning of next, depending on when your meetings with me are.
The next session is tomorrow (Tuesday), from 4:00 - 5:30. Watch for an announcement with the Zoom link tomorrow afternoon.
Based on “Derivatives of the Sine and Cosine Functions,” and “Derivatives of Other Trigonometric Functions” in section 3.5 of the textbook, and the trigonometric derivatives discussion.
A rider on a carnival ferris wheel with radius 12 feet will have a height above ground level (or at least boarding level), in feet, of
h(t) = 12 - 12 cos t
where t represents time, measured in units of however long it takes the ferris wheel to rotate 1 radian. What is the rider’s vertical speed as a function of time? Vertical acceleration?
Vertical speed is the derivative of height, so use the constant and constant multiple rules for derivatives, along with the derivative of cosine, to get...
Then acceleration is the derivative of speed, or the second derivative of height, so...
There is an old farm silo behind my house, which I estimate to be about 30 feet high. When the sun is rising, this silo casts a shadow across the ground whose length in feet, L, is given by the equation
L = 30 / tan Θ
where Θ is the sun’s angle above the ground in radians. How “fast” is the length of this shadow changing, as a function of Θ?
There are at least 2 ways to solve this. One is to use the quotient rule on 3 / tan Θ, the other is to realize that 1 / tan Θ =cot Θ, and rewrite 30 / tan Θ = 30 cot Θ and differentiate that. The second of these seemed simpler, so we started with it, remembering, or looking up, that the derivative of cotangent is negative cosecant squared:
For comparison, we also worked out the approach based on the quotient rule:
(There’s yet another way of solving this in the trigonometric derivatives discussion, too.)
Now that we have differentiation rules for trigonometric functions, we automatically have antiderivative rules also, at least where it is reasonable to “run” the differentiation rules backwards. For example...
What is the antiderivative of cos x?
What is the antiderivative of -sin x? How about sin x?
We worked each of these out by asking “what would have a derivative equal to...?” We also added an arbitrary constant, denoted by “+ C,” to each, since any constant would go to zero when taking its derivative. We found the following antiderivatives:
There is an interesting pattern to the higher-order derivatives of sine (and cosine). For example, what are the second, third, and fourth derivatives of sin x?
In general, the results repeat after every 4 derivatives. So the (4n+k)th derivative (i.e., the derivative whose number is 4 times some integer plus another integer) will always be the same as the kth derivative:
The units in the ferris wheel (time per radian) and silo (feet per radian) examples really were awkward. That’s because we don’t have a way to take derivatives of a function (sine or tangent) applied to another function (time per radian to radians, radians per minute to minutes). For example, a nicer expression for the height of the ferris wheel is
h(t) = 12 - 12 cos( t/20 )
where t is time in seconds and the wheel happens to be turning at a speed of 1/20 radian per second (roughly 1 revolution every 2 minutes). But at the moment you don’t officially know how to correctly find the derivative of cos( t/20 ). You got a preview of this problem when we derived the “reciprocal rule” for derivatives, and found that we can’t just take the derivative of 1/f(x) by applying the power rule to f(x)-1.
So it’s time to learn about the chain rule, which allows you to take such derivatives.
Please read “Deriving the Chain Rule” in section 3.6 of the textbook, and participate in this discussion of the chain rule.